text: (cid:107) (cid:107) ≤ ≥ (ii) Let c > 0 be a constant. For ∆ c and n /n γ > 0 for all k and n, F k − (cid:104)∇ (cid:105) ≥ γ (cid:107) (cid:107) ≡ (cid:107) (cid:107) (cid:96)˜ (Θ + ∆) + (cid:96)˜ (Θ ) + (cid:96)˜ (Θ ), ∆ { } ∆ 2 κ ∆ 2 . (11) n 0 n 0 n 0 2 λ + c 2 F (cid:96)n,c F Θ M M⊥ (iii) The map r is a seminorm, convex, and decomposable with respect to ( , ) in the ∈ M ∈ M⊥ sense that r(Θ + Θ ) = r(Θ ) + r(Θ ) for every Θ and Θ . Moreover, 1 2 1 2 1 2 − ≥ − r(Θ + ∆) r(Θ ) r(∆M⊥) r(∆M). 0 0 ∈ R × (iv) For ∆ ˜(pK) (pK), |(cid:104)∇ (cid:105)| ≤ (cid:107) − (cid:107) (cid:96)˜ (Θ ), ∆ r(∆) max Ψˆ (k) Ψ(k) ∞. (12) n 0 ≤ ≤ n 0 1 k K ∈ R × (cid:16) (cid:17) (v) For Θ ˜(pK) (pK), ≤ (cid:107) (cid:107) (cid:107) (cid:107) r(ΘM) (s + 1)1/2 1 + ρ L 1/2 ΘM . 2 2 F Proof. (i) The result follows by taking derivatives blockwise. (ii) Rothman et al. (2008) (page 500-502) showed that − (cid:88) (cid:16) − (cid:104)−∇ (cid:105) (cid:17) (cid:96)˜ (Θ + ∆) + (cid:96)˜ (Θ ) (cid:96)˜ (Θ ), ∆ n 0 n 0 n 0 K n − (cid:104) (cid:105) = (cid:88) nk log det(Θ(0k) + ∆(k)) + log det(Θ(0k)) + Ψ(0k), ∆(k) k=1 (cid:110)(cid:13) (cid:13) (cid:13) (cid:13) (cid:111) (cid:13)(cid:107) (cid:13)(cid:107) (cid:13) (cid:13) ≥ K n (cid:13) ∆(k(cid:13)) 2 k F . k=1 n 2 min0≤v≤1 Θ(0k) + v ∆(k) 2 2 2 (cid:107) (cid:107) ≤ (cid:107) (cid:107) ≥ (cid:107) (cid:107) ≤ Since A A , n /n γ and ∆ c, this is further bounded below by 2 F k F (cid:88) (cid:110) (cid:13) (cid:13) (cid:111) (cid:107) (cid:13)(cid:107) (cid:13) K γ ∆(k) 2 ≥ (cid:107) (cid:107) F κ ∆ 2 . k=1 2 (cid:107)Θ(0k)(cid:107)2 + ∆(k) F 2 (cid:96)n,c F 32 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (ii)

text: Let

text: c

text: be

text: a

text: constant.

text: For

text: c

text: and

text: n

text: /n

text: γ

text: for

text: all

text: k

text: and

text: n,

text: F

text: k

text: γ

text: (Θ

text: (Θ

text: (Θ

text: κ

text: n

text: n

text: n

text: λ

text: c

text: F

text: F

text: Θ

text: M

text: M⊥

text: (iii)

text: The

text: map

text: r

text: is

text: a

text: seminorm,

text: convex,

text: and

text: decomposable

text: with

text: respect

text: to

text: in

text: the

text: M

text: M⊥

text: sense

text: that

text: r(Θ

text: Θ

text: r(Θ

text: r(Θ

text: for

text: every

text: Θ

text: and

text: Θ

text: Moreover,

text: r(Θ

text: r(Θ

text: r(∆M⊥)

text: r(∆M).

text: R

text: (iv)

text: For

text: ˜(pK)

text: (pK),

text: (Θ

text: r(∆)

text: max

text: Ψˆ

text: (k)

text: Ψ(k)

text: n

text: n

text: k

text: K

text: R

text: (v)

text: For

text: Θ

text: ˜(pK)

text: (pK),

text: r(ΘM)

text: (s

text: ρ

text: L

text: ΘM

text: F

text: Proof.

text: (i)

text: The

text: result

text: follows

text: by

text: taking

text: derivatives

text: blockwise.

text: (ii)

text: Rothman

text: et

text: al.

text: (page

text: showed

text: that

text: (Θ

text: (Θ

text: (Θ

text: n

text: n

text: n

text: K

text: n

text: nk

text: log

text: ∆(k))

text: log

text: ∆(k)

text: K

text: n

text: k

text: F

text: n

text: v

text: ∆(k)

text: Since

text: A

text: A

text: n

text: /n

text: γ

text: and

text: c,

text: this

text: is

text: further

text: bounded

text: below

text: by

text: F

text: k

text: F

text: K

text: γ

text: ∆(k)

text: F

text: κ

text: ∆(k)

text: F

text: F

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:107) − (cid:107) Next, we obtain an upper bound for max ≤ ≤ Ψˆ (k) Ψ(k) ∞, which holds with high- 1 k K n 0 probability assuming the tail conditions of the random vectors. ≥ Lemma 4. Suppose that n /n γ > 0 for all k and n. k ≥ − (i) Suppose that Condition 1 holds. Then for n 6γ 1 log p we have (cid:32) (cid:115) (cid:33) √ (cid:107) − (cid:107) ≥ − log p ≤ P Σˆ Σ ∞ 23 6(1 + 4c2)2γ 1/2 max σ(k) 2K/p. (13) n 0 1 k,i ii γn (cid:113) ≤ (ii) Suppose that Condition 2 holds with c2, c3 > 0 and p c7nc2. Then we have for τ > { } { } max (27 + 23 1 + 2432c max σ(k) 2)/(9c max σ(k) 2) k 4 k,i(cid:32) ii 4 k,i ii (cid:33) (cid:88) K (cid:107) − (cid:107) ≥ ≤ P Σˆ Σ ∞ δ(k) Kν (14) n 0 n n k=1 where ≡ | | δ(k) (1 + 2 max µ(k),i )(2δ(k) + δ(k)) + (δ(k))2 + (δ(k))2 + 2δ(k), n n,1 n,2 n,1 n,2 n,3 i with δ(k) ≡ max E |X(k),iX(k),j|I(|X(k),iX(k),j| ≥ n1/2(log p)−1/2), n,1 l l l l k i,j ≡ { { } } δ(k) c max σ(k) 2(4 + τ) log p/n 1/2, n,2 4 ii k k,i δ(k) ≡ max E |X(k),i|I(|X(k),i| ≥ n1/2(log p)−1/2). n,3 l l k i (cid:107) − (cid:107) ≥ (iii) Suppose that Condition 3 holds and that P ( Σˆ Σ ∞ b ) = o(1) and b = o(1) as n 0 n n → ∞ (cid:107) − (cid:107) ≥ n . Then P ( Ψˆ Ψ ∞ C b ) = o(1). n 0 1 n Proof. (i) This was proved by Ravikumar et al. (2011). (ii) Note that 34 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Next,

text: we

text: obtain

text: an

text: upper

text: bound

text: for

text: max

text: Ψˆ

text: (k)

text: Ψ(k)

text: which

text: holds

text: with

text: high-

text: k

text: K

text: n

text: probability

text: assuming

text: the

text: tail

text: conditions

text: of

text: the

text: random

text: vectors.

text: Lemma

text: Suppose

text: that

text: n

text: /n

text: γ

text: for

text: all

text: k

text: and

text: n.

text: k

text: (i)

text: Suppose

text: that

text: Condition

text: holds.

text: Then

text: for

text: n

text: 6γ

text: log

text: p

text: we

text: have

label: equation

text: log

text: p

text: P

text: Σˆ

text: Σ

text: max

text: σ(k)

text: n

text: k,i

text: ii

text: γn

text: (ii)

text: Suppose

text: that

text: Condition

text: holds

text: with

text: and

text: p

text: Then

text: we

text: have

text: for

text: τ

text: max

text: max

text: σ(k)

text: max

text: σ(k)

text: k

text: ii

text: k,i

text: ii

text: K

text: P

text: Σˆ

text: Σ

text: δ(k)

text: Kν

text: n

text: n

text: n

text: where

text: δ(k)

text: max

text: µ(k),i

text: δ(k))

text: n

text: i

text: with

text: δ(k)

text: max

text: E

text: |X(k),iX(k),j|I(|X(k),iX(k),j|

text: l

text: l

text: l

text: l

text: k

text: i,j

text: δ(k)

text: c

text: max

text: σ(k)

text: τ)

text: log

text: p/n

text: ii

text: k

text: k,i

text: δ(k)

text: max

text: E

text: |X(k),i|I(|X(k),i|

text: l

text: l

text: k

text: i

text: (iii)

text: Suppose

text: that

text: Condition

text: holds

text: and

text: that

text: P

text: Σˆ

text: Σ

text: b

text: and

text: b

text: as

text: n

text: n

text: n

text: n

text: Then

text: P

text: Ψˆ

text: Ψ

text: C

text: b

text: n

text: n

text: Proof.

text: (i)

text: This

text: was

text: proved

text: by

text: Ravikumar

text: et

text: al.

text: (ii)

text: Note

text: that

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:80) (cid:80) (cid:80) span of u , . . . , u − . Thus, the subgradient v can be written as v = p α u . Thus, using 1 j0 1 (cid:80) j=j0 j j { } the spectral decomposition of A as A = p λ u uT , we can write f(y) = p λ β2 1/2. j=j0 j j j j=j0 j j On the other hand, (cid:104)y, v(cid:105) = p α β. Thus, the inequality (22) further reduces to (cid:88)j=j0 j j (cid:88)  1/2 p p ≥ ∀ ∈ R λ β2 α β , β . j j j j j j=j0 j=j0 It follows from the Cauchy-Schwartz inequality that the left hand side of the inequality is bounded from above;     (cid:88) (cid:88) (cid:88)  (cid:88)   1/2  1/2 p p α ≤ p α2 p α β = j λ1/2β j λ β2 . j j 1/2 j j λ j j λ j j=j0 j=j0 j j=j0 j=j0 Thus,    (cid:88) (cid:88)    p p αj2 ≤ ∈ R ∂f(x) = v : v = α v , 1, α . j j j λ j j=j0 j=j0 It is easy to see that this set is the image of the map UΛ1/2 on the closed ball of radius 1. (cid:107) (cid:107) ≤ (cid:107) (cid:107) Given that x ∞ x , to establish the bound in the (cid:96)∞-norm, we compute the bound (cid:13) (cid:13) (cid:54) (cid:13) (cid:13) in the Euclidean norm. We use the same notation as in (i). For x with Ax = 0, (cid:13) (cid:13) (cid:13) (cid:13) (cid:107) (cid:107) √ Ax UΛ1/2Λ1/2UT x ≤ (cid:107) (cid:107) = (cid:107) (cid:107) UΛ1/2 . 2 xT Ax Λ1/2UT x (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) But UΛ1/2 = sup(cid:107) (cid:107) UΛ1/2x = sup ∈R UΛ1/2(UT x) / UT x = A 1/2, because 2 x =1 x K 2 (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107)(cid:107) ≤ (cid:107) (cid:107) UT x = x . For x with Ax = 0, Λ1/2y/ y A 1/2 for every y. Because of the form of 2 (cid:107) (cid:107) (cid:107) (cid:107) the subdiﬀerential and the fact that Ux = x , the result follows. (ii) Let B be a product of elementary matrices for row and column exchange such that S − B g (x) = (x, 0). Notice that B = B 1 and that B = BT since B only rearranges S S S S S S S (cid:107) (cid:107) ≤ elements of vectors and exchanges rows by multiplication from the left. Note also that B S 2 (cid:107) (cid:107) (cid:107) (cid:107) ≤ (cid:107) (cid:107) B ∞ ∞ = 1, since C C ∞ ∞ for C = CT and each row of B has only one element S / 2 / S with value 1. Because { } h (x) 2 = g (x)T Ag (x) = (B g (x))T (B AB )(B g (x)) = xT A x, A,S S S S S S S S S SS 43 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: span

text: of

text: u

text: u

text: Thus,

text: the

text: subgradient

text: v

text: can

text: be

text: written

text: as

text: v

text: p

text: α

text: u

text: Thus,

text: using

text: j

text: j

text: the

text: spectral

text: decomposition

text: of

text: A

text: as

text: A

text: p

text: λ

text: u

text: uT

text: we

text: can

text: write

text: f(y)

text: p

text: λ

text: β2

text: j

text: j

text: j

text: j

text: j

text: On

text: the

text: other

text: hand,

text: p

text: α

text: β.

