text: Purity and entropy evolution speed limits for open quantum systems Raam Uzdin1, Eric Lutz2, and Ronnie Kosloﬀ1 1Fritz Haber Research Center for Molecular Dynamics, Hebrew University of Jerusalem, Jerusalem 91904, Israel and 2Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany We derive generic upper bounds on the rate of purity change and entropy increase for open quantum systems. These bounds depend solely on the generators of the nonunitary dynamics and are independent of the particular states of the systems. They are thus perfectly suited to investigate dephasing and thermalization processes of arbitrary systems. We apply these results to single and multiple dephasing channels, to a problem of quantum control in the presence of noise, and to cooling. PACS numbers: 03.65.-w, 03.65.Yz A Determining the maximal rate of evolution of an open tion of both the state and the generator of the quan- system is of crucial importance in quantum physics. Any tum evolution – the generator of unitary evolution is the quantum system unavoidably couples to external degrees Hamiltonian H of a closed system, while the generator of of freedom (the environment) that lead to loss of phase nonunitary Markovian evolution of an open system may coherence and/or to thermalization [1]. In most applica- be given by a Lindblad generator L [1]. The dependence tions, the main challenge is to minimize the eﬀect of the of the bound on the system state ρ is especially useful environment. In quantum computing [2] and coherent when the evolution speed of diﬀerent states are to be control [3], for example, it is vital to achieve low dephas- compared, as for example in quantum metrology when ing rates in order to protect the system against deco- the optimal state for phase estimation is needed [18, 19]. herence. On the other hand, there are many instances Here we consider another class of quantum speed lim- where it is of advantage to maximize the inﬂuence of the | | ≤ A surroundings. A case in point is the cooling of a quan- its of the form d G(ρ) f( ), where the function f t A depends solely on the generator . Since these bounds tum system and the preparation of pure states, where high cooling rates are sought after [4]. Additionally, in do not refer to any particular state ρ, they present a quantum thermodynamics, a bound on the rate of ther- clear separation between the geometric part G(ρ) that describes the state of the system and the kinetic part malization of a heat engine with its reservoirs will limit A its cycle time and therefore put a restriction on maximal f( ) that is controlled by the generator. These bounds are particularly well suited to investigate the impact of an power output [5]. external environment on generic quantum systems whose Bounds on the rate of quantum evolution are useful to states either not known in detail, or are too complex to assess if a process can be completed in a given time, with- be determined exactly, e.g. those of an interacting many- out having to explicitly solve the (usually complicated) body system. Bounds of this type have been studied for equations of motion [6–11]. Quantum speed limits, de- Hermitian [23] and non-Hermitian [24] Hamiltonians that ﬁned as the time derivative of a geometric quantity, for in- often appear in systems with absorption as in optics. Our stance an angle between two states, are often introduced aim in this paper is to provide state-independent speed P to characterize the maximal rate of evolution of a quan- limits for the purity = trρ2 and the von Neumann en- − tum system. Two prominent examples of quantum speed tropy S = trρ ln ρ of a generic open quantum system, limits for closed systems are the Mandelstam-Tamm two quantities that are commonly used to quantify the bound, |d θ| ≤ ∆E /(cid:126) [12], and the Margolus-Levitin action of the environment [1]. In the following, we ﬁrst bound, |dtθ| ≤ (cid:104)E(cid:105)ψ/(cid:126) [13]. Here ∆E is the energy illustrate our method in the usual Hilbert space of den- t ψ (cid:104) (cid:105)ψ width of the initial pure state ψ(0), E its mean en- sity matrices by deriving a bound for the purity speed in ψ |(cid:104) | (cid:105)| ergy above the ground state, and θ = arccos ψ(0) ψ(t) terms of the Hilbert-Schmidt norm of the Lindblad op- the geometric angle between initial and ﬁnal states (we erators of the nonunitary dynamics. We then extend our will set (cid:126) = 1 in the sequel). In the last years, the approach to Liouville space (space of density "vectors" Mandelstam-Tamm and Margolus-Levitin bounds have as described later on) and obtain a tighter inequality in been extended to mixed [14] and driven [15–17] closed terms of the spectral norm of the corresponding Hamil- quantum systems as well as to open quantum systems ton superoperator. We further considerably improve the [18–20]. These speed limits depend explicitly on the state bound by introducing a purity deviation that is obtained ρ of the system and can be written in the general form by subtracting the steady-state dynamics of the time- | | ≤ A d G(ρ) f(ρ, ), where G(ρ) is a geometric quantity dependent system. Remarkably, we show that this bound t that characterizes the state of the system and f is a func- is tight, for all states and at all times, for a single qubit

text: Purity

label: title

text: and

text: entropy

text: evolution

text: speed

text: limits

text: for

text: open

text: quantum

text: systems

text: Raam

label: author

text: Eric

text: and

text: Ronnie

text: Haber

text: Research

text: Center

text: for

text: Molecular

text: Dynamics,

text: Hebrew

text: University

text: of

text: Jerusalem,

text: Jerusalem

text: Israel

text: and

text: of

text: Physics,

text: Friedrich-Alexander-Universität

text: Erlangen-Nürnberg,

text: Erlangen,

text: Germany

text: We

label: abstract

text: derive

text: generic

text: upper

text: bounds

text: on

text: the

text: rate

text: of

text: purity

text: change

text: and

text: entropy

text: increase

text: for

text: open

text: quantum

text: systems.

text: These

text: bounds

text: depend

text: solely

text: on

text: the

text: generators

text: of

text: the

text: nonunitary

text: dynamics

text: and

text: are

text: independent

text: of

text: the

text: particular

text: states

text: of

text: the

text: systems.

