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Quantifying spatial correlations of general quantum dynamics

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2015 New J. Phys. 17 062001

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New J. Phys. 17 (2015) 079602 doi:10.1088/1367-2630/17/7/079602

ERRATUM

Erratum: Quantifying spatial correlations of general quantumdynamics (2015New J. Phys.17 062001)

Ángel Rivas andMarkusMüllerDepartamento de Física Teórica I, UniversidadComplutense, E-28040Madrid, Spain

E-mail: anrivas@ucm.es

Onpage 4, an error wasmade in the statement ‘Theorem 1’. The correct sentence is as follows:

Theorem1. If for a map S ( )the property I 1S = holds such a map must be unitary, ( )S ⋅ =U U U U( ) ,S S S S

† †⋅ = .

Due to a typesetting error, on page 9, line below equation (C5), commaswere incorrectly inserted. The

correct equation is:( )kdkV

1 1

(2 )3

3 ∫∑ →π

.

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New J. Phys. 17 (2015) 062001 doi:10.1088/1367-2630/17/6/062001

FAST TRACK COMMUNICATION

Quantifying spatial correlations of general quantum dynamics

Ángel Rivas andMarkusMüllerDepartamento de Física Teórica I, UniversidadComplutense, E-28040Madrid, Spain

E-mail: anrivas@ucm.es

Keywords: open quantum systems, spatially correlated dynamics, correlation quantifiers

AbstractUnderstanding the role of correlations in quantum systems is both a fundamental challenge as well asof high practical relevance for the control ofmulti-particle quantum systems.Whereas a lot of researchhas been devoted to study the various types of correlations that can be present in the states of quantumsystems, in this workwe introduce a general and rigorousmethod to quantify the amount ofcorrelations in the dynamics of quantum systems. Using a resource-theoretical approach, weintroduce a suitable quantifier and characterize the properties of correlated dynamics. Furthermore,we benchmark ourmethod by applying it to the paradigmatic case of two atomsweakly coupled to theelectromagnetic radiation field, and illustrate its potential use to detect and assess spatial noisecorrelations in quantumcomputing architectures.

1. Introduction

Quantum systems can display awide variety of dynamical behaviors, in particular depending on how the systemis affected by its coupling to the surrounding environment. One interesting feature which has attractedmuchattention is the presence ofmemory effects (non-Markovianity) in the time evolution. These typically arise forstrong enough coupling between the system and its environment, or when the environment is structured, suchthat the assumptions of thewell-knownweak-coupling limit [1–3] are no longer valid.Whereasmemory effects(or time correlations) can be present in any quantum system exposed to noise, another extremely relevantfeature, whichwewill focus on in this work, are correlations in the dynamics of different parts ofmulti-partitequantum systems. Since different parties of a partition are commonly, though not always, identifiedwithdifferent places in space, without loss of generality wewill in the following refer to these correlations betweendifferent subsystems of a larger system as spatial correlations.

Spatial correlations in the dynamics give rise to awide plethora of interesting phenomena ranging fromsuper-radiance [4] and super-decoherence [5] to sub-radiance [6] and decoherence-free subspaces [7–11].Moreover, clarifying the role of spatial correlations in the performance of a large variety of quantumprocesses,such as e.g. quantum error correction [12–17], photosynthesis and excitation transfer [18–28], dissipative phasetransitions [29–33] and quantummetrology [34] has been and still is an active area of research.

Along the last few years, numerousworks have aimed at quantifying up towhich extent quantumdynamicsdeviates from theMarkovian behavior, see e.g. [35–43].However,much less attention has been paid to developquantifiers of spatial correlations in the dynamics, although someworks e.g. [44, 45] have addressed this issuefor some specificmodels. Thismay be partially due to thewell-known fact that undermany, though not allpractical circumstances, dynamical correlations can be detected by studying the time evolution of correlationfunctions of properly chosen observables A and B , acting respectively on the two partiesA andB of interest.For instance, in the context of quantum computing, sophisticatedmethods towitness the correlated character ofquantumdynamics, have been developed and implemented in the laboratory [45]. Indeed, any correlation

C ( , )A B A B A B= ⟨ ⊗ ⟩ − ⟨ ⟩⟨ ⟩ detected during the time evolution of an initial product state,

A Bρ ρ ρ= ⊗ , witnesses the correlated character of the dynamics. However, note that there exist highlycorrelated dynamics, which cannot be realized by a combination of local processes, which do not generate any

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such correlation, e.g. the swap process between two parties. Such dynamics can either act on internal degrees offreedom, induced e.g. by the action of a swap gate acting on two qubits [46], or can correspond to (unwanted)external dynamics, caused e.g. by correlated hopping of atoms in an optical lattice [47, 48] or crystalmelting andsubsequent recooling dynamics in trapped-ion architectures [49].