text: Thus,

text: the

text: inequality

text: further

text: reduces

text: to

text: j

text: j

text: p

text: p

text: R

text: λ

text: β2

text: α

text: β

text: β

text: j

text: j

text: j

text: j

text: j

text: It

text: follows

text: from

text: the

text: Cauchy-Schwartz

text: inequality

text: that

text: the

text: left

text: hand

text: side

text: of

text: the

text: inequality

text: is

text: bounded

text: from

text: above;

text: p

text: p

text: α

text: p

text: α2

text: p

text: α

text: β

text: j

text: j

text: λ

text: β2

text: j

text: j

text: j

text: j

text: λ

text: j

text: j

text: λ

text: j

text: j

text: Thus,

text: p

text: p

text: R

text: ∂f(x)

text: v

text: v

text: α

text: v

text: α

text: j

text: j

text: j

text: λ

text: j

text: It

text: is

text: easy

text: to

text: see

text: that

text: this

text: set

text: is

text: the

text: image

text: of

text: the

text: map

text: on

text: the

text: closed

text: ball

text: of

text: radius

text: Given

text: that

text: x

text: x

text: to

text: establish

text: the

text: bound

text: in

text: the

text: we

text: compute

text: the

text: bound

text: in

text: the

text: Euclidean

text: norm.

text: We

text: use

text: the

text: same

text: notation

text: as

text: in

text: (i).

text: For

text: x

text: with

text: Ax

text: Ax

text: x

text: xT

text: Ax

text: x

text: But

text: sup

text: ∈R

text: x)

text: UT

text: x

text: A

text: because

text: x

text: x

text: K

text: UT

text: x

text: x

text: For

text: x

text: with

text: Ax

text: y

text: A

text: for

text: every

text: y.

text: Because

text: of

text: the

text: form

text: of

text: the

text: subdiﬀerential

text: and

text: the

text: fact

text: that

text: Ux

text: x

text: the

text: result

text: follows.

text: (ii)

text: Let

text: B

text: be

text: a

text: product

text: of

text: elementary

text: matrices

text: for

text: row

text: and

text: column

text: exchange

text: such

text: that

text: S

text: B

text: g

text: (x)

text: (x,

text: Notice

text: that

text: B

text: B

text: and

text: that

text: B

text: BT

text: since

text: B

text: only

text: rearranges

text: S

text: S

text: S

text: S

text: S

text: S

text: S

text: elements

text: of

text: vectors

text: and

text: exchanges

text: rows

text: by

text: multiplication

text: from

text: the

text: left.

text: Note

text: also

text: that

text: B

text: S

text: B

text: since

text: C

text: C

text: for

text: C

text: CT

text: and

text: each

text: row

text: of

text: B

text: has

text: only

text: one

text: element

text: S

text: S

text: with

text: value

text: Because

text: h

text: (x)

text: g

text: (x)T

text: Ag

text: (x)

text: (B

text: g

text: (x))T

text: (B

text: AB

text: )(B

text: g

text: (x))

text: xT

text: A

text: x,

text: A,S

text: S

text: S

text: S

text: S

text: S

text: S

text: S

text: S

text: SS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: D the matrices involved with eigenvalues in h (A) are uniformly bounded in , and hence the k k eigenvalues are also uniformly bounded. ∈ D (k) To show that f (A) , ﬁrst we show that f (A) + Θ is positive semideﬁnite. This k k k 0 ∈ R follows because for any x p xT (f (A) + Θ(k))x k 0 { − } = xT (f˜ (A)) I x + xT (A + Θ(k))x + xT x k S(k) 0 ≥ { } − (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) ≥ h (A)λ ( Θ(k)g (A + Θ(k) + δI)Θ(k) I) x 2 + λ (A + Θ(k)) x 2 + x 2 0. k 1 0 k 0 0 S(k) 1 0 ≡ { } − (k) (k) (k) To see this, note that if λ λ ( Θ g (A + Θ + δI)Θ I) is positive, then the A 1 0 k 0 0 S(k) − inequality easily follows. On the other hand, if λ < 1, we have A (cid:107) (cid:107) ≥ − − { − }(cid:107) (cid:107) − − (cid:107) (cid:107) h (A)λ x 2 2 1 min λ (A + Θ(k)), 2 1 x 2 2 1 x 2 k A 1 0 ≥ − (cid:107) (cid:107) (λ (A + Θ(k))/2 + 1/2) x 2. 1 0 − ≤ Lastly, if 1 λ < 0, we have A (cid:107) (cid:107) ≥ −| | − { − } (cid:107) (cid:107) h (A)λ x 2 λ [2 1 min λ (A + Θ(k)), 2 1 + 1/2] x 2 k A A 1 0 ≥ −| | (cid:107) (cid:107) λ (λ (A + Θ(k))/2 + 1/2) x 2. A 1 0 (cid:107) (cid:107) ≤ Next, we show that f (A) ∞ r. Because diag(f (A)) = diag(A), we have only to k (k) k S (cid:107) (cid:107) ≤ ≤ { { } show f (A) ∞ r. Since δ + r 1/ 6d max κ , κ3 κ , we have k S(k) Ψ Ψ Γ (cid:107) (cid:107) ≤ (cid:107) (cid:107) ≤ ≤ Ψ(k)(A + δI) ∞ ∞ κ d A + δI ∞ κ d(r + δ) 1/3. 0 / Ψ Ψ It follows from Lemma 9 that − − { } R(A + δI) = (A + δI + Θ(k)) 1 Ψ(k) + Ψ(k)(A + δI)Ψ(k) = Ψ(k)(A + δI) 2H(k)Ψ(k). 0 0 0 0 0 0 49 ##LTLine##

text: D

text: the

text: matrices

text: involved

text: with

text: eigenvalues

text: in

text: h

text: (A)

text: are

text: uniformly

text: bounded

text: in

text: and

text: hence

text: the

text: k

text: k

text: eigenvalues

text: are

text: also

text: uniformly

text: bounded.

text: D

text: (k)

text: To

text: show

text: that

text: f

text: (A)

text: ﬁrst

text: we

text: show

text: that

text: f

text: (A)

text: Θ

text: is

text: positive

text: semideﬁnite.

text: This

text: k

text: k

text: k

text: R

text: follows

text: because

text: for

text: any

text: x

text: p

text: xT

text: (f

text: (A)

text: Θ(k))x

text: k

text: xT

text: (f˜

text: (A))

text: I

text: x

text: xT

text: (A

text: Θ(k))x

text: xT

text: x

text: k

text: S(k)

text: h

text: (A)λ

text: Θ(k)g

text: (A

text: Θ(k)

text: δI)Θ(k)

text: I)

text: x

text: λ

text: (A

text: Θ(k))

text: x

text: x

text: k

text: k

text: S(k)

text: (k)

text: (k)

text: (k)

text: To

text: see

text: this,

text: note

text: that

text: if

text: λ

text: λ

text: Θ

text: g

text: (A

text: Θ

text: δI)Θ

text: I)

text: is

text: positive,

text: then

text: the

text: A

text: k

text: S(k)

text: inequality

text: easily

text: follows.

text: On

text: the

text: other

text: hand,

text: if

text: λ

text: we

text: have

text: A

text: h

text: (A)λ

text: x

text: min

text: λ

text: (A

text: Θ(k)),

text: x

text: x

text: k

text: A

text: (λ

text: (A

text: x

text: Lastly,

text: if

text: λ

text: we

text: have

text: A

text: h

text: (A)λ

text: x

text: λ

text: min

text: λ

text: (A

text: Θ(k)),

text: x

text: k

text: A

text: A

text: λ

text: (λ

text: (A

text: x

text: A

text: Next,

text: we

text: show

text: that

text: f

text: (A)

text: r.

text: Because

text: diag(f

text: (A))

text: diag(A),

text: we

text: have

text: only

text: to

text: k

text: (k)

text: k

text: S

text: show

text: f

text: (A)

text: r.

text: Since

text: δ

text: r

text: max

text: κ

text: κ3

text: κ

text: we

text: have

text: k

text: S(k)

text: Ψ

text: Ψ

text: Γ

text: Ψ(k)(A

text: δI)

text: κ

text: d

text: A

text: δI

text: κ

text: d(r

text: δ)

text: Ψ

text: Ψ

text: It

text: follows

text: from

text: Lemma

text: that

text: R(A

text: δI)

text: (A

text: δI

text: Θ(k))

text: Ψ(k)

text: Ψ(k)(A

text: δI)Ψ(k)

text: Ψ(k)(A

text: δI)

text: ##LTLine##


text: (k) Thus, adding and subtracting Ψ yields 0 { } − − − f˜ (A) + A = h (A)Θ(k)((n /n) Ψ(k)(A + δI) 2H(k)Ψ(k) (n /n)Ξ(k) ρ Uˇ(k) ρ ρ Uˇ(k))Θ(k) k k 0 k 0 0 k n 1 n 2 2 0 − − +(1 (n /n)h (A))A (n /n)δh (A)I. k k k k Vectorization and restriction on S(k) gives (cid:107) (cid:107) (cid:107) (cid:107) vec(f (A)) ∞ = vec(f˜ (A) + A) ∞ k S(k) k S(k) ≤ (cid:107){ − } { } (cid:107) (n /n)h (A) (Γ(k)) 1 vec( Ψ(k)(A + δI) 2H(k)Ψ(k)) ∞ k k S(k)S(k) 0 0 S(k) (cid:107){ − } { }(cid:107) +h (A) (Γ(k)) 1 vec((n /n)Ξ(k)) + ρ vec(Uˇ(k)) + ρ ρ vec(Uˇ(k)) ∞ k S(k)S(k) k S(k) n 1 S(k) n 2 2 S(k) − (cid:107) (cid:107) +(1 (n /n)h (A)) vec(A) ∞ + (n /n)δ, (29) k k S(k) k { − } ⊗ ≤ where (Γ(k)) 1 = (Θ(k) Θ(k)) . Here we used h (A) (1/4 + 1/2)/1 = S(k)S(k) 0 S(k)S(k) k (cid:107) (cid:107) ≤ 3/4. For the ﬁrst term of the upper bound in (29), it follows by the inequality Ax ∞ (cid:107) (cid:107) (cid:107) (cid:107) ∈ R × ∈ R ≤ A ∞ ∞ x ∞ for A p p and x p, Lemma 9 and the choice of δ satisfying δ + r / { { } 1/ 6d max κ , κ3 κ that Ψ Ψ Γ (cid:107){ − } { } (cid:107) (Γ(k)) 1 vec( Ψ(k)(A + δI) 2H(k)Ψ(k)) ∞ S(k)S(k) 0 0 S(k) ≤ (cid:107) (cid:107) ≤ 3 (cid:107) (cid:107) ≤ 3 (cid:107) (cid:107) κ R(k)(A + δI) ∞ κ d A + δI 2∞κ3 κ d A + δI ∞(r + δ)κ3 Γ Γ 2 Ψ Γ 2 Ψ ≤ (r + δ)/4. (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) For the second term, it follows by the assumption, the inequality that Ax ∞ A ∞ ∞ x ∞ / ∈ R × ∈ R for A p p and x p, and Lemma 6 that (cid:107){ − } { }(cid:107) (Γ(k)) 1 (n /n)vec(Ξ(k)) + ρ vec(Uˇ(k)) + ρ ρ vec(Uˇ(k)) ∞ S(k)S(k) k S(k) n 1 S(k) n 2 2 S(k) ≤ (cid:107) (cid:107) (cid:107) ≤ κ ( Ξ(k) + ρ + ρ ρ L 1/2) = (min π )r/4 (n /n)r/2. Γ n n 2 2 k k k 50 ##LTLine## ##LTLine##

text: (k)

text: Thus,

text: adding

text: and

text: subtracting

text: Ψ

text: yields

text: f˜

text: (A)

text: A

text: h

text: (A)Θ(k)((n

text: /n)

text: Ψ(k)(A

text: δI)

text: (n

text: /n)Ξ(k)

text: ρ

text: Uˇ(k)

text: ρ

text: ρ

text: Uˇ(k))Θ(k)

text: k

text: k

text: k

text: k

text: n

text: n

text: (n

text: /n)h

text: (A))A

text: (n

text: /n)δh

text: (A)I.