text: They

text: are

text: thus

text: perfectly

text: suited

text: to

text: investigate

text: dephasing

text: and

text: thermalization

text: processes

text: of

text: arbitrary

text: systems.

text: We

text: apply

text: these

text: results

text: to

text: single

text: and

text: multiple

text: dephasing

text: channels,

text: to

text: a

text: problem

text: of

text: quantum

text: control

text: in

text: the

text: presence

text: of

text: noise,

text: and

text: to

text: cooling.

text: PACS

text: numbers:

text: A

text: Determining

text: the

text: maximal

text: rate

text: of

text: evolution

text: of

text: an

text: open

text: tion

text: of

text: both

text: the

text: state

text: and

text: the

text: generator

text: of

text: the

text: quan-

text: system

text: is

text: of

text: crucial

text: importance

text: in

text: quantum

text: physics.

text: Any

text: tum

text: evolution

text: the

text: generator

text: of

text: unitary

text: evolution

text: is

text: the

text: quantum

text: system

text: unavoidably

text: couples

text: to

text: external

text: degrees

text: Hamiltonian

text: H

text: of

text: a

text: closed

text: system,

text: while

text: the

text: generator

text: of

text: of

text: freedom

text: (the

text: environment)

text: that

text: lead

text: to

text: loss

text: of

text: phase

text: nonunitary

text: Markovian

text: evolution

text: of

text: an

text: open

text: system

text: may

text: coherence

text: and/or

text: to

text: thermalization

text: In

text: most

text: applica-

text: be

text: given

text: by

text: a

text: Lindblad

text: generator

text: L

text: The

text: dependence

text: tions,

text: the

text: main

text: challenge

text: is

text: to

text: minimize

text: the

text: eﬀect

text: of

text: the

text: of

text: the

text: bound

text: on

text: the

text: system

text: state

text: ρ

text: is

text: especially

text: useful

text: environment.

text: In

text: quantum

text: computing

text: and

text: coherent

text: when

text: the

text: evolution

text: speed

text: of

text: diﬀerent

text: states

text: are

text: to

text: be

text: control

text: for

text: example,

text: it

text: is

text: vital

text: to

text: achieve

text: low

text: dephas-

text: compared,

text: as

text: for

text: example

text: in

text: quantum

text: metrology

text: when

text: ing

text: rates

text: in

text: order

text: to

text: protect

text: the

text: system

text: against

text: deco-

text: the

text: optimal

text: state

text: for

text: phase

text: estimation

text: is

text: needed

text: herence.

text: On

text: the

text: other

text: hand,

text: there

text: are

text: many

text: instances

text: Here

text: we

text: consider

text: another

text: class

text: of

text: quantum

text: speed

text: lim-

text: where

text: it

text: is

text: of

text: advantage

text: to

text: maximize

text: the

text: inﬂuence

text: of

text: the

text: A

text: surroundings.

text: A

text: case

text: in

text: point

text: is

text: the

text: cooling

text: of

text: a

text: quan-

text: its

text: of

text: the

text: form

text: d

text: G(ρ)

text: f(

text: where

text: the

text: function

text: f

text: t

text: A

text: depends

text: solely

text: on

text: the

text: generator

text: Since

text: these

text: bounds

text: tum

text: system

text: and

text: the

text: preparation

text: of

text: pure

text: states,

text: where

text: high

text: cooling

text: rates

text: are

text: sought

text: after

text: Additionally,

text: in

text: do

text: not

text: refer

text: to

text: any

text: particular

text: state

text: ρ,

text: they

text: present

text: a

text: quantum

text: thermodynamics,

text: a

text: bound

text: on

text: the

text: rate

text: of

text: ther-

text: clear

text: separation

text: between

text: the

text: geometric

text: part

text: G(ρ)

text: that

text: describes

text: the

text: state

text: of

text: the

text: system

text: and

text: the

text: kinetic

text: part

text: malization

text: of

text: a

text: heat

text: engine

text: with

text: its

text: reservoirs

text: will

text: limit

text: A

text: its

text: cycle

text: time

text: and

text: therefore

text: put

text: a

text: restriction

text: on

text: maximal

text: f(

text: that

text: is

text: controlled

text: by

text: the

text: generator.

text: These

text: bounds

text: are

text: particularly

text: well

text: suited

text: to

text: investigate

text: the

text: impact

text: of

text: an

text: power

text: output

text: external

text: environment

text: on

text: generic

text: quantum

text: systems

text: whose

text: Bounds

text: on

text: the

text: rate

text: of

text: quantum

text: evolution

text: are

text: useful

text: to

text: states

text: either

text: not

text: known

text: in

text: detail,

text: or

text: are

text: too

text: complex

text: to

text: assess

text: if

text: a

text: process

text: can

text: be

text: completed

text: in

text: a

text: given

text: time,

text: with-

text: be

text: determined

text: exactly,

text: e.g.

text: those

text: of

text: an

text: interacting

text: many-

text: out

text: having

text: to

text: explicitly

text: solve

text: the

text: (usually

text: complicated)

text: body

text: system.

text: Bounds

text: of

text: this

text: type

text: have

text: been

text: studied

text: for

text: equations

text: of

text: motion

text: Quantum

text: speed

text: limits,

text: de-

text: Hermitian

text: and

text: non-Hermitian

text: Hamiltonians

text: that

text: ﬁned

text: as

text: the

text: time

text: derivative

text: of

text: a

text: geometric

text: quantity,

text: for

text: in-

text: often

text: appear

text: in

text: systems

text: with

text: absorption

text: as

text: in

text: optics.