Thus, it is of eminent importance to developmethodswhich allow us to detect the presence or absence ofspatial correlations in the dynamics, without a priori knowledge of the underlyingmicroscopic dynamics, anddo not require us to resort to adequately chosen ‘test’ observables and initial ‘test’ quantum states. Suchmethodsshould furthermore provide a rigorous ground to quantitatively compare the amount of spatial correlations indifferent dynamical processes. These characteristics are essential for a ‘good’ correlation quantifier that can beused to study spatial correlations in quantumdynamics from a fundamental point of view [50–52], to clarifytheir role in physical processes [12–34], as well as tomeasure and quantify spatial correlations in the dynamics ofexperimental quantum systems.

It is the aim of this work to introduce amethod to quantify the degree of correlation in general quantumdynamics from a fundamental view point. Specifically,

(i) we propose a theoretical framework and formulate a general measure to assess the amount of spatialcorrelations of quantumdynamics without resorting to any specific physicalmodel. To this end, we adopt aresource theory approach, and formulate a fundamental law that any faithfulmeasuremust satisfy.

(ii) Within this framework, we study the properties that a dynamics has to fulfill to be considered asmaximallycorrelated.

(iii) We apply our measure to the paradigmatic quantum-optical model of two two-level atoms radiating intothe electromagnetic vacuum. This case exemplifies theworking principle of ourmeasure and quantitativelyconfirms the expectation that spatial dynamical correlations decaywith increasing interatomic distanceand for long times.

(iv) Finally, we illustrate this formalism with a second example in the context of quantum computing, wherequantum error correction protocols rely on certain assumptions on (typically sufficiently small) noisestrengths and noise correlations. Specifically, we consider two qubits subject to local thermal baths thatsuffer some residual interactionwhich induces a correlated noisy dynamics. Ourmethod reveals theremarkable fact that, under keeping the overall error probability for the two qubits constant, the degree ofspatial correlations decays very rapidly as the bath temperature increases. This suggests that, in somesituations, noise addition as e.g. by amoderate increase of the environmental temperature, can be beneficialto tailor specific desired noise characteristics.

2.Measure of correlations for dynamics

2.1. Uncorrelated dynamicsLet us consider a bipartite quantum system S AB= undergoing some dynamics given by a completely positiveand trace preserving (CPT)map S [without loss of generality we shall assume ddim( ) dim( )A B= = and

so d ddim( )S S2≔ = ]. This dynamics is said to be uncorrelatedwith respect to the subsystems A and B if it

can be decomposed as S A B= ⊗ , withCPTmaps A and B acting on A and B, respectively. Otherwise itis said to be correlated.

The central tool of our construction is the Choi–Jamiołkowski isomorphism [53, 54], which provides a one-to-onemap of a given quantumdynamics to an equivalent representation in the formof a quantum state in anenlargedHilbert space. Thismapping allows us to use tools developed for the quantification of correlations inquantum states for our purpose of quantifying correlations in quantumdynamics. Thus, consider a second d2

-dimensional bipartite system S A B′ = ′ ′, and let SSΦ∣ ⟩′ be themaximally entangled state between S and S′,

djj

dk k

1 1. (1)SS

j

d

SSk

d

AB A B

1 , 1

2

∑ ∑Φ ℓ ℓ≔ = ⊗ℓ

′=

′=

′ ′

Here, j∣ ⟩denotes the state vector with 1 at the jth position and zero elsewhere (canonical basis). TheChoi–Jamiołkowki representation of someCPTmap S on S is given by the d4-dimensional state

( ), (2)S S S SS SSCJρ Φ Φ≔ ⊗ ′ ′ ′

where S′ denotes the identitymap acting on S′. The entire information about the dynamical process S iscontained in this unique state.

2

New J. Phys. 17 (2015) 062001

2.2. Construction of the correlationmeasureIn order to formulate a faithfulmeasure of spatial correlations for dynamics, we adopt a resource theoryapproach [55–62]. This is, wemay consider correlated dynamics as a resource to performwhatever task thatcannot be implemented solely by (composing) uncorrelated evolutions A B⊗ . Then, suppose that the systemSundergoes some dynamics given by themap S , and consider the (left and right) composition of S with someuncorrelatedmaps A B⊗ and A B⊗ , so that the total dynamics is given by S′= ( )A B S⊗ ( )A B⊗ . It is clear that any task that we can dowith S′ by compositionwith uncorrelatedmaps can also beachievedwith S by compositionwith uncorrelatedmaps.Hence, we assert that the amount of correlation in Sis at least as large as in S′ . In other words, the amount of correlations of some dynamics does not increase undercompositionwith uncorrelated dynamics. This is the fundamental law of this resource theory, and any faithfulmeasure of correlations should satisfy it. For the sake of comparison, in the resource theory of entanglement,entanglement is the resource, and the fundamental law is that entanglement cannot increase under applicationof local operations and classical communication (LOCC) [55].