text: k

text: k

text: k

text: k

text: Vectorization

text: and

text: restriction

text: on

text: S(k)

text: gives

text: vec(f

text: (A))

text: vec(f˜

text: (A)

text: A)

text: k

text: S(k)

text: k

text: S(k)

text: (n

text: /n)h

text: (A)

text: (Γ(k))

text: vec(

text: Ψ(k)(A

text: δI)

text: k

text: k

text: S(k)S(k)

text: S(k)

text: +h

text: (A)

text: (Γ(k))

text: vec((n

text: /n)Ξ(k))

text: ρ

text: vec(Uˇ(k))

text: ρ

text: ρ

text: vec(Uˇ(k))

text: k

text: S(k)S(k)

text: k

text: S(k)

text: n

text: S(k)

text: n

text: S(k)

text: (n

text: /n)h

text: (A))

text: vec(A)

text: (n

text: /n)δ,

text: k

text: k

text: S(k)

text: k

text: where

text: (Γ(k))

text: (Θ(k)

text: Θ(k))

text: Here

text: we

text: used

text: h

text: (A)

text: S(k)S(k)

text: S(k)S(k)

text: k

text: For

text: the

text: ﬁrst

text: term

text: of

text: the

text: upper

text: bound

text: in

text: it

text: follows

text: by

text: the

text: inequality

text: Ax

text: R

text: R

text: A

text: x

text: for

text: A

text: p

text: p

text: and

text: x

text: p,

text: Lemma

text: and

text: the

text: choice

text: of

text: δ

text: satisfying

text: δ

text: r

text: max

text: κ

text: κ3

text: κ

text: that

text: Ψ

text: Ψ

text: Γ

text: (Γ(k))

text: vec(

text: Ψ(k)(A

text: δI)

text: S(k)S(k)

text: S(k)

text: κ

text: R(k)(A

text: δI)

text: κ

text: d

text: A

text: δI

text: 2∞κ3

text: κ

text: d

text: A

text: δI

text: ∞(r

text: δ)κ3

text: Γ

text: Γ

text: Ψ

text: Γ

text: Ψ

text: (r

text: For

text: the

text: second

text: term,

text: it

text: follows

text: by

text: the

text: assumption,

text: the

text: inequality

text: that

text: Ax

text: A

text: x

text: R

text: R

text: for

text: A

text: p

text: p

text: and

text: x

text: p,

text: and

text: Lemma

text: that

text: (Γ(k))

text: (n

text: /n)vec(Ξ(k))

text: ρ

text: vec(Uˇ(k))

text: ρ

text: ρ

text: vec(Uˇ(k))

text: S(k)S(k)

text: k

text: S(k)

text: n

text: S(k)

text: n

text: S(k)

text: κ

text: Ξ(k)

text: ρ

text: ρ

text: ρ

text: L

text: (min

text: π

text: (n

text: Γ

text: n

text: n

text: k

text: k

text: k

text: ##LTLine##

text: ##LTLine##


text: (cid:54) note that Θˇ = 0 with probability approaching 1 and that the rearrangement in Θ and ρn,ij ij corresponding exchange of rows and columns of L for each i, j does not change the original and restricted optimization problems (3) and (21). Thus, with the appropriate rearrangement of (cid:54) elements and exchange of rows and columns, U˜(k) with ω(k) = 0 is in fact Uˇ(k). Thus for such 2,ij 0,ij 2,ij k the equation holds because of the corresponding KKT condition in the restricted problem (21). For other k, the equation holds by construction. We thus conclude the oracle estimator satisﬁes the KKT condition of the original problem (3). This completes the proof. Proof of Corollary 1. In the proof of Theorem 2, the (cid:96)∞-bound of the error yields (cid:107) − (cid:107) Θˆ (k) Θ(k) ∞ = O (κ ρ ) . ρn 0 P Γ n (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) Note that if one of two matrices A and B is diagonal, AB ∞ A ∞ B ∞. Thus, we can proceed in the same way as in the proof of Theorem 2 of Rothman et al. (2008) to conclude that (cid:107) − (cid:107) Ωˆ(k) Ω(k) ∞ = O (κ ρ ) . n 0 P Γ n The result follows from a similar argument to the proof of Corollary 3 in Ravikumar et al. (2011). (cid:107) − Proof of Corollary 2. It follows from Condition 8 and Lemma 1 applied to Θˇ that Θˇ (k) ρn ρn (cid:107) ≤ (k) Θ 1/(2λ ). Then we can apply Lemma 10 instead of Lemma 9. The rest is similar to 0 2 Θ the proof of Theorem 2. Hierarchical Clustering For simplicity, we prove Theorem 3 for the case of K = 2; the proof can be easily gen- eralized to K > 2. Let X and Y be the random variable from the ﬁrst and subpopulation, ∼ respectively. Suppose that X = (X , . . . , X )T N(µ , Σ ) with µ = (µ , . . . , µ ) and 1 p X X X 1,X p,X the spectral decomposition Σ = Q Λ QT of Σ where λ , . . . , λ are the eigenval- X X X X X 1,X p,X ∼ ues of Σ and that Y N(µ , Σ ) with µ = (µ , . . . , µ ) and the spectral decom- X Y Y Y 1,Y p,Y position Σ = Q Λ QT of Σ where λ , . . . , λ are the eigenvalues of Σ . Deﬁne Y Y Y Y Y 1,Y p,Y Y − ∼ Z = (X Y ) = (Z , . . . , Z )T N(µ , Σ ) with µ = (µ , . . . , µ ) and the spec- 1 p Z Z Z 1,Z p,Z tral decomposition Σ = Q Λ QT of Σ where λ , . . . , λ are the eigenvalues of Σ . Z Z Z Z Z 1,Z p,Z Z − − Let X˜ = (X˜ , . . . , X˜)T = Λ1/2QT Σ 1/2X, Y˜ = (Y˜ , . . . , Y˜ )T = Λ1/2QT Σ 1/2Y and Z˜ = 1 X X X 1 Y Y Y 54 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: note

text: that

text: Θˇ

text: with

text: probability

text: approaching

text: and

text: that

text: the

text: rearrangement

text: in

text: Θ

text: and

text: ρn,ij

text: ij

text: corresponding

text: exchange

text: of

text: rows

text: and

text: columns

text: of

text: L

text: for

text: each

text: i,

text: j

text: does

text: not

text: change

text: the

text: original

text: and

text: restricted

text: optimization

text: problems

text: and

text: Thus,

text: with

text: the

text: appropriate

text: rearrangement

text: of

text: elements

text: and

text: exchange

text: of

text: rows

text: and

text: columns,

text: U˜(k)

text: with

text: ω(k)

text: is

text: in

text: fact

text: Uˇ(k).

text: Thus

text: for

text: such

text: k

text: the

text: equation

text: holds

text: because

text: of

text: the

text: corresponding

text: KKT

text: condition

text: in

text: the

text: restricted

text: problem

text: For

text: other

text: k,

text: the

text: equation

text: holds

text: by

text: construction.

text: We

text: thus

text: conclude

text: the

text: oracle

text: estimator

text: satisﬁes

text: the

text: KKT

text: condition

text: of

text: the

text: original

text: problem

text: This

text: completes

text: the

text: proof.

text: Proof

text: of

text: Corollary

text: In

text: the

text: proof

text: of

text: Theorem

text: the

text: of

text: the

text: error

text: yields

text: Θˆ

text: (k)

text: Θ(k)

text: O

text: (κ

text: ρ

text: ρn

text: P

text: Γ

text: n

text: Note

text: that

text: if

text: one

text: of

text: two

text: matrices

text: A

text: and

text: B

text: is

text: diagonal,

text: AB

text: A

text: B

text: Thus,

text: we

text: can

text: proceed

text: in

text: the

text: same

text: way

text: as

text: in

text: the

text: proof

text: of

text: Theorem

text: of

text: Rothman

text: et

text: al.

text: to

text: conclude

text: that

text: Ωˆ(k)

text: Ω(k)

text: O

text: (κ

text: ρ

text: n

text: P

text: Γ

text: n

text: The

text: result

text: follows

text: from

text: a

text: similar

text: argument

text: to

text: the

text: proof

text: of

text: Corollary

text: in

text: Ravikumar

text: et

text: al.

text: Proof

text: of

text: Corollary

text: It

text: follows

text: from

text: Condition

text: and

text: Lemma

text: applied

text: to

text: Θˇ

text: that

text: Θˇ

text: (k)

text: ρn

text: ρn

text: (k)

text: Θ

text: Then

text: we

text: can

text: apply

text: Lemma

text: instead

text: of

text: Lemma

text: The

text: rest

text: is

text: similar

text: to

text: Θ

text: the

text: proof

text: of

text: Theorem

text: Hierarchical

text: Clustering

text: For

text: simplicity,

text: we

text: prove

text: Theorem

text: for

text: the

text: case

text: of

text: K

text: the

text: proof

text: can

text: be

text: easily

text: gen-

text: eralized

text: to

text: K

text: Let

text: X

text: and

text: Y

text: be

text: the

text: random

text: variable

text: from

text: the

text: ﬁrst

text: and

text: subpopulation,

text: respectively.