text: Our

text: stance

text: an

text: angle

text: between

text: two

text: states,

text: are

text: often

text: introduced

text: aim

text: in

text: this

text: paper

text: is

text: to

text: provide

text: state-independent

text: speed

text: P

text: to

text: characterize

text: the

text: maximal

text: rate

text: of

text: evolution

text: of

text: a

text: quan-

text: limits

text: for

text: the

text: purity

text: and

text: the

text: von

text: Neumann

text: en-

text: tum

text: system.

text: Two

text: prominent

text: examples

text: of

text: quantum

text: speed

text: tropy

text: S

text: trρ

text: ln

text: ρ

text: of

text: a

text: generic

text: open

text: quantum

text: system,

text: limits

text: for

text: closed

text: systems

text: are

text: the

text: Mandelstam-Tamm

text: two

text: quantities

text: that

text: are

text: commonly

text: used

text: to

text: quantify

text: the

text: bound,

text: |d

text: θ|

text: ∆E

text: and

text: the

text: Margolus-Levitin

text: action

text: of

text: the

text: environment

text: In

text: the

text: following,

text: we

text: ﬁrst

text: bound,

text: |dtθ|

text: Here

text: ∆E

text: is

text: the

text: energy

text: illustrate

text: our

text: method

text: in

text: the

text: usual

text: Hilbert

text: space

text: of

text: den-

text: t

text: ψ

text: width

text: of

text: the

text: initial

text: pure

text: state

text: E

text: its

text: mean

text: en-

text: sity

text: matrices

text: by

text: deriving

text: a

text: bound

text: for

text: the

text: purity

text: speed

text: in

text: ψ

text: ergy

text: above

text: the

text: ground

text: state,

text: and

text: θ

text: arccos

text: ψ(t)

text: terms

text: of

text: the

text: Hilbert-Schmidt

text: norm

text: of

text: the

text: Lindblad

text: op-

text: the

text: geometric

text: angle

text: between

text: initial

text: and

text: ﬁnal

text: states

text: (we

text: erators

text: of

text: the

text: nonunitary

text: dynamics.

text: We

text: then

text: extend

text: our

text: will

text: set

text: in

text: the

text: sequel).

text: In

text: the

text: last

text: years,

text: the

text: approach

text: to

text: Liouville

text: space

text: (space

text: of

text: density

text: "vectors"

text: Mandelstam-Tamm

text: and

text: Margolus-Levitin

text: bounds

text: have

text: as

text: described

text: later

text: on)

text: and

text: obtain

text: a

text: tighter

text: inequality

text: in

text: been

text: extended

text: to

text: mixed

text: and

text: driven

text: closed

text: terms

text: of

text: the

text: spectral

text: norm

text: of

text: the

text: corresponding

text: Hamil-

text: quantum

text: systems

text: as

text: well

text: as

text: to

text: open

text: quantum

text: systems

text: ton

text: superoperator.

text: We

text: further

text: considerably

text: improve

text: the

text: These

text: speed

text: limits

text: depend

text: explicitly

text: on

text: the

text: state

text: bound

text: by

text: introducing

text: a

text: purity

text: deviation

text: that

text: is

text: obtained

text: ρ

text: of

text: the

text: system

text: and

text: can

text: be

text: written

text: in

text: the

text: general

text: form

text: by

text: subtracting

text: the

text: steady-state

text: dynamics

text: of

text: the

text: time-

text: A

text: d

text: G(ρ)

text: f(ρ,

text: where

text: G(ρ)

text: is

text: a

text: geometric

text: quantity

text: dependent

text: system.