In this spirit, we introduce ameasure of correlations for dynamics via the (normalized) quantummutualinformation of theChoi–Jamiołkowski state S

CJρ , equation (2),

( )

( ) ( ) ( )

II

d

dS S S

¯( )4 log

1

4 log, (3)

SS

S AA S BB S

CJ

CJ CJ CJ⎡⎣ ⎤⎦

ρ

ρ ρ ρ

≔

≔ + −′ ′

with S ( · ) Tr[( · )log( · )]≔ − the vonNeumann entropy evaluated for the reduced density operatorsTr ( )S AA BB S

CJ CJρ ρ∣ ≔′ ′ and Tr ( )S BB AA SCJ CJρ ρ∣ ≔′ ′ , and S

CJρ ; seefigure 1. The quantity I ( )S is a faithfulmeasure of how correlated the dynamics given by S is, as it satisfies the following properties:

(i) I ( ) 0S = if and only if S is uncorrelated, S A B= ⊗ . This follows from the fact that the Choi–Jamiołkowski state of an uncorrelatedmap is a product state with respect to the bipartition AA BB′∣ ′, seeappendix A.

(ii) I ( ) [0, 1]S ∈ . It is clear that I ( ) 0S ⩾ , moreover it reaches its maximum value when S ( )SCJρ is minimal

and ( ) ( )S SS AA S BBCJ CJρ ρ∣ + ∣′ ′ ismaximal. Both conditionsmeet when S

CJρ is amaximally entangled state

with respect to the bipartition AA BB′∣ ′, leading to I d( ) 2 logSCJ 2ρ = .

(iii) The fundamental law is satisfied,

I I¯( ) ¯[( ) ( )], (4)S A B S A B⩾ ⊗ ⊗ where the equality is reached for uncorrelated unitaries U U( · ) ( · )A A A

†= , U U( · ) ( · )B B B†= ,

V V( · ) ( · )A A A†= , and V V( · ) ( · )B B B

†= . This result follows from themonotonicity of thequantummutual information under local CPTmaps (which in turn follows from themonotonicity ofquantum relative entropy [63]) and the fact that for anymatrixA, A AS SS S SS

tΦ Φ⊗ ∣ ⟩ = ⊗ ∣ ⟩′ ′ ′ wherethe superscript ‘t’ denotes the transposition in the Schmidt basis of themaximally entangled state SSΦ∣ ⟩′ .

2.3.Maximally correlated dynamicsBefore computing I for some cases it is worth studyingwhich dynamics achieve themaximumvalue I 1max = .From the resource theory point of view, these dynamics can be considered asmaximally correlated since they

Figure 1. Schematics of themethod. Left: the system S is prepared in amaximally entangled state SSΦ∣ ⟩′ with the auxiliary system S′(this state is just a product ofmaximally entangled states between AA′ and BB′, see equation (1)).Middle: the systemundergoes somedynamics S . Right: if and only if this process is correlatedwith respect toA andB, the total system SS′ becomes correlatedwithrespect to the bipartition AA BB′∣ ′ and the degree of correlation can bemeasured by the normalizedmutual information, equation (3).

3

New J. Phys. 17 (2015) 062001

cannot be constructed fromothermaps by compositionwith uncorrelatedmaps (because of equation (4)).Wehave the following results:

Theorem1. If for amap S the property I ( )S holds, such amapmust be unitary U U( · ) ( · )S S S†= ,U US S

† = .

Proof.As aforementioned, themaximumvalue, I ( ) 1S = , is reached if and only if SCJρ is amaximally

entangled state with respect to the bipartition AA BB′∣ ′, AA BB( ) ( )Ψ∣ ⟩′ ∣ ′ . Then

( ) (5)S S SS SS AA BB AA BB( ) ( ) ( ) ( )Φ Φ Ψ Ψ⊗ =′ ′ ′ ′ ′ ′ ′

is a pure state. Therefore S must be unitary as theChoi–Jamiołkowski state is pure if and only if it represents aunitarymap. □

Despite the connectionwithmaximally entangled states, the set ofmaximally correlated operationsU I U{ ; ¯ ( ) 1}S SC ≔ = , can not be so straightforwardly characterized as itmay seem.Note that not allmaximally

entangled states AA BB( ) ( )Ψ∣ ⟩′ ∣ ′ are valid Choi–Jamiołkowski states. In appendix Bwe provide a detailed proof ofthe next theorem.