text: Suppose

text: that

text: X

text: (X

text: X

text: )T

text: N(µ

text: Σ

text: with

text: µ

text: (µ

text: µ

text: and

text: p

text: X

text: X

text: X

text: p,X

text: the

text: spectral

text: decomposition

text: Σ

text: Q

text: Λ

text: QT

text: of

text: Σ

text: where

text: λ

text: λ

text: are

text: the

text: eigenval-

text: X

text: X

text: X

text: X

text: X

text: p,X

text: ues

text: of

text: Σ

text: and

text: that

text: Y

text: N(µ

text: Σ

text: with

text: µ

text: (µ

text: µ

text: and

text: the

text: spectral

text: decom-

text: X

text: Y

text: Y

text: Y

text: p,Y

text: position

text: Σ

text: Q

text: Λ

text: QT

text: of

text: Σ

text: where

text: λ

text: λ

text: are

text: the

text: eigenvalues

text: of

text: Σ

text: Deﬁne

text: Y

text: Y

text: Y

text: Y

text: Y

text: p,Y

text: Y

text: Z

text: (X

text: Y

text: (Z

text: Z

text: )T

text: N(µ

text: Σ

text: with

text: µ

text: (µ

text: µ

text: and

text: the

text: spec-

text: p

text: Z

text: Z

text: Z

text: p,Z

text: tral

text: decomposition

text: Σ

text: Q

text: Λ

text: QT

text: of

text: Σ

text: where

text: λ

text: λ

text: are

text: the

text: eigenvalues

text: of

text: Σ

text: Z

text: Z

text: Z

text: Z

text: Z

text: p,Z

text: Z

text: Let

text: X˜

text: (X˜

text: X˜)T

text: Σ

text: Y˜

text: (Y˜

text: Y˜

text: )T

text: Σ

text: and

text: Z˜

text: X

text: X

text: X

text: Y

text: Y

text: Y

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: One-Mode Wigner Function for ECS 2 bipartite entanglement of ECS’s with the behavior of peaks of Wigner functions. More- over, we investigate decoherence eﬀects on Wigner function, arising from transmitting ECS’s through noisy channels. PACs Index:03.65.Ud 1 Introduction Phase-space representations of quantum states have been important tools for exploring the connections between quantum and classical physics. In 1932, Wigner introduced a distribution function in mechanics that permitted a description of mechanical phenomena in a phase space [1, 2]. Some important characteristics of the spatial Wigner function of entangled photon pairs were analyzed in [3]. In [4] the negativity of the Wigner function was discussed as a measure of the non-classicality. Negativity of the Wigner function is the reason why the Wigner function can not be regarded as a real probability distribution but a quasi-probability distribution function and it is a good indication of the possibility of the occurrence of nonclassical properties. Wigner functions have been especially used for describing the quadratures of the electrical ﬁeld with coherent and squeezed states or single photon states [5, 6, 7]. The four-dimensional chronocyclic Wigner function of the parametric downconversion state was discussed in [8]. In 1926, Schrodinger introduced coherent states [9] then advanced studies were done in [10, 11]. In recent years, there has been the considerable interest in studying multi-mode quantum states of radiation ﬁelds because they have widely role in quantum information theory. The problem of generating various quantum states of an electromagnetic ﬁeld was discussed in [12, 13, 14, 15, 16, 17, 18, 19]. In [20, 21] it was proposed a scheme for producing a superposition of two arbitrary Glauber coherent states via parity and displacement operator. The other scheme for preparation of superposition of coherent states in cavity QED was studied in [22, 23] in which an atom either ﬂying through or trapped within a cavity, is controlled

text: One-Mode

text: Wigner

text: Function

text: for

text: ECS

text: bipartite

label: abstract

text: entanglement

text: of

text: ECS’s

text: with

text: the

text: behavior

text: of

text: peaks

text: of

text: Wigner

text: functions.

text: More-

text: over,

text: we

text: investigate

text: decoherence

text: eﬀects

text: on

text: Wigner

text: function,

text: arising

text: from

text: transmitting

text: ECS’s

text: through

text: noisy

text: channels.

text: PACs

label: section

text: Introduction

text: Phase-space

text: representations

text: of

text: quantum

text: states

text: have

text: been

text: important

text: tools

text: for

text: exploring

text: the

text: connections

text: between

text: quantum

text: and

text: classical

text: physics.

text: In

text: Wigner

text: introduced

text: a

text: distribution

text: function

text: in

text: mechanics

text: that

text: permitted

text: a

text: description

text: of

text: mechanical

text: phenomena

text: in

text: a

text: phase

text: space

text: Some

text: important

text: characteristics

text: of

text: the

text: spatial

text: Wigner

text: function

text: of

text: entangled

text: photon

text: pairs

text: were

text: analyzed

text: in

text: In

text: the

text: negativity

text: of

text: the

text: Wigner

text: function

text: was

text: discussed

text: as

text: a

text: measure

text: of

text: the

text: non-classicality.

text: Negativity

text: of

text: the

text: Wigner

text: function

text: is

text: the

text: reason

text: why

text: the

text: Wigner

text: function

text: can

text: not

text: be

text: regarded

text: as

text: a

text: real

text: probability

text: distribution

text: but

text: a

text: quasi-probability

text: distribution

text: function

text: and

text: it

text: is

text: a

text: good

text: indication

text: of

text: the

text: possibility

text: of

text: the

text: occurrence

text: of

text: nonclassical

text: properties.

text: Wigner

text: functions

text: have

text: been

text: especially

text: used

text: for

text: describing

text: the

text: quadratures

text: of

text: the

text: electrical

text: ﬁeld

text: with

text: coherent

text: and

text: squeezed

text: states

text: or

text: single

text: photon

text: states

text: The

text: four-dimensional

text: chronocyclic

text: Wigner

text: function

text: of

text: the

text: parametric

text: downconversion

text: state

text: was

text: discussed

text: in

text: In

text: Schrodinger

text: introduced

text: coherent

text: states

text: then

text: advanced

text: studies

text: were

text: done

text: in

text: In

text: recent

text: years,

text: there

text: has

text: been

text: the

text: considerable

text: interest

text: in

text: studying

text: multi-mode

text: quantum

text: states

text: of

text: radiation

text: ﬁelds

text: because

text: they

text: have

text: widely

text: role

text: in

text: quantum

text: information

text: theory.

text: The

text: problem

text: of

text: generating

text: various

text: quantum

text: states

text: of

text: an

text: electromagnetic

text: ﬁeld

text: was

text: discussed

text: in

text: In

text: it

text: was

text: proposed

text: a

text: scheme

text: for

text: producing

text: a

text: superposition

text: of

text: two

text: arbitrary

text: Glauber

text: coherent

text: states

text: via

text: parity

text: and

text: displacement

text: operator.

text: The

text: other

text: scheme

text: for

text: preparation

text: of

text: superposition

text: of

text: coherent

text: states

text: in

text: cavity

text: QED

text: was

text: studied

text: in

text: in

text: which

text: an

text: atom

text: either

text: ﬂying

text: through

text: or

text: trapped

text: within

text: a

text: cavity,

text: is

text: controlled


text: One-Mode Wigner Function for ECS 12 in which M(3) is normalization factor. In [22] the optimal values of ε , ε , and ε determined to 0 1 2 − − †− ∗ be 0.8200, 2.1184, and 0.4720, respectively. Acting displacement operator Dˆ(β) = eβaˆ β aˆ | (cid:105) | (cid:105) ˆ with property D(β) α = α + β , onto state Eq.(3.25) leads to the following superposition of coherent states | (cid:48) (cid:105) √ 1 | (cid:105) | (cid:105) | − (cid:105) ˆ D(β) ψ = ( 2α + β + 1.35 β + 2α + β ). (3.26) 2 2e−8α2 + 5.4e−2α2 + 3.8225 − By transmitting this state Eq.(3.26) through 50 50 beam splitter we have − − | (cid:105) √ 1 |2α√+ β (cid:105)|2α√+ β (cid:105) |√β (cid:105)|√β (cid:105) | 2√α + β (cid:105)| 2√α + β (cid:105) Φ(3) = ( + 1.35 + ). (3.27) M(3) 2 2 2 2 2 2 In the following, we concern the one-mode Wigner function of qutrit like ECS’s. Firstly, we | (cid:105) | (cid:105) | (cid:105) | (cid:105) will consider the general qutrit like ECS state Ψ(3) = √ 1 ( αα + µ ββ + µ γγ ) then as (cid:48) 1 2 M (3) | (cid:105) an especial example we will consider the state Eq.(3.27). One can show that the state Ψ(3) is {| (cid:105) | (cid:105) | (cid:105)} a two-mode qutrit like ECS. To this end we assume the set α , β , γ are in general linearly {| (cid:105) | (cid:105) | (cid:105)} independent, i.e. they span the three dimensional Hilbert space 0 , 1 2 . Therefore we can deﬁne three orthonormal basis as | (cid:105) | (cid:105) 0 = α , (cid:113) (cid:16) (cid:17) | (cid:105) √ | (cid:105) − | (cid:105) 1 = 1 ( β p α ), (3.28) − 1 1 p2 1 | (cid:105) − | (cid:105) − | (cid:105) − | (cid:105) 2 = − − 1− p21 γ + (p1p−3 p2 ) β + (p1p−2 p3 ) α , 1 p21 p22 p23+2p1p2p3 1 p21 1 p21 (cid:104) | (cid:105) (cid:104) | (cid:105) (cid:104) | (cid:105) where p = α β , p = γ β , p = γ α and again for simplicity we assumed that all the 1 2 3 | (cid:105) parameters are real. By substituting Eq. (3.28) in state Ψ(3) we have | (cid:105) { | (cid:105) | (cid:105) | (cid:105) Ψ(3) = √ 1 (1 + µ p2 + µ p2) 00 + N2(µ + µ x2) 11 + µ N2 22 M(cid:48)(3) 1 1 2 3 1 1 2 2 2 − | (cid:105) | (cid:105) − | (cid:105) | (cid:105) + N (µ p µ xp )( 10 + 01 ) µ xN N ( 21 + 12 ) (3.29) 1 1 1 2 3 2 1 2 (cid:112)| (cid:105) | (cid:105) } (cid:112) + µ N p ( 20 + 02 ) , 2 2 3 − − − − (cid:48) where x = p1p−3 p2 , N = 1 p2, N = 1 p2 x2N2 and M (3) = 1 + µ2 + µ2 + 2µ p2 + 1 p2 1 1 2 3 1 1 2 1 1 1 2µ µ p2 + 2µ p2 is normalization factor. This state is clearly a two qutrit like state. 1 2 2 2 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: One-Mode Wigner Function for ECS 17 | (cid:105) Figure 8: (Color online) Wigner function of reduced density matrix of Φ(3) as a function of x for β = 7. in ﬁgure (9b). In order to study the noise eﬀect on entanglement between modes 1 and 2, | (cid:48) (cid:105)(cid:104) (cid:48) | we should trace out the environment mode E by partial trace, i.e. ρ(2) = T r ( Ψ (2) Ψ (2) ). 12 E | (cid:105) | (cid:105) By assumption α and β are linearly independent the reduced density matrix in orthogonal   | (cid:105) | (cid:105) basis 0 and 1 reads        a a a a   11 12 12 14    ρ(2) = 1  a12 a22 a22 a24  , (4.39) 12 2 + 2p2 a a a a 12 22 22 24 a a a a 14 24 24 44 where (cid:112) a = 1 + 2p2 + p4η, 11 − − a = p η 1 p2η(p2 + p4η), 12 − − − a = p2 2η + p2η p4η p2, 14 (4.40) − a = p2η p4η, 22 − a = pη(1 p2η)3/2, 24 − a = (1 p2η)2, 44 ##LTFigure## ##LTLine## ##LTLine##

text: One-Mode

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text: One-Mode Wigner Function for ECS 3 by the classical Stark eﬀect. ECS’s have many applications in quantum optics and quantum information processing [24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. In [34, 35] the required conditions | (cid:105) | (cid:105)| (cid:105) | (cid:105)| (cid:105) for the maximal entangled states of the form ψ = µ α β + ν γ δ have been studied and | (cid:105) | (cid:105)| (cid:105) | (cid:105)| (cid:105) | (cid:105)| (cid:105) | (cid:105)| (cid:105) then have been generalized to the state ψ = µ α β + λ α δ + ρ γ β + ν γ δ in Ref. [36]. Generation of multipartite ECS’s and entanglement of multipartite states constructed by linearly independent coherent states are investigated in [37, 38]. In [39] it was considered the production and entanglement properties of the generalized balanced N-mode Glauber coherent (cid:88) (cid:113) states of the form − (cid:124) (cid:123)(cid:122) (cid:125) | (cid:105) 1 d 1 | (cid:105) · · · | (cid:105) Ψ(d) = µ α α , (1.1) N i i i M(d) i=0 N N modes | (cid:105) | (cid:105)| (cid:105) which is a general form of the balanced two-mode entangled coherent state Ψ(2) = √ 1 ( α α + bal M(2) | (cid:105)| (cid:105) | (cid:105) | (cid:105)| (cid:105) | (cid:105)| (cid:105) | (cid:105)| (cid:105) µ β β ) and Ψ(3) = √ 1 ( α α + µ β β + µ γ γ ). By assumption that the coherent bal 1 2 M(3) states are linearly independent, these states recast in two qubit and qutrit form respectively. Then the entanglement of this states was evaluated by concurrence measure. Another problem which has been investigated extensively in quantum information process- ing is noise eﬀect or decoherence which arise from the coupling of the system to its surroundings [38, 40]. van Enk in [41] introduced the eﬀect of noise on coherent states with the modes 1 or 2 √ √ | (cid:105) | (cid:105) → | (cid:105) | − (cid:105) after traveling through a noisy channel as α 0 ηα 1 ηα where the second 1(2) E 1(2) E state now refers to the environment and η is the noise parameter, which gives the fraction of photons that survives the noisy channel. The eﬀect of noise on entanglement between modes 1 and 2 in qubit and qutrit like ECS’s was investigated in Ref.[42]. In this paper we use the displacement operator together with parity operation to construct | (cid:105) | (cid:105) the superposition of two coherent states α + µ β . By transmitting this superposition from | (cid:105) 50-50 beam splitter the two-mode qubit like ECS, Ψ(2) , is generated. Moreover we introduce a method for producing qutrit like ECS using atom-ﬁeld interaction in cavity QED and beam splitter. The Wigner functions for reduced density matrices of both two-mode qubit and qutrit like ECS’s yield some information about the quantum entanglement of ECS’s. This ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: One-Mode