text: Remarkably,

text: we

text: show

text: that

text: this

text: bound

text: t

text: that

text: characterizes

text: the

text: state

text: of

text: the

text: system

text: and

text: f

text: is

text: a

text: func-

text: is

text: tight,

text: for

text: all

text: states

text: and

text: at

text: all

text: times,

text: for

text: a

text: single

text: qubit


text: 2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:88) dephasing channel. We ﬁnally derive a speed limit for (cid:12) (cid:12) ˆ (cid:12) (cid:12) the rate of entropy decrease, and apply our formalism to P quantum control in the presence of noise and to cooling. ln P(tf) ≤ 4 tf (cid:107)A (cid:107)2 dt. (4) Purity bound in Hilbert space. We begin by deriving (ti) ti k k 2 an upper bound to the purity speed limit in the density − P matrix formulation. It will be convenient to consider an The quantity ln is known as the Rényi entropy of integral bound of the type, order two or the collision entropy [28]. We note that the ˆ unitary evolution term, i[ρ, H ], can be omitted, since the ρ norm is unitarily invariant. Equation (4) provides an up- |G(ρ ) − G(ρ )| ≤ tf f(A )dt, (1) per bound to the speed of variation of the Rényi entropy f i t of order two of the system in terms of the Hilbert-Schmidt ti ´ ´ | | ≤ norm of the Lindblad operators. Its practical usefulness Awhich follows from the di|ﬀerential bo|u≤nd, dt|G(ρ) | stems from the fact that the operators Ak can be deter- f( ), with the inequality tf d G(ρ) tf d G(ρ) . mined via quantum process tomography [29]. This tech- Thist form is often more appeatliingtas it relatetsithetinitial nique has been successfully demonstrated experimentally and ﬁnal states of interest without reference to the in- in NMR systems [30], solid state qubits [31], vibrational termediate dynamics. The right-hand side will typically states of atoms in optical lattices [32], quantum gate op- have the form of an "action" integral [25]. erations [33–35] and quantum memory [36], as well as We consider a possibly driven N-level quantum sys- to relaxing photon ﬁelds inside cavities [37] and super- tem with Hamiltonian H ∈ CN×N. We describe the conducting quantum circuits [38]. It is worth noticing ρ nonunitary time evolution of its density matrix ρ by a that the purity has been measured directly in some cases, Markovian master equatio(cid:88)n of the Lindblad-type [1], without having recourse to full quantum state tomogra- phy [39–41]. However, it is notoriously diﬃcult to mea- sure nonlinear functions of the density operator [42] and † − {1 † } dtρ = Lt(ρ) = i[Hρ, ρ] + AkρAk 2AkAk, ρ , (2) Eq. (4) allows to estimate the rate of change of the Rényi k entropy from the more accessible Lindblad operators. where the operators A ∈ CN×N describe the inter- Purity bounds in Liouville space. Quantum dynamics k is traditionally described in Hilbert space. However, it action with the external environment (L is the time de- t is sometimes convenient, in particular for open quantum pendent Lindblad generator of the nonunitary dynamics). systems, to introduce an extended space where density Master equations of the form (2) are the tool of choice operators are simple vectors and time evolution is gen- to investigate the dynamics of systems weakly coupled to erated by superoperators. This space is usually referred a reservoir in quantum optics and solid state physics [1]. to as Liouville space [43]. We denote the "density vec- ApucrciotrydfionrgatnoyaρthoenolyredmecbryeaLsiedsairf,aSnhdaboannlyi aifn[dAA,liAck†i],=th0e tor" by r ∈ C1×N2. It can be obtained by reshaping the for all k [26]. This condition provides useful meakns tko sep- den∈sit{y matrix ρ} into a larger single vector with index arate between purely dephasing processes that can only α 1, 2, ....N2 . The one-to-one mappin{g of} t→he two reduce the purity and processes that have the capability matrix indices(cid:80)into a single vector index i, j α is arbitrary, but has to be used consistently. The equation reducing the entropy and cooling the system. of motion of the density vector in Liouville space follows We use the cyclic property of the trace to write d ln tr(ρ2) = 2tr(ρL (ρ))/tr(ρ2). Integrating over time from dtρα = β ρβ∂(dtρα)/∂ρβ. Using this equation t t one can verify that the dynamics of the density vector and using th(cid:12)e triangle(cid:12) inequality, we have, (cid:12)(cid:12) (cid:12)(cid:12) ˆ r is governed by a Schrödinger-like equation in the new (cid:12) (cid:12) space, P | | (t ) ≤ 2 tr(ρL(ρ)) ln P f dt. (3) (t ) tr(ρ2) i i∂ r = H r, (5) t r P (cid:107) (cid:107) We next exploit the fact that (t) = tr(ρ2) = ρ 2, (cid:107)·(cid:107) 2 where 2 denotes the Hilbert-Schmidt norm. An up- where the Hamiltonian superoperator H ∈ CN2×N2 is per bound to Eq. (3) can be derived with the help of r given by, elementary matrix algebra [27]. Combining the Cauchy- | | ≤ (cid:107) (cid:107) (cid:107) (cid:107) Schwarz inequality, tr(ρL (ρ)) ρ L (ρ) , the tri- (cid:80) t 2 t 2 angle inequality together with the submultiplicativity ∂(d ρ ) H = i t α . (6) property of the norm and the master equation (2), we r,αβ ∂ρ (cid:107) (cid:107) (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) β ﬁnd ρ L (ρ) 2 A 2 ρ 2. Inserting this ex- 2 t 2 k k 2 2 pression into Eq. (3), we obtain a "norm action" integral The vector r is not normalized to unity in general. Its P (cid:104) | (cid:105) inequality of type (1) for the logarithm of the purity: norm is equal to the purity, = tr(ρ2) = r r , where ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

label: equation

text: dephasing

text: channel.

text: We

text: ﬁnally

text: derive

text: a

text: speed

text: limit

text: for

text: ˆ

text: the

text: rate

text: of

text: entropy

text: decrease,

text: and

text: apply

text: our

text: formalism

text: to

text: P

text: quantum

text: control

text: in

text: the

text: presence

text: of

text: noise

text: and

text: to

text: cooling.

text: ln

text: P(tf)

text: tf

text: dt.

text: Purity

text: bound

text: in

text: Hilbert

text: space.

text: We

text: begin

text: by

text: deriving

text: (ti)

text: ti

text: k

text: k

text: an

text: upper

text: bound

text: to

text: the

text: purity

text: speed

text: limit

text: in

text: the

text: density

text: P

text: matrix

text: formulation.

text: It

text: will

text: be

text: convenient

text: to

text: consider

text: an

text: The

text: quantity

text: ln

text: is

text: known

text: as

text: the

text: Rényi

text: entropy

text: of

text: integral

text: bound

text: of

text: the

text: type,

text: order

text: two

text: or

text: the

text: collision

text: entropy

text: We

text: note

text: that

text: the

text: ˆ

text: unitary

text: evolution

text: term,

text: i[ρ,

text: H

text: can

text: be

text: omitted,

text: since

text: the

text: ρ

text: norm

text: is

text: unitarily

text: invariant.

text: Equation

text: provides

text: an

text: up-

text: |G(ρ

text: G(ρ

text: tf

text: f(A

text: )dt,

text: per

text: bound

text: to

text: the

text: speed

text: of

text: variation

text: of

text: the

text: Rényi

text: entropy

text: f

text: i

text: t

text: of

text: order

text: two

text: of

text: the

text: system

text: in

text: terms

text: of

text: the

text: Hilbert-Schmidt

text: ti

text: norm

text: of

text: the

text: Lindblad

text: operators.