Theorem2.AunitarymapUS C∈ if and only if it fulfills the equation

ki U mj nj U i . (6)i j

S S k mn

,

†∑ ℓ δ δ= ℓ

Examples ofmaximally correlated dynamics are the swap operation exchanging the states of the two partiesAandB,U US A B= ↔ , and thus also any unitary of the formof U U U V V( ) ( )A B A B A B⊗ ⊗↔ . However, not everyUS C∈ falls into this class. For example, the unitary operation of two qubitsU 21 12 iS′ = ∣ ⟩⟨ ∣ + ( 11 21∣ ⟩⟨ ∣

12 11+∣ ⟩⟨ ∣ 22 22 )+∣ ⟩⟨ ∣ belongs to C and it cannot bewritten as U U U( )A B A B⊗ ↔ V V( )A B⊗ , since thatwouldimply vanishing I U U¯( )S A B′ ↔ whereas I U U¯( ) 1 2 0S A B′ = ≠↔ . Interestingly, operations able to create highlycorrelated states such as the two-qubit controlled-NOT gate [46] aswell as the two-qubit dynamicalmapsdescribing the dissipative generation of Bell states [64, 65] achieve a correlation value of 1/2 and thus do notcorrespond tomaximally correlated dynamics. Note that whereas a controlled-NOTgate creates forappropriately chosen two-qubit initial statesmaximally entangled states, there are other states which are leftcompletely uncorrelated under its action. Themeasure I captures—completely independently of initial statesand ofwhether possibly created correlations are quantumor classical—the fact that correlated dynamics cannotbe realized by purely local dynamics.

Let us point out that in some resource theories, such as bi-partite entanglement, themaximal element cangenerate any other element by applying the operations which fulfill its fundamental law, e.g. LOCC. This is notthe case here, i.e.maximally correlated evolutions cannot generate any arbitrary dynamics by compositionwithuncorrelated operations. Indeed, if S

max were able to generate any other dynamics, in particular it would be able

to generate any unitary evolutionUS, U U( ) ( )( · ) ( · )A B S A B S Smax †⊗ ⊗ = . However, this would

imply that A B⊗ , Smax and ( )A B⊗ are unitary evolutions aswell, so that U U( )A B⊗ US

max

V V U( )A SB⊗ = , with U U( · ) ( · )S S Smax max max †= . Since I ( )S is invariant under the composition of

uncorrelated unitaries, this result would imply that for any correlated unitaryUS, I U¯( )S would take the samevalue I U[¯( )]S

max , and this is not true as can be easily checked.

3. Applications

3.1. Two-level atoms in the electromagnetic vacuumTo illustrate the behavior of I ( )S , consider the paradigmatic example of two identical two-level atomswithtransition frequencyω interacting with the vacuumof the electromagnetic radiation field (see appendix C).Under a series of standard approximations, the dynamics of the reduced densitymatrix of the atoms Sρ isdescribed by themaster equation

4

New J. Phys. 17 (2015) 062001

( ){ }t

a

d

d( ) i ,

, , (7)

SS

z zS

j k

jk k S j j k S

2 1 2

, 1,2

1

2

⎡⎣ ⎤⎦∑

ρρ σ σ ρ

σ ρ σ σ σ ρ

= = − +

+ −

ω

=

− + + −

where jzσ is the Pauli z-matrix for the jth atom, and e g( )j j j

†σ σ= = ∣ ⟩ ⟨ ∣+ − the electronic raising and loweringoperators, describing transitions between the exited e j∣ ⟩ and ground g j∣ ⟩ states. The coefficients ajk depend onthe spatial separation r between the atoms. In the limit of r 1 ω≫ they reduce to a jk jk0γ δ≃ , whereas forr 1 ω≪ they take the form a jk 0γ≃ . Here 0γ is the decay rate of the individual transition between e∣ ⟩ and g∣ ⟩. Inthefirst regime the two-level atoms interact effectively with independent environments, while in the second, thetransitions are collective and lead to theDickemodel of super-radiance [4].