text: Wigner

text: Function

text: for

text: ECS

text: by

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text: Quantum speed limit for noisy dynamics ∗ Haidong Yuan Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong † Chi-Hang Fred Fung Canada Research Centre, Huawei Technologies Canada, Ontario, Canada (Dated: May 28, 2020) The laws of quantum physics place a limit on the speed of computation, in particular the evolution time of a system cannot be arbitrarily fast. Bounds on the speed of evolution for unitary dynamics have been long studied. A few bounds on the speed of evolution for noisy dynamics have also been obtained recently, these bounds, however, are in general not tight. In this article we present a new framework for quantum speed limit of noisy dynamics. With this framework we obtain the exact maximal angle that a noisy dynamics can achieve at any given time, this then provides tight bounds on the evolution time for noisy dynamics. The obtained bound reveals that noisy dynamics are generically diﬀerent from unitary dynamics, in particular we show that the ’orthogonalization’ time, which is the minimum time needed to evolve any state to its orthogonal states, is in general not applicable to noisy dynamics. Quantum information processing may be regarded as Our framework is based on a method that gives the ex- the transformation of quantum states encoding the infor- act maximal rotation angle for a given dynamics, which mation to be processed or computed. The time for which ensures that the bound is achievable. And the bound the states transform dictates the speed of the quantum is obtained directly from the Kraus operators of the dy- computation. Quantum physics imposes a limit on the namics, which allows ease of computation. The obtained transformation time. This quantum speed limit (QSL) [1] bound reveals that noisy dynamics are generically diﬀer- arises because the energy of the system and environment ent from unitary dynamics, in particular we show that is ﬁnite and the system state may evolve according to a the ’orthogonalization’ time, a concept heavily used in slow dynamics. QSL, is in general not applicable to noisy dynamics. The transformation of quantum states may be mea- Our framework builds on a distance measure on quan- × sured by the Bures angle which characterizes the distance tum channels [31] which we describe brieﬂy. For a m m − between two states. In a time period t, a quantum pro- unitary matrix U, we denote by e iθj the eigenvalues ∈ − ≤ ≤ cess may rotate an initial state by an angle θ. In QSL, the of U, where θ ( π, π] for 1 j m and we call j reverse question is asked. Given a certain angle θ, we ask θ the eigen-angles of U. We deﬁne(see also[27–29]) (cid:107)j (cid:107) | | (cid:107) (cid:107) what is the minimum time t needed to rotate any state U = max ≤ ≤ θ , and U as the minimum (cid:107) max (cid:107) 1 j m j g by θ. The ﬁrst major result of QSL was by Mandelstam of eiγU over equivalent unitary operators with dif- max (cid:107) (cid:107) (cid:107) (cid:107) and Tamm [2]. Since then, many results on unitary dy- ferent global phases, i.e., U = min ∈R eiγU . g γ max namics followed (see e.g. [3–17]), while studies on noisy We then deﬁne (cid:26) dynamics are only carried out quite recently [18–26]. A few lower bounds have been obtained for the evolution (cid:107) (cid:107) (cid:107) (cid:107) ≤ U , if U π, time of noisy dynamics [18–20], these bounds, however, Cθ(U) = π, g if (cid:107) U (cid:107)g> π2 . (1) are in general not achievable. In this article we present a 2 g 2 new framework for QSL of noisy dynamics. While previ- Essentially C (U) represents the maximal angle that ous studies mostly focus on the rotating speed of a given θ (cid:113) U can rotate a state away from itself[29, 31], i.e., state under a dynamics, here we study the maximal speed † cos[C (U)] = min F (ρ, UρU ), where F (ρ , ρ ) is the of evolution that a dynamics can generate on all quan- θ ρ B B 1 2 1 1 tum states, which requires an optimization over all states. ﬁdelity between two states F (ρ , ρ ) = Tr ρ2 ρ ρ2 . ≥ ≥ ·B· · 1≥ 2 1 2 1 The obtained speed of evolution represents the limit of If θ = θ θ θ = θ are ar- max 1 2 m min − the quantum speed that a given dynamics can possibly ranged in d−ecreasi≤ng order, then Cθ(U) = θmax2 θmin induce on any quantum states, which is then a funda- when θ θ π[29]. A metric on unitary opera- max min † mental limit of the dynamics and can be used to provide tors can be induced by C (U) as d(U , U ) = C (U U ); θ 1 2 θ 1 2 bounds on the computation speed of a quantum device. d(U , U ) represents the maximal angle U and U 1 2 1 2 While the QSL on a ﬁxed state tells little about the abil- can generate on the same input state, cos d(U , U ) = † † 1 2 ity of a dynamics to rotating the states in general, the min F (U ρU , U ρU )[31]. ρ B 1 1 2 2 maximal speed of evolution provides a way to gauge the This metric can be generalized to noisy dynam- dynamics. ics as d(K , K ) = min d(U , U ), where 1 2 UES2 ES1 ES2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Quantum

label: title

text: speed

text: limit

text: for

text: noisy

text: dynamics

text: Haidong

label: author

text: Yuan

text: Department

text: of

text: Mechanical

text: and

text: Automation

text: Engineering,

text: The

text: Chinese

text: University

text: of

text: Hong

text: Kong,

text: Shatin,

text: Hong

text: Kong

text: Chi-Hang

text: Fred

text: Fung

text: Canada

text: Research

text: Centre,

text: Huawei

text: Technologies

text: Canada,

text: Ontario,

text: Canada

text: (Dated:

text: May

label: date

text: The

text: laws

text: of

text: quantum

text: physics

text: place

text: a

text: limit

text: on

text: the

text: speed

text: of

text: computation,

text: in

text: particular

text: the

text: evolution

text: time

text: of

text: a

text: system

text: cannot

text: be

text: arbitrarily

text: fast.

text: Bounds

text: on

text: the

text: speed

text: of

text: evolution

text: for

text: unitary

text: dynamics

text: have

text: been

text: long

text: studied.

text: A

text: few

text: bounds

text: on

text: the

text: speed

text: of

text: evolution

text: for

text: noisy

text: dynamics

text: have

text: also

text: been

text: obtained

text: recently,

text: these

text: bounds,

text: however,

text: are

text: in

text: general

text: not

text: tight.

text: In

text: this

text: article

text: we

text: present

text: a

text: new

text: framework

text: for

text: quantum

text: speed

text: limit

text: of

text: noisy

text: dynamics.

text: With

text: this

text: framework

text: we

text: obtain

text: the

text: exact

text: maximal

text: angle

text: that

text: a

text: noisy

text: dynamics

text: can

text: achieve

text: at

text: any

text: given

text: time,

text: this

text: then

text: provides

text: tight

text: bounds

text: on

text: the

text: evolution

text: time

text: for

text: noisy

text: dynamics.

text: The

text: obtained

text: bound

text: reveals

text: that

text: noisy

text: dynamics

text: are

text: generically

text: diﬀerent

text: from

text: unitary

text: dynamics,

text: in

text: particular

text: we

text: show

text: that

text: the

text: ’orthogonalization’

text: time,

text: which

text: is

text: the

text: minimum

text: time

text: needed

text: to

text: evolve

text: any

text: state

text: to

text: its

text: orthogonal

text: states,

text: is

text: in

text: general

text: not

text: applicable

text: to

text: noisy

text: dynamics.

text: Quantum

text: information

text: processing

text: may

text: be

text: regarded

text: as

text: Our

text: framework

text: is

text: based

text: on

text: a

text: method

text: that

text: gives

text: the

text: ex-

text: the

text: transformation

text: of

text: quantum

text: states

text: encoding

text: the

text: infor-

text: act

text: maximal

text: rotation

text: angle

text: for

text: a

text: given

text: dynamics,

text: which

text: mation

text: to

text: be

text: processed

text: or

text: computed.

text: The

text: time

text: for

text: which

text: ensures

text: that

text: the

text: bound

text: is

text: achievable.

text: And

text: the

text: bound

text: the

text: states

text: transform

text: dictates

text: the

text: speed

text: of

text: the

text: quantum

text: is

text: obtained

text: directly

text: from

text: the

text: Kraus

text: operators

text: of

text: the

text: dy-

text: computation.

text: Quantum

text: physics

text: imposes

text: a

text: limit

text: on

text: the

text: namics,

text: which

text: allows

text: ease

text: of

text: computation.

text: The

text: obtained

text: transformation

text: time.

text: This

text: quantum

text: speed

text: limit

text: (QSL)

text: bound

text: reveals

text: that

text: noisy

text: dynamics

text: are

text: generically

text: diﬀer-

text: arises

text: because

text: the

text: energy

text: of

text: the

text: system

text: and

text: environment

text: ent

text: from

text: unitary

text: dynamics,

text: in

text: particular

text: we

text: show

text: that

text: is

text: ﬁnite

text: and

text: the

text: system

text: state

text: may

text: evolve

text: according

text: to

text: a

text: the

text: ’orthogonalization’

text: time,

text: a

text: concept

text: heavily

text: used

text: in

text: slow

text: dynamics.

text: QSL,

text: is

text: in

text: general

text: not

text: applicable

text: to

text: noisy

text: dynamics.

text: The

text: transformation

text: of

text: quantum

text: states

text: may

text: be

text: mea-

text: Our

text: framework

text: builds

text: on

text: a

text: distance

text: measure

text: on

text: quan-

text: sured

text: by

text: the

text: Bures

text: angle

text: which

text: characterizes

text: the

text: distance

text: tum

text: channels

text: which

text: we

text: describe

text: brieﬂy.

text: For

text: a

text: m

text: m

text: between

text: two

text: states.

text: In

text: a

text: time

text: period

text: t,

text: a

text: quantum

text: pro-

text: unitary

text: matrix

text: U,

text: we

text: denote

text: by

text: e

text: iθj

text: the

text: eigenvalues

text: cess

text: may

text: rotate

text: an

text: initial

text: state

text: by

text: an

text: angle

text: θ.

text: In

text: QSL,

text: the

text: of

text: U,

text: where

text: θ

text: π,

text: π]

text: for

text: j

text: m

text: and

text: we

text: call

text: j

text: reverse

text: question

text: is

text: asked.

text: Given

text: a

text: certain

text: angle

text: θ,

text: we

text: ask

text: θ

text: the

text: eigen-angles

text: of

text: U.

text: We

text: deﬁne(see

text: what

text: is

text: the

text: minimum

text: time

text: t

text: needed

text: to

text: rotate

text: any

text: state

text: U

text: max

text: θ

text: and

text: U

text: as

text: the

text: minimum

text: max

text: j

text: m

text: j

text: g

text: by

text: θ.