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text: The

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text: entropy

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text: of

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text: density

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text: use

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text: β

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text: Using

text: this

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text: t

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text: one

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text: verify

text: that

text: the

text: dynamics

text: of

text: the

text: density

text: vector

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text: using

text: inequality,

text: we

text: have,

text: ˆ

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text: by

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text: P

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text: ln

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text: f

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text: i

text: i∂

text: r

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text: fact

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text: (t)

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text: denotes

text: the

text: Hilbert-Schmidt

text: norm.

text: An

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text: where

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text: Eq.

text: can

text: be

text: derived

text: with

text: the

text: help

text: of

text: r

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text: elementary

text: matrix

text: algebra

text: Combining

text: the

text: Cauchy-

text: Schwarz

text: inequality,

text: tr(ρL

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text: ρ

text: L

text: (ρ)

text: the

text: tri-

text: t

text: t

text: angle

text: inequality

text: together

text: with

text: the

text: submultiplicativity

text: ∂(d

text: ρ

text: H

text: i

text: t

text: α

text: property

text: of

text: the

text: norm

text: and

text: the

text: master

text: equation

text: we

text: r,αβ

text: ∂ρ

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text: ﬁnd

text: ρ

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text: (ρ)

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text: k

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text: we

text: obtain

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text: "norm

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text: the

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text: r

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 [1] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, (Oxford, Oxford, 2007). [2] M. A. Nielsen and I. L. Chuang, Quantum Computa- tion and Quantum Information, (Cambridge, Cambridge, 2000). [3] M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes, (Wiley, New York, 2003). [4] X. Wang, S. Vinjanampathy, F. W. Strauch, and K. Ja- cobs, Phys. Rev. Lett. 110, 157207 (2013). [5] R. Kosloﬀ and A. Levy, Annu. Rev. Phys. Chem. 65, 365 (2014). [6] G. N. Fleming, Nuovo Cimento A 16, 232 (1973). [7] K. Bhattacharyya, J. Phys. A 16, 2993 (1983). Figure 2: Entropy bounds for a qubit decay channel with [8] J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 A = σ− = (σ − iσ )/2 in a fully mixed state at t = 0. The (1990). bound (14) (cxurve iiyi) is better than the entropy boui nds based [9] L. Vaidman, Am. J. Phys. 60, 182 (1991). on Eq. (10) in Liouville space (curve iv) and on Eq. (4) in [10] J. Uﬃnk, Am. J. Phys. 61, 935 (1993). Hilbert space (curve v). Curve i and ii respectively correspond [11] D. C. Brody, J. Phys. A 36, 5587 (2003). to the exact entropy S(t) and exact log-purity − ln P(t). [12] L. Mandelstam and I. Tamm, J. Phys. (USSR) 9, 249 (1945). [13] N. Margolus and L. B. Levitin, Physica D 120, 188 (1998). for the speed limit in Liouville space. This is no longer [14] V. Giovannetti, S. Lloyd and L. Maccone, Phys. Rev. A a norm and is valid only when purity is growing. Yet, 67, 052109 (2003). [15] P. Pfeifer, Phys. Rev. Lett. 70, 3365 (1993). when valid, it can be signiﬁcantly better than the spec- [16] P. Pfeifer and J. Fröhlich, Rev. Mod. Phys. 67, 759 tral norm bound, as shown in Fig. 2 for the simple decay − − (1995). channel characterized by A = σ = (σx iσy)/2. We [17] S. Deﬀner and E. Lutz, J. Phys. A 46, 335302 (2013). may moreover obtain a bound on the decrease of the von [18] M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. Neumann entropy with the help of the Jensen inequality, de Matos Filho, Phys. Rev. Lett. 110, 050402 (2013). − ≥ − P S = trρ ln ρ ln (t), and Eq. (10), [19] A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Phys. Rev. Lett. 110, 050403 (2013). [20] S. Deﬀner and E. Lutz, Phys. Rev. Lett. 111, 010402 ˆ (2013). ≥ − P − tf − − † [21] C. A. Rodriguez-Rosario, G. Kimura, H. Imai, and A. S(t ) ln (t ) max[ i eig(H H )]dt. (14) Aspuru-Guzik, Phys. Rev. Lett. 106, 050403 (2011). f i r r ti [22] A. Hutter and S. Wehner, Phys. Rev. Lett. 108, 070501 (2012). [23] D. A. Lidar, P. Zanardi, and K. Khodjasteh, Phys. Rev. For given initial and ﬁnal entropies (or purities), ex- A 78, 012308 (2008). pression (14) provides a bound on how fast (minimal time [24] R. Uzdin, U. Günther, S. Rahav, and N. Moiseyev, J. interval t −t ) a quantum system can be cooled in terms Phys. A 45, 415304 (2012). f i [25] R. Uzdin, J. of Phys. A 46, 145302 (2013). of the generators of the open dynamics. The appearance [26] D. Lidar, A. Shabani, and R. Alicki, Chem. Phys. 322, of the entropy instead of the purity establishes an impor- 82 (2006). tant link to thermodynamics and highlights its relevance [27] R. Bhatia, Matrix Analysis, (Springer, Berlin, 1997). for the investigation of e.g. quantum heat engines. [28] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, J. Math. Phys. 54, 122203 (2013). Conclusions. We have derived state-independent [29] I.L. Chuang and M.A. Nielsen, J. Mod. Opt. 44, 2455 quantum speed limits for the purity and the entropy of (1997). Markovian open quantum systems. We have obtained [30] A. M. Childs, I. L. Chuang, and D. W. Leung, Phys. Rev. increasingly tighter bounds by considering the Liouville A 64, 012314 (2001). space instead of the usual Hilbert space. We have ad- [31] M. Howard, J. Twamley, C. Wittmann, T. Gaebel, F. ditionally shown that these bounds can be signiﬁcantly Jelezko, and J. Wrachtrup, New J. Phys. 8, 33 (2006). improved by introducing a purity deviation obtained by [32] S. H. Myrskog, J. K. Fox, M. W. Mitchell, and A. M. Steinberg, Phys. Rev. A 72, 013615 (2005). subtraction of the steady-state contribution. We have ﬁ- [33] J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, nally emphasized the usefulness of these results for the in- N. K. Langford, T. C. Ralph, and A. G. White, Phys. vestigation of decoherence and thermalization processes Rev. Lett. 93, 080502 (2004). in general, and applied them to concrete problems of de- [34] Y. Nambu and K. Nakamura, Phys. Rev. Lett. 94, 010404 phasing, noisy coherent control and cooling. (2005). ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