To quantitatively assess this behavior of uncorrelated/correlated dynamics as a function of r, we compute themeasure of correlations I , equation (3) (see appendix C for details). The results are shown infigure 2.Despitethe fact that the value of I depends on time (the dynamicalmap is et

S = ), I decreases as r increases, asexpected. Furthermore, the value of I approaches zero for t large enough (see inset plot), except in the limitingcase r=0, because for r 0≠ the dynamics becomes uncorrelated in the asymptotic limit, lim et

t = ⊗→∞ ,

where K K K K( · ) ( · ) ( · )1 1†

2 2†= + withKraus operators ( )K 0 0

1 01 = and ( )K 0 00 12 = ; however for

r=0, lim ett

→∞ is a correlatedmap. Thus, we obtain perfect agreement between the rigorousmeasure of

correlations I and the physically expected behavior of two distant atoms undergoing independent noise.

3.2. Spatial noise correlations in quantum computingFault-tolerant quantum computing is predicted to be achievable provided that detrimental noise is sufficientlyweak andnot too strongly correlated [66]. However, even if noise correlations decay sufficiently fast in space,associated (provable) bounds for the accuracy threshold values can decrease by several orders ofmagnitude ascompared to uncorrelated noise [14]. Thus, it is of both fundamental and practical importance [45] to be able todetect, quantify and possibly reducewithout a priori knowledge of the underlyingmicroscopic dynamics theamount of correlated noise.Here, we exemplify how the proposedmeasure can be employed in this context byapplying it to a simple, though paradigmaticmodel systemof two representative qubits from a larger qubitregister.We assume that the qubits are exposed to local thermal (bosonic) baths, such as realized e.g. by couplingdistant atomic qubits to the surrounding electromagnetic radiationfield, and that they interact via aweakZZ-coupling, which could be caused, e.g., by undesired residual dipolar or van-der-Waals type interactions betweenthe atoms. The ‘error’ dynamics of this system is described by themaster equation

Figure 2.Maximumvalue of I as a function of the distance r for two two-level atoms radiating in the electromagnetic vacuum.Asexpected, the amount of correlations in the dynamics decreases with r. In the inset, I is represented as a function of time for differentdistances r between atoms ( d 2 1ω = ∣ ∣ = , 0θ = , see appendixC).

5

New J. Phys. 17 (2015) 062001

( )

( )

{ }

{ }

tJ

n

n

d

d( ) i ( ) ,

( ¯ 1) ,

¯ , , (8)

SS

z z z zS

j

j S j j j S

j

j S j j j S

2 1 2 1 2

1,20

1

2

1,20

1

2

⎡⎣ ⎤⎦∑

∑

ρρ σ σ σ σ ρ

γ σ ρ σ σ σ ρ

γ σ ρ σ σ σ ρ

= = − + +

+ + −

+ −

ω

=

− + + −

=

+ − − +

whereω is the energy difference between the qubit states, J the strength of the residualHamiltonian coupling, 0γis again the decay rate between upper and lower energy level of each individual qubit and n T¯ [exp( ) 1] 1ω= − −

ismean number of bosonswith frequencyω in the two local baths of temperatureT (assumed to be equal).We assume J and 0γ to be out of our control and aim at studying the spatial correlations of the errors induced

by the interplay of the residualZZ-coupling and the baths as a function of the bath temperatureT and elapsedtime t, which in the present contextmight be interpreted as the time for executing one round of quantum errorcorrection [66, 67]. Since the overall probability that some error occurs on the two qubits will increase underincreasing t andT, we need tofix it for a fair assessment of the correlation of the dynamics. A natural way to dothis is by defining the error probability in terms of how close the dynamicalmap induced by equation (8)(excluding the term ( )z z

2 1 2σ σ+ω , as this is not considered a source of error) is to the identitymap (the case of no

errors). Particularly, we can use thefidelity between bothChoi–Jamiołkowski states, SCJρ for the ‘error’map and

SSΦ∣ ⟩′ for the identitymap, P 1 SS SSerror SCJΦ ρ Φ= − ⟨ ∣ ∣ ⟩′ ′ . Figure 3 shows the value of amount of dynamical

correlations asmeasured by I along a t–T line onwhich the error probability is constant (P 0.1error = , green linein the inset plot). The numerical data shows, despite this fixing of the overall error rate, that as the temperatureincreases the correlatedness of errors decreases very rapidly. This remarkable result suggests that by increasingthe effective, surrounding temperature one can strongly decrease the non-local character of the noise at theexpense of a slightly higher error rate per fixed time t, or constant error rates if the time t for an error correctionround can be reduced. Thus, the proposed quantifiermight prove useful tomeet and certify in a given physicalarchitecture the noise levels and noise correlation characteristics which are required to reach the regimewherefault-tolerant scalable quantum computing becomes feasible in practice.