text: The

text: ﬁrst

text: major

text: result

text: of

text: QSL

text: was

text: by

text: Mandelstam

text: of

text: eiγU

text: over

text: equivalent

text: unitary

text: operators

text: with

text: dif-

text: max

text: and

text: Tamm

text: Since

text: then,

text: many

text: results

text: on

text: unitary

text: dy-

text: ferent

text: global

text: phases,

text: i.e.,

text: U

text: min

text: ∈R

text: eiγU

text: g

text: γ

text: max

text: namics

text: followed

text: (see

text: e.g.

text: while

text: studies

text: on

text: noisy

text: We

text: then

text: deﬁne

text: dynamics

text: are

text: only

text: carried

text: out

text: quite

text: recently

text: A

text: few

text: lower

text: bounds

text: have

text: been

text: obtained

text: for

text: the

text: evolution

text: U

text: if

text: U

text: π,

text: time

text: of

text: noisy

text: dynamics

text: these

text: bounds,

text: however,

text: Cθ(U)

text: π,

text: g

text: if

text: U

text: π2

text: are

text: in

text: general

text: not

text: achievable.

text: In

text: this

text: article

text: we

text: present

text: a

text: g

text: new

text: framework

text: for

text: QSL

text: of

text: noisy

text: dynamics.

text: While

text: previ-

text: Essentially

text: C

text: (U)

text: represents

text: the

text: maximal

text: angle

text: that

text: ous

text: studies

text: mostly

text: focus

text: on

text: the

text: rotating

text: speed

text: of

text: a

text: given

text: θ

text: U

text: can

text: rotate

text: a

text: state

text: away

text: from

text: i.e.,

text: state

text: under

text: a

text: dynamics,

text: here

text: we

text: study

text: the

text: maximal

text: speed

text: cos[C

text: (U)]

text: min

text: F

text: (ρ,

text: UρU

text: where

text: F

text: (ρ

text: ρ

text: is

text: the

text: of

text: evolution

text: that

text: a

text: dynamics

text: can

text: generate

text: on

text: all

text: quan-

text: θ

text: ρ

text: B

text: B

text: tum

text: states,

text: which

text: requires

text: an

text: optimization

text: over

text: all

text: states.

text: ﬁdelity

text: between

text: two

text: states

text: F

text: (ρ

text: ρ

text: Tr

text: ρ2

text: ρ

text: ρ2

text: ·B·

text: 1≥

text: The

text: obtained

text: speed

text: of

text: evolution

text: represents

text: the

text: limit

text: of

text: If

text: θ

text: θ

text: θ

text: θ

text: θ

text: are

text: ar-

text: max

text: m

text: min

text: the

text: quantum

text: speed

text: that

text: a

text: given

text: dynamics

text: can

text: possibly

text: ranged

text: in

text: d−ecreasi≤ng

text: order,

text: then

text: Cθ(U)

text: θmin

text: induce

text: on

text: any

text: quantum

text: states,

text: which

text: is

text: then

text: a

text: funda-

text: when

text: θ

text: θ

text: A

text: metric

text: on

text: unitary

text: opera-

text: max

text: min

text: mental

text: limit

text: of

text: the

text: dynamics

text: and

text: can

text: be

text: used

text: to

text: provide

text: tors

text: can

text: be

text: induced

text: by

text: C

text: (U)

text: as

text: d(U

text: U

text: C

text: (U

text: U

text: θ

text: θ

text: bounds

text: on

text: the

text: computation

text: speed

text: of

text: a

text: quantum

text: device.

text: d(U

text: U

text: represents

text: the

text: maximal

text: angle

text: U

text: and

text: U

text: While

text: the

text: QSL

text: on

text: a

text: ﬁxed

text: state

text: tells

text: little

text: about

text: the

text: abil-

text: can

text: generate

text: on

text: the

text: same

text: input

text: state,

text: cos

text: d(U

text: U

text: ity

text: of

text: a

text: dynamics

text: to

text: rotating

text: the

text: states

text: in

text: general,

text: the

text: min

text: F

text: (U

text: ρU

text: U

text: ρU

text: ρ

text: B

text: maximal

text: speed

text: of

text: evolution

text: provides

text: a

text: way

text: to

text: gauge

text: the

text: This

text: metric

text: can

text: be

text: generalized

text: to

text: noisy

text: dynam-

text: dynamics.

text: ics

text: as

text: d(K

text: K

text: min

text: d(U

text: U

text: where

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11 counter-intuitive insight that the DC scenario presents better where x denotes the transmitting source which is excluded communication performance than the CC scenario in randomly from(cid:18)the aggregate(cid:19)d interf(cid:34)ere(cid:89)nce. Sin(cid:104)ce t(cid:0)he fading(cid:1) is iid(cid:105),(cid:35) deployed dense networks. Nevertheless, thanks to its higher lifetime, the CC scenario could be proved ideal for applications ∗ sbuocdhieasswinithcoasuetsewashyearecctehses,nwodheesrearleonegmebveitdydiesdminorbeuiimldpinogrtsaonrt LIr 1(cid:34)−µγ(cid:89)φγr∗αrα (cid:0)= EΦ i∈(cid:1)Φ/x E(cid:35)h e− 1−µγφ∗γr∗αrα hr−α = than high data rates. E µ VII. CONCLUSION = Φ ∈ µ + −µγ∗r∗α r−α . This paper has studied the impact of WEH on the infor- i Φ/x 1 φγ rα mation exchange in large-scale networks. The purpose of the Using the probability generating functional for the PPP Φ randomly deployed WSN nodes is to exchange successfully [27,(cid:18)4.6], we ob(cid:19)tain (cid:18) (cid:90) (cid:19) their messages locally with their neighbors, either directly ∗ ∞ ((dcoiroepcetractoivmemcoumnimcautinoincatsiocennsacreion)arioor). tThhroeudgihffearenrtelsacyennaroidoes LIr 1(cid:32)−µγφγr∗αrα (cid:0) = exp −(cid:1) r 1 +(cid:0)uα(12/π(λγϕ∗rα) − φ)udu =(cid:1)(cid:33) were compared in terms of message exchange probability, spa- (32) ttFiiaoinlnastlhlywr,oeutrhgeehpvcuaoltmidapanatderdisnoebntywooefrxktthelneifsetivwtiemo Mes.coeTnnhtaeer-iCothsaerohlorigethsiilcmiagluhltdaeetdiroivtnhase-. = exp − 2πλϕ γ∗1rα − φ −(α2/−α 22)F(cid:0)11−1r,γαα∗γα−∗2φ;(cid:1)22−−ααα2 ; rαγγ∗∗φ−1 , (33) importance of WEH in large-scale networks and revealed that the direct communication scenario presents better communi- where the integral in (32) is derived with the aid of a compu- cation performance than the cooperative scenario in randomly tational software program2 and the hypergeometric function | | deployed dense networks. However, the cooperative scenario is F (a, b; c; z) is valid for z < 1 which holds for realistic 2 1 more advisable in applications where longevity matters, since WSN scenarios. Combining (33) with (31), yields the result ∗| ≥ it is superior in terms of lifetime. to the probability Pr(γ > γ h ψ). APPENDIX A APPENDIX B PROOF OF PR(γ > γ∗|h ≥ ψ) IN THEOREM 1 PROOF OF LEMMA 1 For the special case that the path loss exponent is α = 4, In this section, we will derive the probability Pr(γ > ∗| ≥ Theorem 1 can be further simpliﬁed. Considering (8), the γ h ψ). Conditioning on the nearest transmitting source at ∗| probability Pr(γ > γ h < ψ) for α = 4 is derived in a distance r, the probability of successful message reception ∗| ≥ ≥ [29]. Regarding Pr(γ > γ h ψ), we will simplify it by given that h ψ is given by using Euler’s transformation formula for the hypergeometric (cid:90) function F [36, 15]: ∗| ≥ E ∗| ≥ 2 1 Pr(γ > γ h ψ) = [Pr(γ > γ h ψ, r)] = ∞ r = (cid:90) Pr((cid:18)γ > γ∗|h ≥ ψ, r)f ((cid:12)(cid:12)r)(cid:19)dr = 2F1(a, b; c; z) = (1 − z)c−a−b · 2F1(c − a, c − b; c; z). r (cid:12) (cid:90)0∞ (cid:18)P hr−αv(ψ) ∗ (cid:12) (cid:19) = Pr t > γ r (cid:12)f (r)dr = T(cid:18)herefore, in our case (cid:19) (cid:18) (cid:19) v(ψ)I + N (cid:12) r 0∞ r ∗ = 0 Pr h > 1 γ− φrαγI∗rrα > γ∗ r fr(r)dr, 2F1 1, 12; 32; P ψγ−∗Prt4ψγ∗N =2F1 12, 1; 32; P ψγ−∗Prt4ψγ∗N t t where f (r) denotes the probability density function (PDF) of r, givenr in [27, 2.9.1] and φ = N/(P ψ). Since h follows Moreover, by applying the hy(cid:18)pergeometric r(cid:19)epresentation of t an exponential distribution, we have (cid:90) (cid:20) (cid:18) (cid:12) (cid:19) arctan (cid:12) (cid:12) Pr(γ(cid:90) >∞γ∗|h(cid:20)≥ ψ)(cid:18)= 0∞ E∗Ir Pr h(cid:19)>(cid:12)(cid:12)(cid:12)(cid:12) (cid:21)1 γ−∗φrαγI∗rrα > γ∗ r fr(r)dr = arctazn(z) =2 F1 12, 1; 32; −z2 , = (cid:90)0∞ EIr (cid:18)exp ∗− 1 −µ(cid:19)γφγr∗αrα Ir r fr(r)dr = we obtain the result of Lemma 1. L µγ rα = 0 Ir 1 − φγ∗rα fr(r)dr, (31) APPENDIX C L where (s) deﬁnes the Laplace transform of the interfer- PROOF OF LEMMA 2 ence. WeIarim to calculate the Laplace transform by applying the(cid:18)following s(cid:19)teps: (cid:0) (cid:1) (cid:20) (cid:0) (cid:1) (cid:21) In order to derive the probability p∈CC{in (}11), we need to (cid:89) calculate the probabilities p , ϕ 1, 2 , that the relay ∗ ∗ ∗ has successfully received fromCCaRmϕ essage from a type ϕ source LIr 1 −µγφγr∗αrα = Ee− 1−µγφγr∗αrα Ir = E ∈ e− 1−µγφγr∗αrα rhα , 2Wolfram Research, Inc., Mathematica, Version 10.0, Champaign, IL, 2014. i Φ/x ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: counter-intuitive

text: insight

text: that

text: the

text: DC

text: scenario

text: presents

text: better

text: where

text: x

text: denotes

text: the

text: transmitting

text: source

text: which

text: is

text: excluded

text: communication

text: performance

text: than

text: the

text: CC

text: scenario

text: in

text: randomly

text: is

text: deployed

text: dense

text: networks.

text: Nevertheless,

text: thanks

text: to

text: its

text: higher

text: lifetime,

text: the

text: CC

text: scenario

text: could

text: be

text: proved

text: ideal

text: for

text: applications

text: LIr

text: EΦ

text: e−

text: 1−µγφ∗γr∗αrα

text: hr−α

text: than

text: high

text: data

text: rates.

text: E

text: µ

text: VII.

text: CONCLUSION

text: Φ

text: µ

text: −µγ∗r∗α

text: r−α

text: This

text: paper

text: has

text: studied

text: the

text: impact

text: of

text: WEH

text: on

text: the

text: infor-

text: i

text: Φ/x

text: φγ

text: rα

text: mation

text: exchange

text: in

text: large-scale

text: networks.