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text: and

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text: its

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text: Matrix

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text: the

text: purity

text: and

text: the

text: entropy

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text: quantum

text: systems.

text: We

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text: A.

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text: tighter

text: bounds

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text: the

text: Liouville

text: A

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text: instead

text: of

text: the

text: usual

text: Hilbert

text: space.

text: We

text: have

text: ad-

text: M.

text: Howard,

text: J.

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text: Location Prediction: Communities Speak Louder than Friends Jun Pang Yang Zhang University of Luxembourg University of Luxembourg FSTC & SnT FSTC jun.pang@uni.lu yang.zhang@uni.lu ABSTRACT groups, i.e., social communities, using diﬀerent criteria and considerations. By deﬁnition, a community is a social unit Humans are social animals, they interact with diﬀerent com- of any size that shares common values.1 Typical commu- munities of friends to conduct diﬀerent activities. The lit- nities include family, close friends, colleagues, etc. In daily erature shows that human mobility is constrained by their life, humans are engaged in various social environments, and social relations. In this paper, we investigate the social im- they interact with diﬀerent communities depending on the pact of a person’s communities on his mobility, instead of environments. For our speciﬁc behaviors, social inﬂuences, all friends from his online social networks. This study can in most of cases, are not from all our friends but from certain be particularly useful, as certain social behaviors are inﬂu- communities. For example, we listen to similar types of mu- enced by speciﬁc communities but not all friends. To achieve sic as our close friends, but not as our parents; we have lunch our goal, we ﬁrst develop a measure to characterize a per- together with our colleagues on weekdays, but not with our son’s social diversity, which we term ‘community entropy’. college friends living in another city; on weekends we spend Through analysis of two real-life datasets, we demonstrate more time with family, but not with our colleagues. that a person’s mobility is inﬂuenced only by a small frac- Location-based social network services (LBSNs) have been tion of his communities and the inﬂuence depends on the booming during the past ﬁve years. Nowadays, it is com- social contexts of the communities. We then exploit ma- mon for a user to attach his location when he publishes a chine learning techniques to predict users’ future movement photo or a status using his online social network account. based on their communities’ information. Extensive experi- Moreover, users may just share their locations, called check- ments demonstrate the prediction’s eﬀectiveness. in, to tell their friends where they are or to engage in social Categories and Subject Descriptors games as in Foursquare. Since these large amount of lo- cation and social relation data become available, studying H.2.8 [Database Management]: Database Applications— human mobility and its connection with social relationships Data mining becomes quantitatively achievable (e.g., [16, 10, 9, 34, 15, 7, 8]). Understanding human mobility can lead to compelling General Terms applications including location recommendation [44, 41, 43, 14, 20], urban planning [42], immigration patterns [5], etc. Algorithms, theory, experiments Previous works, including [1, 6, 10, 33], show that human mobility is inﬂuenced by social factors. However, there is one Keywords common shortcoming: they all treat friends of users equally. Similar to other social behaviors, in most cases mobility is Human mobility, social networks, network communities inﬂuenced by speciﬁc communities but not all friends. For example, the aforementioned colleagues can inﬂuence the 1. INTRODUCTION place a user goes for lunch but probably have nothing to do Humans are social animals, everyone is a part of the so- with his weekend plans. Meanwhile, where a user visits on ciety and gets inﬂuences from it. For example, our daily weekends largely depends on his friends or family, but not his behaviors, such as what types of music we listen to, where colleagues. Therefore, the impact on a user’s mobility should we have lunch on weekdays and what activities we conduct be considered from the perspectives of communities instead on weekends, are largely dependent on our social relations. of all friends. In a broader view, community is arguably the Normally, we categorize our social relations into diﬀerent most useful resolution to study social networks [39]. Contributions. In this paper, we aim to study the im- pact from communities on a user’s mobility and predict his locations based on his community information. First, we partition each users’ friends into communities and propose a notion namely community entropy to quantify a user’s social diversity. Second, we analyze communities’ in- ﬂuences on users’ mobility and our main conclusions include: COSN’15, November 2–3, 2015, Palo Alto, California, USA. . 1http://en.wikipedia.org/wiki/Community ##LTLine##

text: Location

font: VMNXAU+NimbusSanL-Bold

text: Prediction:

text: Communities

text: Speak

text: Louder

text: than

text: Friends

text: Jun

font: MLIQBW+NimbusSanL-Regu

text: Pang

text: Yang

text: Zhang

text: University

text: of

text: Luxembourg

text: University

text: of

text: Luxembourg

text: FSTC

text: SnT

text: FSTC

text: jun.pang@uni.lu

label: footer

text: yang.zhang@uni.lu

text: ABSTRACT

text: groups,

text: i.e.,

text: social

text: communities,

text: using

text: diﬀerent

text: criteria

text: and

text: considerations.