4. Conclusion

In this work, we have formulated a generalmeasure for the spatial correlations of quantumdynamics withoutrestriction to any specificmodel. To that aimwe have adopted a resource theory approach and obtained afundamental law that any faithful quantifier of spatial correlationmust satisfy.We have characterized themaximally correlated dynamics, and applied ourmeasure to the paradigmatic example of two atoms radiating in

Figure 3.Amount of spatial correlations I along the t–T line corresponding to constant error probability P 0.1error = .We see the rapiddecreasing of I asT increases (J = 1 and 4 30γ = in units ofω). The inset shows t–T isolines for various values of the error probabilityPerror, which increases with both t andT.

6

New J. Phys. 17 (2015) 062001

the electromagnetic field, where spatial correlations are naturally related to the separation between atoms.Furthermore, we have illustrated the applicability of themeasure in the context of quantum computing, where itcan be employed to quantify and potentially control spatial noise correlationswithout a priori knowledge of theunderlying dynamics.

Beyond the scope of this work it will be interesting from a fundamental point of view to study howmanyindependent (up to local unitaries)maximally correlated dynamics there are, and how to deal with the case ofmulti-partite or infinite dimensional systems. From a practical point of view, it is also interesting to developefficientmethods to estimate the proposedmeasure, in particular in high-dimensional quantum systems, e.g. bythe construction of witnesses or bounds, in analogy to entanglement estimators [68] that have been developedbased on the resource theory of entanglement. In this regard, it is our hope that the present results provide auseful tool to study rigorously the role of spatial correlations in a variety of physical processes, including noiseassisted transport, quantum computing and dissipative phase transitions.

Acknowledgments

Weacknowledge interesting discussions with TMonz andDNigg, as well asfinancial support by the SpanishMINECOgrant FIS2012-33152, the CAMresearch consortiumQUITEMADgrant S2009-ESP-1594, theEuropeanCommission PICC: FP7 2007-2013, GrantNo. 249958, theUCM-BS grantGICC-910758 and theUSArmyResearchOffice through grantW911NF-14-1-0103.

AppendixA. Choi–Jamiołkowski state of uncorrelatedmaps

First of all, letUB A↔ ′ be the commutationmatrix (or unitary swap operation) [69, 70] betweenHilbertsubspaces B and A′ of the totalHilbert space A B A B⊗ ⊗ ⊗′ ′ :

( )U M M M M U (A1)B A B A1 2 3 4†⊗ ⊗ ⊗↔ ′ ↔ ′

M M M M . (A2)1 3 2 4= ⊗ ⊗ ⊗

whereM1,M2,M3 andM4 are operators acting on the respectiveHilbert subspaces in the decomposition

A B A B⊗ ⊗ ⊗′ ′ . This is,M1 acts on A ,M4 on B′ , andM2 andM3 act on B and A′ on the left-hand side and on A′ and B on the right-hand side of the equality respectively. Note thatU UB A B A =↔ ′ ↔ ′ andthenU UB A B A

†=↔ ′ ↔ ′.Now, it turns out that the evolution given by some dynamicalmap S is uncorrelatedwith respect to the

subsystems A and B, S A B= ⊗ , if and only if its Choi–Jamiołkowski state ( )S S S SS SSCJρ Φ Φ≔ ⊗ ∣ ⟩⟨ ∣′ ′ ′

is

( )U U , (A3)S B A A B B ACJ CJ CJρ ρ ρ= ⊗↔ ′ ↔ ′

where ACJρ and B

CJρ are theChoi–Jamiołkowski states of themaps A and B , respectively.Indeed, if S A B= ⊗ , we have (omitting for the sake of clarity the subindexes in the basis expansion of

SSΦ∣ ⟩′ ):

dk mn k mn

dk m n k mn

( )1

( )

1( ) ( ) , (A4)

S S S SS SS

k m n

d

S

k m n

d

A B

CJ2

, , , 1

2, , , 1

∑

∑

ρ Φ Φ ℓ ℓ

ℓ ℓ

= ⊗ = ⊗

= ⊗ ⊗

ℓ

ℓ

′ ′ ′=

=

then

U Ud

k m k m n n

dkk mm

dnn

1( ) ( )

1( )

1( )

. (A5)

B A S B A

k m n

d

A B

k m

d

A

n

d

B

A B

CJ2

, , , 1

, 1 , 1

CJ CJ

∑

∑ ∑

ρ ℓ ℓ

ℓℓ

ρ ρ

= ⊗ ⊗ ⊗

= ⊗ ⊗ ⊗

= ⊗

ℓ

ℓ

↔ ′ ↔ ′=

= =

Conversely, if equation (A3) holds, then the dynamics has to be uncorrelated because the correspondencebetweenChoi–Jamiołkowski states and dynamicalmaps is one-to-one.