text: The

text: purpose

text: of

text: the

text: Using

text: the

text: probability

text: generating

text: functional

text: for

text: the

text: PPP

text: Φ

text: randomly

text: deployed

text: WSN

text: nodes

text: is

text: to

text: exchange

text: successfully

text: we

text: their

text: messages

text: locally

text: with

text: their

text: neighbors,

text: either

text: directly

text: tThhroeudgihffearenrtelsacyennaroidoes

text: LIr

text: exp

text: r

text: φ)udu

text: were

text: compared

text: in

text: terms

text: of

text: message

text: exchange

text: probability,

text: spa-

text: exp

text: 2πλϕ

text: γ∗1rα

text: φ

text: −(α2/−α

text: rαγγ∗∗φ−1

text: importance

text: of

text: WEH

text: in

text: large-scale

text: networks

text: and

text: revealed

text: that

text: the

text: direct

text: communication

text: scenario

text: presents

text: better

text: communi-

text: where

text: the

text: integral

text: in

text: is

text: derived

text: with

text: the

text: aid

text: of

text: a

text: compu-

text: cation

text: performance

text: than

text: the

text: cooperative

text: scenario

text: in

text: randomly

text: tational

text: software

text: and

text: the

text: hypergeometric

text: function

text: deployed

text: dense

text: networks.

text: However,

text: the

text: cooperative

text: scenario

text: is

text: F

text: (a,

text: b;

text: c;

text: z)

text: is

text: valid

text: for

text: z

text: which

text: holds

text: for

text: realistic

text: more

text: advisable

text: in

text: applications

text: where

text: longevity

text: matters,

text: since

text: WSN

text: scenarios.

text: Combining

text: with

text: yields

text: the

text: result

text: it

text: is

text: superior

text: in

text: terms

text: of

text: lifetime.

text: to

text: the

text: probability

text: Pr(γ

text: γ

text: h

text: ψ).

text: APPENDIX

text: A

text: APPENDIX

text: B

text: PROOF

text: OF

text: PR(γ

text: γ∗|h

text: ψ)

text: IN

text: THEOREM

text: PROOF

text: OF

text: LEMMA

text: For

text: the

text: special

text: case

text: that

text: the

text: path

text: loss

text: exponent

text: is

text: α

text: In

text: this

text: section,

text: we

text: will

text: derive

text: the

text: probability

text: Pr(γ

text: Theorem

text: can

text: be

text: further

text: simpliﬁed.

text: Considering

text: the

text: γ

text: h

text: ψ).

text: Conditioning

text: on

text: the

text: nearest

text: transmitting

text: source

text: at

text: probability

text: Pr(γ

text: γ

text: h

text: ψ)

text: for

text: α

text: is

text: derived

text: in

text: a

text: distance

text: r,

text: the

text: probability

text: of

text: successful

text: message

text: reception

text: Regarding

text: Pr(γ

text: γ

text: h

text: ψ),

text: we

text: will

text: simplify

text: it

text: by

text: given

text: that

text: h

text: ψ

text: is

text: given

text: by

text: using

text: Euler’s

text: transformation

text: formula

text: for

text: the

text: hypergeometric

text: function

text: F

text: E

text: Pr(γ

text: γ

text: h

text: ψ)

text: [Pr(γ

text: γ

text: h

text: ψ,

text: r)]

text: r

text: γ∗|h

text: ψ,

text: r)f

text: b;

text: c;

text: z)

text: z)c−a−b

text: a,

text: c

text: b;

text: c;

text: z).

text: r

text: hr−αv(ψ)

text: Pr

text: t

text: γ

text: r

text: (r)dr

text: in

text: our

text: case

text: v(ψ)I

text: N

text: r

text: 0∞

text: r

text: Pr

text: h

text: γ−

text: φrαγI∗rrα

text: γ∗

text: r

text: fr(r)dr,

text: P

text: ψγ−∗Prt4ψγ∗N

text: P

text: ψγ−∗Prt4ψγ∗N

text: t

text: t

text: where

text: f

text: (r)

text: denotes

text: the

text: probability

text: density

text: function

text: (PDF)

text: of

text: r,

text: givenr

text: in

text: and

text: φ

text: N/(P

text: ψ).

text: Since

text: h

text: follows

text: Moreover,

text: by

text: applying

text: the

text: of

text: t

text: an

text: exponential

text: distribution,

text: we

text: have

text: arctan

text: 0∞

text: E∗Ir

text: Pr

text: γ−∗φrαγI∗rrα

text: γ∗

text: r

text: fr(r)dr

text: arctazn(z)

text: EIr

text: Ir

text: r

text: fr(r)dr

text: we

text: obtain

text: the

text: result

text: of

text: Lemma

text: L

text: µγ

text: rα

text: Ir

text: φγ∗rα

text: fr(r)dr,

text: APPENDIX

text: C

text: L

text: where

text: (s)

text: deﬁnes

text: the

text: Laplace

text: transform

text: of

text: the

text: interfer-

text: PROOF

text: OF

text: LEMMA

text: ence.

text: WeIarim

text: to

text: calculate

text: the

text: Laplace

text: transform

text: by

text: applying

text: In

text: order

text: to

text: derive

text: the

text: probability

text: p∈CC{in

text: we

text: need

text: to

text: calculate

text: the

text: probabilities

text: p

text: ϕ

text: that

text: the

text: relay

text: has

text: successfully

text: received

text: fromCCaRmϕ

text: essage

text: from

text: a

text: type

text: ϕ

text: source

text: LIr

text: −µγφγr∗αrα

text: Ee−

text: 1−µγφγr∗αrα

text: Ir

text: E

text: e−

text: 1−µγφγr∗αrα

text: rhα

text: Research,

text: Inc.,

text: Mathematica,

text: Version

text: Champaign,

text: IL,

text: i

text: Φ/x

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 (cid:90) (cid:20) (cid:18) (cid:90) (cid:19) (cid:21) ∞ ∞ ∗ p = πλ (1(cid:90)− e−µψ(cid:20)) exp −(cid:0)πλ r 1 + γ∗2/α (cid:1)(cid:0) 1 du (cid:1)− µ(cid:0)γ N rα/2 dr(cid:1)+ (cid:21) DCϕ ϕ ϕ ∗− 1 + ua/2 ψP 0 γ 2/α t + 2πλ e−µψ ∞ exp − 2πλ 2F1 1, αα−2; 2αα−2; Ptψγ−∗Prαtψγ∗N −γ∗1rα − PNtψ −α2 Ptψγ−∗Prαtψγ∗N αα−2 − πλ r2 rdr (5) ϕ ϕ α 2 ϕ 0 end-to-end performance of the network. Consequently, in the the average harvested power and, hence, the network lifetime following, we perform an analysis on the network lifetime and becomes inﬁnite (i.e., the perpetual network operation). other end-to-end performance metrics (e.g., spatial through- In the following theorem, the average harvested power put) to identify trade-offs between the two scenarios. P¯ EH of a source node is provided, in order to complete the d derivation of the average network lifetime with EH in the DC scenario L¯ EH, given in (15). d IV. NETWORK LIFETIME One of the most important metrics for a WSN is its operat- Theorem 2. The average harvested power in one CP of a type S source node at the DC scenario, while taking into ing lifetime. In this section, the analysis for the derivation of ϕ account DPS and before the RF-to-DC conversion efﬁciency the network lifetime and the average harvested power is given is described by (cid:34) (cid:20) (cid:21)(cid:35) for all scenarios. In this way, it becomes possible to determine (cid:8) (cid:9) the gains of WEH using DPS. P¯DPSdϕ = Pte−µψ µ(παα−λϕˆ2) +ψe−µψEi[−µψ] παα−λϕ2ˆ −E rc−ϕˆα , A. Direct Communication Scenario ∈ N whereas the actual average harvested power after applying After wd 0 communication periods and without taking the RF-to-DC conv(cid:104)ersi(cid:0)on ef(cid:1)ﬁciency(cid:0)is giv(cid:1)en by (cid:0) (cid:1) (cid:105) EH into account, the average battery level of a source node in the DC scenario is deﬁned by the amount of energy E con consumed per CP and it is given by P¯ EH = P¯ a P¯ 3 + a P¯ 2 + a P¯ + a , dϕ DPSdϕ 3 log 2 log 1(cid:82) log 0 (16) − − B¯d(wd) = BI wdEcon = BI wdts(Pr + Pt), (12) (cid:8)wher(cid:9)e P¯ = 10 log P¯DPSdϕ , Ei[x] = − −∞ e−t dt for log 10 1mW x t nonzero values of x denotes the exponential integral and E − where B is the initial energy level, t is the duration of a r α denotes the expected value of the path loss to the I s cϕˆ time slot, Pr is the power consumption at the reception mode, nearest type Sϕˆ transmitter for different path loss exponent and P is the power consumption at the transmission mode. values α > 2, given within the proof. t In the case that the source nodes have EH capabilities, their battery level is increased in each CP by the average harvested power per CP denoted as P¯ EH. Thus, Proof. The proof of Theorem 2 is provided in Appendix D. d − B¯ EH(wEH) = B wEHt (P +P )+wEHt P¯ EH. (13) d d I d s r t d s d Remark 3. At this point, it should be mentioned that the The roots of (12) and (13) (i.e., the values of w and average lifetim(cid:14)e L¯dEH is limited by the set of sources with d the least average harvested power. Thus, it holds that wEH that the battery is discharged) provide the source node’s d average lifetime in CPs L¯d and L¯dEH, respectively L¯dEH = BI [ts(Pr + Pt − min{P¯d1EH, P¯d2EH})]+. (17) B L¯ = I (14) d t (P + P ) This happens because when a set of sources consumes all s r t of its energy, then we assume that the system has reached its and lifetime. B L¯dEH = [t (P + P −I P¯ EH)] , (15) s r t d + B. Cooperative Communication Scenario where [ξ] = max(ξ, 0). + In the cooperative communication scenario, a set of relay Remark 2. In the extreme case that the denominator of (15) nodes assists the source nodes to exchange their messages. is equal to zero, the consumed power is lower or equal than Therefore, without taking EH into account, the battery level ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: p

text: πλ

text: exp

text: r

text: γ∗2/α

text: du

text: N

text: DCϕ

text: ϕ

text: ϕ

text: ψP

text: γ

text: t

text: 2πλ

text: e−µψ

text: exp

text: 2πλ

text: αα−2;

text: 2αα−2;

text: Ptψγ−∗Prαtψγ∗N

text: −γ∗1rα

text: PNtψ

text: −α2

text: Ptψγ−∗Prαtψγ∗N

text: αα−2

text: πλ

text: rdr

text: ϕ

text: ϕ

text: α

text: ϕ

text: end-to-end

text: performance

text: of

text: the

text: network.

text: Consequently,

text: in

text: the

text: the

text: average

text: harvested

text: power

text: and,

text: hence,

text: the

text: network

text: lifetime

text: following,

text: we

text: perform

text: an

text: analysis

text: on

text: the

text: network

text: lifetime

text: and

text: becomes

text: inﬁnite

text: (i.e.,

text: the

text: perpetual

text: network

text: operation).

text: other

text: end-to-end

text: performance

text: metrics

text: (e.g.,

text: spatial

text: through-

text: In

text: the

text: following

text: theorem,

text: the

text: average

text: harvested

text: power

text: put)

text: to

text: identify

text: trade-offs

text: between

text: the

text: two

text: scenarios.

text: P¯

text: EH

text: of

text: a

text: source

text: node

text: is

text: provided,

text: in

text: order

text: to

text: complete

text: the

text: d

text: derivation

text: of

text: the

text: average

text: network

text: lifetime

text: with

text: EH

text: in

text: the

text: DC

text: scenario

text: L¯

text: EH,

text: given

text: in

text: d

text: IV.

text: NETWORK

text: LIFETIME

text: One

text: of

text: the

text: most

text: important

text: metrics

text: for

text: a

text: WSN

text: is

text: its

text: operat-

text: Theorem

text: The

text: average

text: harvested

text: power

text: in

text: one

text: CP

text: of

text: a

text: type

text: S

text: source

text: node

text: at

text: the

text: DC

text: scenario,

text: while

text: taking

text: into

text: ing

text: lifetime.