text: By

text: deﬁnition,

text: a

text: community

text: is

text: a

text: social

text: unit

text: Humans

text: are

text: social

text: animals,

text: they

text: interact

text: with

text: diﬀerent

text: com-

text: of

text: any

text: size

text: that

text: shares

text: common

text: Typical

text: commu-

text: munities

text: of

text: friends

text: to

text: conduct

text: diﬀerent

text: activities.

text: The

text: lit-

text: nities

text: include

text: family,

text: close

text: friends,

text: colleagues,

text: etc.

text: In

text: daily

text: erature

text: shows

text: that

text: human

text: mobility

text: is

text: constrained

text: by

text: their

text: life,

text: humans

text: are

text: engaged

text: in

text: various

text: social

text: environments,

text: and

text: social

text: relations.

text: In

text: this

text: paper,

text: we

text: investigate

text: the

text: social

text: im-

text: they

text: interact

text: with

text: diﬀerent

text: communities

text: depending

text: on

text: the

text: pact

text: of

text: a

text: person’s

text: communities

text: on

text: his

text: mobility,

text: instead

text: of

text: environments.

text: For

text: our

text: speciﬁc

text: behaviors,

text: social

text: inﬂuences,

text: all

text: friends

text: from

text: his

text: online

text: social

text: networks.

text: This

text: study

text: can

text: in

text: most

text: of

text: cases,

text: are

text: not

text: from

text: all

text: our

text: friends

text: but

text: from

text: certain

text: be

text: particularly

text: useful,

text: as

text: certain

text: social

text: behaviors

text: are

text: inﬂu-

text: communities.

text: For

text: example,

text: we

text: listen

text: to

text: similar

text: types

text: of

text: mu-

text: enced

text: by

text: speciﬁc

text: communities

text: but

text: not

text: all

text: friends.

text: To

text: achieve

text: sic

text: as

text: our

text: close

text: friends,

text: but

text: not

text: as

text: our

text: parents;

text: we

text: have

text: lunch

text: our

text: goal,

text: we

text: ﬁrst

text: develop

text: a

text: measure

text: to

text: characterize

text: a

text: per-

text: together

text: with

text: our

text: colleagues

text: on

text: weekdays,

text: but

text: not

text: with

text: our

text: son’s

text: social

text: diversity,

text: which

text: we

text: term

text: ‘community

text: entropy’.

text: college

text: friends

text: living

text: in

text: another

text: city;

text: on

text: weekends

text: we

text: spend

text: Through

text: analysis

text: of

text: two

text: real-life

text: datasets,

text: we

text: demonstrate

text: more

text: time

text: with

text: family,

text: but

text: not

text: with

text: our

text: colleagues.

text: that

text: a

text: person’s

text: mobility

text: is

text: inﬂuenced

text: only

text: by

text: a

text: small

text: frac-

text: Location-based

text: social

text: network

text: services

text: (LBSNs)

text: have

text: been

text: tion

text: of

text: his

text: communities

text: and

text: the

text: inﬂuence

text: depends

text: on

text: the

text: booming

text: during

text: the

text: past

text: ﬁve

text: years.

text: Nowadays,

text: it

text: is

text: com-

text: social

text: contexts

text: of

text: the

text: communities.

text: We

text: then

text: exploit

text: ma-

text: mon

text: for

text: a

text: user

text: to

text: attach

text: his

text: location

text: when

text: he

text: publishes

text: a

text: chine

text: learning

text: techniques

text: to

text: predict

text: users’

text: future

text: movement

text: photo

text: or

text: a

text: status

text: using

text: his

text: online

text: social

text: network

text: account.

text: based

text: on

text: their

text: communities’

text: information.

text: Extensive

text: experi-

text: Moreover,

text: users

text: may

text: just

text: share

text: their

text: locations,

text: called

text: check-

text: ments

text: demonstrate

text: the

text: prediction’s

text: eﬀectiveness.

text: in,

text: to

text: tell

text: their

text: friends

text: where

text: they

text: are

text: or

text: to

text: engage

text: in

text: social

text: Categories

text: and

text: Subject

text: Descriptors

text: games

text: as

text: in

text: Foursquare.

text: Since

text: these

text: large

text: amount

text: of

text: lo-

text: cation

text: and

text: social

text: relation

text: data

text: become

text: available,

text: studying

text: [Database

text: Management]:

text: Database

text: Applications—

text: human

text: mobility

text: and

text: its

text: connection

text: with

text: social

text: relationships

text: Data

text: mining

text: becomes

text: quantitatively

text: achievable

text: (e.g.,

text: Understanding

text: human

text: mobility

text: can

text: lead

text: to

text: compelling

text: General

text: Terms

text: applications

text: including

text: location

text: recommendation

text: urban

text: planning

text: immigration

text: patterns

text: etc.