7

New J. Phys. 17 (2015) 062001

From equation (A3) it is straightforward to conclude that I ( ) 0S = if and only if S is uncorrelated,

because the vonNeumann entropy of theChoi-Jamiłkowski state factorizes S ( )SCJρ =

( )S U UB A A B B ACJ CJ⎡⎣ ⎤⎦ρ ρ⊗ =↔ ′ ↔ ′ ( )S A B

CJ CJρ ρ⊗ = ( )S ACJρ ( )S B

CJρ+ if and only if S is uncorrelated.

Appendix B. Proof of theorem2

As commented in section 2.3,US C∈ if

U , (B1)AA BB S SS( ) ( )Ψ Φ= ⊗′ ′ ′

where AA BB( ) ( )Ψ∣ ⟩′ ∣ ′ is amaximally entangled statewith respect to the bipartition AA BB′∣ ′. Note that ifAA BB( ) ( )Ψ∣ ⟩′ ∣ ′ is amaximally entangled state with respect to the bipartition AA BB′∣ ′,UB A AA BB( ) ( )Ψ∣ ⟩↔ ′ ′ ∣ ′ will be a

maximally entangled state state with respect to the bipartition AB A B S S∣ ′ ′ = ∣ ′. Since anymaximally entangledstate with respect to the bipartition S S∣ ′ can bewritten asU U˜ ˜

S S SSΦ⊗ ∣ ⟩′ ′ for some local unitaries US and US′,we canwrite

U U U˜ ˜ . (B2)B A AA BB S S SS( ) ( )Ψ Φ= ⊗↔ ′ ′ ′ ′ ′

Because of equations (B1) and (B2)we conclude thatUS C∈ if and only if there exist unitaries US and US′ suchthat

U U U U( ˜ ˜ ) . (B3)S S SS B A S S SSΦ Φ⊗ = ⊗′ ′ ↔ ′ ′ ′

Next, we prove the followingLemma. A unitarymapUS C∈ if and only if there exists some other unitaryV such that thematrix elements

ofUS can bewritten as

k U mn km V n . (B4)Sℓ ℓ=

Proof. IfUS C∈ , then by taking inner product with respect to the basis element k mnℓ∣ ⟩ in equation (B3)weobtain:

k U mn d km n U U

km U U n km V n

˜ ˜

˜ ˜ , (B5)

S S S SS

S St

ℓ ℓ Φ

ℓ ℓ

= ⊗

= =′ ′

′

forV U U˜ ˜S S

t= ′. Here we have used that A AS SS S SStΦ Φ⊗ ∣ ⟩ = ⊗ ∣ ⟩′ ′ ′ where the superscript ‘t’ denotes the

transposition in the Schmidt basis of themaximally entangled state SSΦ∣ ⟩′ , which has been taken to be thecanonical basis here.

Conversely, assume that there exists a unitaryV satisfying (B4). AsV can always be decomposed as theproduct of two unitaries,V V V1 2= , by settingU V˜

S 1= andU V˜St

2=′ , the same algebra as in equation (B5) leadsus to rewrite equation (B4) as

k mn U k mn U U U˜ ˜ . (B6)S S SS B A S S SSℓ Φ ℓ Φ⊗ = ⊗′ ′ ↔ ′ ′ ′

Since k mnℓ∣ ⟩ are elements of a basis we conclude that equation (B3) holds. □

With these results, the theorem2 is easy to prove.

Proof of theorem2.Note that for any unitaryUS, equation (B4) is satisfied for somematrixV. Thus, whatwehave to prove is that such amatrixV is unitary if and only ifUS fulfills the equation

ki U mj nj U i , (B7)i j

S S k mn

,

†∑ ℓ δ δ= ℓ

and this follows after a straightforward algebraic computation. □

AppendixC. Two two-level atoms coupled to the radiationfield

The freeHamiltonian of the atoms is

H2

( ), (C1)Sz z

1 2ω σ σ= +

8

New J. Phys. 17 (2015) 062001

where jzσ is the Pauli z-matrix for the jth atom. In addition, the environmental freeHamiltonian is given by

k kH a a( ) ( ), (C2)k

kE

1,2

†∑ ∑ ω=λ

λ λ=

where k and λ stand for thewave vector and the two polarization degrees of freedom, respectively.We havetaken natural units c 1 = = . The dispersion relation in the free space is kkω = ∣ ∣, and the field operators