text: In

text: this

text: section,

text: the

text: analysis

text: for

text: the

text: derivation

text: of

text: ϕ

text: account

text: DPS

text: and

text: before

text: the

text: RF-to-DC

text: conversion

text: efﬁciency

text: the

text: network

text: lifetime

text: and

text: the

text: average

text: harvested

text: power

text: is

text: given

text: is

text: described

text: by

text: for

text: all

text: scenarios.

text: In

text: this

text: way,

text: it

text: becomes

text: possible

text: to

text: determine

text: the

text: gains

text: of

text: WEH

text: using

text: DPS.

text: P¯DPSdϕ

text: Pte−µψ

text: µ(παα−λϕˆ2)

text: +ψe−µψEi[−µψ]

text: παα−λϕ2ˆ

text: −E

text: rc−ϕˆα

text: A.

text: Direct

text: Communication

text: Scenario

text: N

text: whereas

text: the

text: actual

text: average

text: harvested

text: power

text: after

text: applying

text: After

text: wd

text: communication

text: periods

text: and

text: without

text: taking

text: the

text: RF-to-DC

text: by

text: EH

text: into

text: account,

text: the

text: average

text: battery

text: level

text: of

text: a

text: source

text: node

text: in

text: the

text: DC

text: scenario

text: is

text: deﬁned

text: by

text: the

text: amount

text: of

text: energy

text: E

text: con

text: consumed

text: per

text: CP

text: and

text: it

text: is

text: given

text: by

text: P¯

text: EH

text: P¯

text: a

text: P¯

text: a

text: P¯

text: a

text: P¯

text: a

text: dϕ

text: DPSdϕ

text: log

text: log

text: log

text: B¯d(wd)

text: BI

text: wdEcon

text: BI

text: wdts(Pr

text: Pt),

text: P¯

text: log

text: P¯DPSdϕ

text: Ei[x]

text: e−t

text: dt

text: for

text: log

text: x

text: t

text: nonzero

text: values

text: of

text: x

text: denotes

text: the

text: exponential

text: integral

text: and

text: E

text: where

text: B

text: is

text: the

text: initial

text: energy

text: level,

text: t

text: is

text: the

text: duration

text: of

text: a

text: r

text: α

text: denotes

text: the

text: expected

text: value

text: of

text: the

text: path

text: loss

text: to

text: the

text: I

text: s

text: cϕˆ

text: time

text: slot,

text: Pr

text: is

text: the

text: power

text: consumption

text: at

text: the

text: reception

text: mode,

text: nearest

text: type

text: Sϕˆ

text: transmitter

text: for

text: different

text: path

text: loss

text: exponent

text: and

text: P

text: is

text: the

text: power

text: consumption

text: at

text: the

text: transmission

text: mode.

text: values

text: α

text: given

text: within

text: the

text: proof.

text: t

text: In

text: the

text: case

text: that

text: the

text: source

text: nodes

text: have

text: EH

text: capabilities,

text: their

text: battery

text: level

text: is

text: increased

text: in

text: each

text: CP

text: by

text: the

text: average

text: harvested

text: power

text: per

text: CP

text: denoted

text: as

text: P¯

text: EH.

text: Thus,

text: Proof.

text: The

text: proof

text: of

text: Theorem

text: is

text: provided

text: in

text: Appendix

text: D.

text: d

text: B¯

text: EH(wEH)

text: B

text: wEHt

text: (P

text: +P

text: )+wEHt

text: P¯

text: EH.

text: d

text: d

text: I

text: d

text: s

text: r

text: t

text: d

text: s

text: d

text: Remark

text: At

text: this

text: point,

text: it

text: should

text: be

text: mentioned

text: that

text: the

text: The

text: roots

text: of

text: and

text: (i.e.,

text: the

text: values

text: of

text: w

text: and

text: average

text: L¯dEH

text: is

text: limited

text: by

text: the

text: set

text: of

text: sources

text: with

text: d

text: the

text: least

text: average

text: harvested

text: power.

text: Thus,

text: it

text: holds

text: that

text: wEH

text: that

text: the

text: battery

text: is

text: discharged)

text: provide

text: the

text: source

text: node’s

text: d

text: average

text: lifetime

text: in

text: CPs

text: L¯d

text: and

text: L¯dEH,

text: respectively

text: L¯dEH

text: BI

text: [ts(Pr

text: Pt

text: B

text: L¯

text: I

text: d

text: t

text: (P

text: P

text: This

text: happens

text: because

text: when

text: a

text: set

text: of

text: sources

text: consumes

text: all

text: s

text: r

text: t

text: of

text: its

text: energy,

text: then

text: we

text: assume

text: that

text: the

text: system

text: has

text: reached

text: its

text: and

text: lifetime.

text: B

text: L¯dEH

text: [t

text: (P

text: P

text: −I

text: P¯

text: EH)]

text: s

text: r

text: t

text: d

text: B.

text: Cooperative

text: Communication

text: Scenario

text: where

text: [ξ]

text: max(ξ,

text: In

text: the

text: cooperative

text: communication

text: scenario,

text: a

text: set

text: of

text: relay

text: Remark

text: In

text: the

text: extreme

text: case

text: that

text: the

text: denominator

text: of

text: nodes

text: assists

text: the

text: source

text: nodes

text: to

text: exchange

text: their

text: messages.

text: is

text: equal

text: to

text: zero,

text: the

text: consumed

text: power

text: is

text: lower

text: or

text: equal

text: than

text: Therefore,

text: without

text: taking

text: EH

text: into

text: account,

text: the

text: battery

text: level

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: consider the case of b < 1, since almost every real number with respect to Lebesgue measure has both irrationality exponent equal to 2 and eﬀective Hausdorﬀ dimension equal to 1. G Deﬁnition 11 (Family of fractals ((cid:126)q, m(cid:126) )). Let m(cid:126) be an increasing sequence of positive integers such that for all k, m divides m − ; let (cid:126)q be a sequence of integers; and let (cid:126)h be a k k 1∈ sequence of integers such that for each k, h [0, q ). k k • G Let 1((cid:126)h, (cid:126)q, m(cid:126) ) be [0, 1]. (cid:20) (cid:21) • G G Given − ((cid:126)h, (cid:126)q, m(cid:126) ), let ((cid:126)h, (cid:126)q, m(cid:126) ) be the collection of intervals which are completely k 1 k G r r 1 contained in intervals from − ((cid:126)h, (cid:126)q, m(cid:126) ) and which are of the form , + k 1 m m m ≡ k k k such that r h mod q . As usual, discard a negligible number of intervals so that each G k k G interval in − ((cid:126)h, (cid:126)q, m(cid:126) ) has the same number of subintervals in ((cid:126)h, (cid:126)q, m(cid:126) ). Further, k 1 k ensure that this number of subintervals is independent of (cid:126)h. G Let ((cid:126)q, m(cid:126) ) be the family of fractals obtained by considering all possible sequences (cid:126)h. In the following Lemma 12 is parallel to Lemmas 6 and 9. Likewise, the next Lemma 13 is parallel to Lemmas 7 and 10. In fact, the estimates here are simpler than the earlier ones. We leave the proofs for the interested reader. Lemma 12. Suppose (cid:126)b is strictly increasing sequence of real numbers with limit b such that 0 < b < 1. There is a function f, computable from (cid:126)b, such that for any sequence ∈ N ≥ m(cid:126) = (m : k ) for which for all k, m f(k, m ), there is a sequence (cid:126)q, such that k∈ G k+1 k for all E ((cid:126)q, m(cid:126) ) and for µ the uniform measure on E, the following conditions hold. − • | | ≤ q − 1 For all k greater than 2 and all intervals I such that I k 1 , m − k 1 | | µ(I) < I bk. • For all integers k, − ≤ 1 ≤ − 2b 1 2b 1 2m k+1 m k+2 . k q k k Further, we can compute q from (b , . . . , b ) and (m , . . . , m ). k 1 k+2 1 k Lemma 13. Suppose (cid:126)b is strictly increasing sequence of real numbers with limit b such that ≥ 0 < b < 1. Let f be as in Lemma 12, m(cid:126) be such that for all k, m f(k, m ), and (cid:126)q G k+1 k be deﬁned from these sequences as in Lemma 12. For every E in ((cid:126)q, m(cid:126) ), G has Hausdorﬀ dimension b. 2 Hausdorﬀ Dimension and Irrationality Exponent In this section, we will put together the pieces and prove Theorem 1. In the proof we will use the following deﬁnition. ∗ Deﬁnition 14. For positive integers d, d , d , a , we let 1 2 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: consider

text: the

text: case

text: of

text: b

text: since

text: almost

text: every

text: real

text: number

text: with

text: respect

text: to

text: Lebesgue

text: measure

text: has

text: both

text: irrationality

text: exponent

text: equal

text: to

text: and

text: eﬀective

text: Hausdorﬀ

text: dimension

text: equal

text: to

text: G

text: Deﬁnition

text: (Family

text: of

text: fractals

text: Let

text: be

text: an

text: increasing

text: sequence

text: of

text: positive

text: integers

text: such

text: that

text: for

text: all

text: k,

text: m

text: divides

text: m

text: let

text: be

text: a

text: sequence

text: of

text: integers;

text: and

text: let

text: be

text: a

text: k

text: k

text: 1∈

text: sequence

text: of

text: integers

text: such

text: that

text: for

text: each

text: k,

text: h

text: q

text: k

text: k

label: list

text: G

text: Let

text: be

text: G

text: G

text: Given

text: let

text: be

text: the

text: collection

text: of

text: intervals

text: which

text: are

text: completely

text: k

text: k

text: G

text: r

text: r

text: contained

text: in

text: intervals

text: from

text: and

text: which

text: are

text: of

text: the

text: form

text: k

text: m

text: m

text: m

text: k

text: k

text: k

text: such

text: that

text: r

text: h

text: mod

text: q

text: As

text: usual,

text: discard

text: a

text: negligible

text: number

text: of

text: intervals

text: so

text: that

text: each

text: G

text: k

text: k

text: G

text: interval

text: in

text: has

text: the

text: same

text: number

text: of

text: subintervals

text: in

text: Further,

text: k

text: k

text: ensure

text: that

text: this

text: number

text: of

text: subintervals

text: is

text: independent

text: of

text: G

text: Let

text: be

text: the

text: family

text: of

text: fractals

text: obtained

text: by

text: considering

text: all

text: possible

text: sequences

text: In

text: the

text: following

text: Lemma

text: is

text: parallel

text: to

text: Lemmas

text: and

text: Likewise,

text: the

text: next

text: Lemma

text: is

text: parallel

text: to

text: Lemmas

text: and

text: In

text: fact,

text: the

text: estimates

text: here

text: are

text: simpler

text: than

text: the

text: earlier

text: ones.

text: We

text: leave

text: the

text: proofs

text: for

text: the

text: interested

text: reader.

text: Lemma

text: Suppose

text: is

text: strictly

text: increasing

text: sequence

text: of

text: real

text: numbers

text: with

text: limit

text: b

text: such

text: that

text: b

text: There

text: is

text: a

text: function

text: f,

text: computable

text: from

text: such

text: that

text: for

text: any

text: sequence

text: N

text: (m

text: k

text: for

text: which

text: for

text: all

text: k,

text: m

text: f(k,

text: m

text: there

text: is

text: a

text: sequence

text: such

text: that

text: k∈

text: G

text: k

text: for

text: all

text: E

text: and

text: for

text: µ

text: the

text: uniform

text: measure

text: on

text: E,

text: the

text: following

text: conditions

text: hold.

text: q

text: For

text: all

text: k

text: greater

text: than

text: and

text: all

text: intervals

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text: µ(I)

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text: For

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text: can

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text: q

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text: (m

text: m

text: k

text: k

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text: is

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text: sequence

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text: numbers

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text: dimension

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text: Hausdorﬀ

text: Dimension

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text: Irrationality

text: Exponent

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text: this

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