text: Algorithms,

text: theory,

text: experiments

text: Previous

text: works,

text: including

text: show

text: that

text: human

text: mobility

text: is

text: inﬂuenced

text: by

text: social

text: factors.

text: However,

text: there

text: is

text: one

text: Keywords

text: common

text: shortcoming:

text: they

text: all

text: treat

text: friends

text: of

text: users

text: equally.

text: Similar

text: to

text: other

text: social

text: behaviors,

text: in

text: most

text: cases

text: mobility

text: is

text: Human

text: mobility,

text: social

text: networks,

text: network

text: communities

text: inﬂuenced

text: by

text: speciﬁc

text: communities

text: but

text: not

text: all

text: friends.

text: For

text: example,

text: the

text: aforementioned

text: colleagues

text: can

text: inﬂuence

text: the

text: INTRODUCTION

text: place

text: a

text: user

text: goes

text: for

text: lunch

text: but

text: probably

text: have

text: nothing

text: to

text: do

text: Humans

text: are

text: social

text: animals,

text: everyone

text: is

text: a

text: part

text: of

text: the

text: so-

text: with

text: his

text: weekend

text: plans.

text: Meanwhile,

text: where

text: a

text: user

text: visits

text: on

text: ciety

text: and

text: gets

text: inﬂuences

text: from

text: it.

text: For

text: example,

text: our

text: daily

text: weekends

text: largely

text: depends

text: on

text: his

text: friends

text: or

text: family,

text: but

text: not

text: his

text: behaviors,

text: such

text: as

text: what

text: types

text: of

text: music

text: we

text: listen

text: to,

text: where

text: colleagues.

text: Therefore,

text: the

text: impact

text: on

text: a

text: user’s

text: mobility

text: should

text: we

text: have

text: lunch

text: on

text: weekdays

text: and

text: what

text: activities

text: we

text: conduct

text: be

text: considered

text: from

text: the

text: perspectives

text: of

text: communities

text: instead

text: on

text: weekends,

text: are

text: largely

text: dependent

text: on

text: our

text: social

text: relations.

text: of

text: all

text: friends.

text: In

text: a

text: broader

text: view,

text: community

text: is

text: arguably

text: the

text: Normally,

text: we

text: categorize

text: our

text: social

text: relations

text: into

text: diﬀerent

text: most

text: useful

text: resolution

text: to

text: study

text: social

text: networks

text: Contributions.

text: In

text: this

text: paper,

text: we

text: aim

text: to

text: study

text: the

text: im-

text: pact

text: from

text: communities

text: on

text: a

text: user’s

text: mobility

text: and

text: predict

text: his

text: locations

text: based

text: on

text: his

text: community

text: information.

text: First,

text: we

text: partition

text: each

text: users’

text: friends

text: into

text: communities

text: and

text: propose

text: a

text: notion

text: namely

text: community

text: entropy

text: to

text: quantify

text: a

text: user’s

text: social

text: diversity.

text: Second,

text: we

text: analyze

text: communities’

text: in-

text: ﬂuences

text: on

text: users’

text: mobility

text: and

text: our

text: main

text: conclusions

text: include:

text: November

text: Palo

text: Alto,

text: California,

text: USA.

text: ##LTLine##


text: K. L. Yeo et al.: Reconstruction of total and spectral solar irradiance (1974-2013) Fig.15. The rotational variability in the reconstruction and in the spectral measurements from UARS and SORCE over the same spectral intervals as in Fig. 14 (except we combined the 120 to 180 nm and 180 to 250 nm intervals, where the relative agreement between the various time series is similar) during 1993 and 2004. The rotational variability was isolated by taking the integrated ﬂux over each spectral interval and subtracting the corresponding 361-moving average. The dashed lines represent the reconstruction uncertainty. ulation. The R2 with the reconstruction and the PMO6V record is very weak (0.170 and 0.183, respectively). In comparison, the integrated ﬂux in the reconstruction over the wavelength range of SIM still replicates over 94% of the variability in PMO6V TSI (R2 = 0.942). The absence of any consistent relation to the solar cycle in the SIM record and the failure of the total ﬂux to replicate TSI variability indicate signiﬁcant unaccounted instrumental trends. It is responsible, at least in part, for the poor R2 with the recon- struction. We are held back by the fact that there are no other continuous and extended SSI records covering a similar spectral range from a more conclusive evaluation of reconstructed visible and infrared SSI as done for the ultraviolet in Sect. . This high- lights the need for a more accurate calibration of SIM SSI and for other measurements in the visible and infrared. 6. Summary Fig.16. The decline in SSI between the solar cycle maximum in 2000 In this paper, we present a SATIRE-S model reconstruction and the minimum in 2008 in the reconstruction (black) and in the of daily total and spectral solar irradiance spanning the period Morrill et al. 2011 (red) model as a function of wavelength in intervals of 1974 to 2013. The reconstruction is based on full-disc obser- of 10 nm. We considered the mean spectrum over the 3-month period vations of continuum intensity and line-of-sight magnetic ﬁeld centred on the stated activity maximum/minimum. The integral between from the KPVT, MDI and HMI. Gaps in the time series from the 240 and 410 nm is given in parentheses. limited availability of suitable magnetograms were plugged by the regression of indices of solar activity to the reconstruction. This work extends the preceding reconstruction based on simi- tered by SIM, as with the integrated ﬂux over the various wave- lar observations from the KPVT and MDI covering the period length intervals discussed above, lacks obvious solar cycle mod- of 1974 to 2009 by Ball et al. (2012, 2014). Apart from the ex- 14 ##LTFigure## ##LTFigure## ##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## ##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## ##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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