ka ( )†λ and ka ( )λ describe the creation and annihilation of photonswithwave vector k and polarization vector

e .λ These fulfill k e· 0=λ and e e· ,δ=λ λ λ λ′ ′.The atom–field interaction is described in dipole approximation by theHamiltonian

d E r d E rH · ( ) * · ( ) . (C3)SE

j

j j j j

1,2

⎡⎣ ⎤⎦∑ σ σ= − +=

− +

Here, d is the dipolematrix element of the atomic transition, rj denotes the position of the jth atom, and

e g( )j j j†σ σ= = ∣ ⟩ ⟨ ∣+ − for its exited e j∣ ⟩ and ground g j∣ ⟩ states. Furthermore, the electric field operator is given by

(Gaussian units)

( )E r e k k ka a( ) i2

( ) ( )e ( )e , (C4)k

k k r k r

,

i · † i ·∑ πω= −

λλ λ λ

−

where denotes the quantization volume. In theMarkovianweak coupling limit [1] themaster equation for theatoms takes the form:

( ){ }t

a

d

d( ) i ,

, , (C5)

SS

z zS

i j

jk k S j j k S

2 1 2

, 1,2

1

2

⎡⎣ ⎤⎦∑

ρρ σ σ ρ

σ ρ σ σ σ ρ

= = − +

+ −

ω

=

− + + −

where, after taking the continuum limit( )k, , , , , d ,k1 1

(2 )3

3 ∫∑ →π and performing the integrals, the coefficients

ajk are given by (section 3.7.5 of [1])

( )a j x P j x( ) cos ( ) , (C6)jk jk jk jk0 0 2 2⎡⎣ ⎤⎦γ θ= +

here d4

303 2γ ω= ∣ ∣ , and j x( )0 and j x( )2 are spherical Bessel functions [71],

j xx

xj x

x xx

xx( )

sin, ( )

3 1sin

3cos , (C7)0 2 3 2

⎜ ⎟⎛⎝

⎞⎠= = − −

and

P (cos )1

2(3 cos 1) (C8)2

2θ θ= −

is a Legendre polynomial, with

( )r rd r r

d r rx , and cos

· ( ). (C9)jk j k jk

j k

j k

2

2

2 2ω θ= − =

−

−

Notice that if the distance between atoms r rr 1 2= ∣ − ∣, ismuch larger than thewavelength associatedwith

the atomic transition r 1 ω≫ , we have a jk ij0γ δ≃ and only the diagonal terms d4

303 2γ ω= ∣ ∣ are relevant.

Then, themaster equation describes two-level atoms interacting with independent environments, and there areno correlations in the emission of photons by the first and the second atom. In the opposite case, when r 1 ω≪ ,everymatrix element approaches the same value aij 0γ≃ , in themaster equation the atomic transitions can be

approximately described by the collective jumpoperators J 1 1σ σ= +±± ±, and the pair of atoms becomes

equivalent to a four-level systemwithHamiltonian J2

( )zz z

1 2ω ω σ σ= + at themean position r r( ) 21 2−interactingwith the electromagnetic vacuum. This emission of photons in a collective way known as super-radiance is effectively described in terms of collective angularmomentumoperators in theDickemodel [4].

Evaluation of the correlationmeasure. In order to numerically compute I for this dynamics, we consider amaximally entangled state SSΦ∣ ⟩′ between two sets S and S′ of two qubits. Namely, S is the set of the two physicalqubits, i.e. the two two-level atoms 1 and 2, and S′ ismade up of two auxiliary qubits 1′ and 2′ as sketched infigure 1.Next, the part S of themaximally entangled state SS SSΦ Φ∣ ⟩⟨ ∣′ ′ is evolved according to themaster

equation (C5)while keeping the part S′ constant, to obtain t( )SCJρ . This can be done, for instance, by

9

New J. Phys. 17 (2015) 062001

numerically integrating themaster equationt

tt

d ( )

d[ ( )]S

CJ

SCJ

ρρ= ⊗ , with the initial condition

(0) SS SSSCJρ Ψ Ψ= ∣ ⟩⟨ ∣′ ′ , where is for the present example specified in equation (C5). Tracing out the qubits 2

and 2′ of t( )SCJρ yields t( )S

CJ11ρ ∣ ′, and similarly tracing out qubits 1 and 1′ yields t( )S

CJ22ρ ∣ ′. Finally, this allows

one to compute the vonNeumann entropies of t( )SCJ

11ρ ∣ ′, t( )SCJ

22ρ ∣ ′ and t( )SCJρ to calculate I t¯( ) according to

equation (3).

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