a general framework for complete positivity - jason m. dominy, alireza shabani, daniel a. lidar

8/13/2019 A General Framework for Complete Positivity - Jason M. Dominy, Alireza Shabani, Daniel a. Lidar

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arXiv:1312

.0908v1

[quant-ph

]3Dec2013

A general framework for complete positivity

Jason M. Dominy(1,4), Alireza Shabani(5), and Daniel A. Lidar(1,2,3,4)

Departments of (1)Chemistry, (2)Electrical Engineering, and (3)Physics,(4)Center for Quantum Information Science & Technology,

University of Southern California, Los Angeles, CA 90089, USA(5)Department of Chemistry, University of California, Berkeley, CA 94720, USA

Complete positivity of quantum dynamics is often viewed as a litmus test for physicality, yet it is well known

that correlated initial states need not give rise to completely positive evolutions. This observation spurred nu-merous investigations over the past two decades attempting to identify necessary and sufficient conditions for

complete positivity. Here we describe a complete and consistent mathematical framework for the discussion and

analysis of complete positivity for correlated initial states of open quantum systems. Our key observation is that

initial system-bath states with the same reduced state on the system must evolve under all admissible unitary

operators to system-bath states with the same reduced state on the system, in order to ensure that the induced

dynamical maps on the system are well-defined. Once this consistency condition is imposed, related concepts

like the assignment map and the dynamical maps are uniquely defined. In general, the dynamical maps may not

be applied to arbitrary system states, but only to those in an appropriately defined physical domain. We show

that the constrained nature of the problem gives rise to not one but three inequivalent types of complete posi-

tivity. Using this framework we elucidate the limitations of recent attempts to provide conditions for complete

positivity using quantum discord and the quantum data-processing inequality. The problem remains open, and

may require fresh perspectives and new mathematical tools. The formalism presented herein may be one step in

that direction.

I. INTRODUCTION

Completely positive (CP) maps have played an important

role in the long and extensive history of the problem of the for-

mulation and characterization of the dynamics of open quan-

tum systems [1, 2]. They have become a widespread tool,

e.g., in quantum information science [3]. Thus it is of interest

to establish under which conditions CP maps can arise from a

complete description of an open system that includes both the

system and its environmentor bath. To arrive at a map one first

identifies an admissible set of initial system-bath states which

is one-to-one with the corresponding set of system states via

the partial trace. Then one jointly and unitarily evolves the

system and bath, and then traces out the bath. It is well known

that if the set of admissible initial system-bath states is com-

pletely uncorrelated (product states) with a fixed bath state

then this standard procedure gives rise to a CP map descrip-

tion of the evolution of the system. The situation is far more

complicated when the set of initial system-bath states contains

correlated states. The basic reason for this is that when cor-

relations are present in the initial system-bath state, a clean

separation between system and bath is no longer possible, and

the standard procedure alone no longer uniquely defines a

map between the set of all initial and final system states.

Subsystem dynamical maps are uniquely defined by the

joint unitary evolution of system and bath and the choice of

the set of initial states and several studies have pointed out that

entangled initial system-bath states can lead to non-CP maps

[47]. Rodriguez-Rosarioet al. highlighted the role of quan-

tum discord [8] and showed that a CP map arises if the set of

joint initial system-bath states is purely classically correlated,

i.e., has vanishing quantum discord [9]. Subsequently Shabani

& Lidar showed that, under certain additional constraints, the

vanishing quantum discord condition is not just sufficient but

also necessary for complete positivity [10,11]. More recently,

Brodutchet al. [12] and Buscemi[13] demonstrated that this

connection between complete positivity and discord does not

generalize to all cases. Brodutch et al. did so by offering

a counterexample in the form of a set of initial system-bath

states, almost all of which are discordant, which nevertheless

exhibit completely positive subdynamics. Buscemi followed

this up by describing a general method for constructing exam-

ples yielding completely positive subdynamics, even though

they may feature highly entangled states. It remains an open

problem, therefore, to fully elucidate the relationship between

structural features of the set of initial system-bath states and

the behavior of the resulting dynamics, including whether or

not the dynamics are completely positive.

We do not offer a complete solution to the problem in these

pages. Rather, it is our purpose here to establish a complete

and consistent mathematical framework for the discussion and

analysis of linear subsystem dynamics, including the question

of complete positivity for correlated initial states. Addition-

ally, we develop the idea that, in the case of correlated initial

states, there are several inequivalent definitions of complete

positivity. Physical, mathematical, and operational concerns

may recommend one flavor of complete positivity over the

others, as we discuss. Our analysis builds on the notion of

what we call G-consistent operator spaces, that representthe set of admissible initial system-bath states.

G-consistency

is necessary to ensure that initial system-bath states with thesame reduced state on the system evolve under all admissible

unitary operators to system-bath states with the same reduced

state on the system, ensuring that the dynamical maps on the

system are well-defined.

The paper is organized as follows. In Section II, we ex-

amine the minimal conditions necessary for the existence of

well-defined subsystem dynamical maps, and formally de-

fineG-consistent subsets. The more specific case of lineardynamical maps is considered in Section III, where we for-
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mally defineG-consistent subspaces, consider properties ofthe uniquely defined assignment maps associated with these

spaces, describe the subsystem dynamical maps related to a G-consistent subspace, and operator sum representations of these

maps. SectionIVdevelops some definitions of complete pos-

itivity, discusses the physical motivation of these ideas, and

extends these definitions to apply to assignment maps. We

present in Section Va number of examples culled from the

existing literature on completely positive dynamics and showhow they may be expressed in the present framework. Finally,

a summary and discussion of open questions is offered in Sec-

tionVI.

II. SUBSYSTEM DYNAMICAL MAPS

The time-evolution of a quantum subsystem is often de-

scribed in terms of dynamical maps that transformthe reduced

state of the system at time t0 to the reduced state at timet1.However, the future state of a quantum system in contact with

its environment depends upon not just the current state of the

system, but the joint state of the system and environment, aswell as the joint evolution. For this reason, such dynamical

maps are not generally well-defined: each possible reduced

state of the system may arise from infinitely many distinct

system-bath states, which after joint evolution, can yield in-

finitely many distinct possible reduced system states at a fu-

ture timet1. We wish to describe a mathematical frameworkwhich is broad enough to contain the various dynamical map

constructions of [913] and others. It is closely related to the

assignment map approach first described by Pechukas [4], but

is more general and places greater emphasis on the space of

admissible initial system-bath states. In order to build this

framework involving dynamical maps for subsystem evolu-

tion, it is necessary to remove the indeterminacy of the fi-nal system state from the problem, requiring that we begin

by making an assumption about the set of admissible initial

system-bath states, which we describe presently.

To set the stage, let us briefly review the setting of open

quantum systems. For any Hilbert spaceH, letB(H)denotethe algebra of all bounded linear operators on H and letU(H)denote the unitary group onH. For anyU U(H), the ad-joint mapAdU B(B(H))is the conjugation superoperatorAU AU. Consider a system Swith a dS-dimensional sys-tem Hilbert-space HS, a bath Bwith a dB-dimensional HilbertspaceHB and the joint system-bath Hilbert spaceHS HB .LetDS B(HS),DB B(HB), andDSB B(HS HB)denote the convex sets of density matrices on

HS,H

B, and

HS HB, respectively. The joint system-bath stateSB (t)DSB evolves under a joint time-dependent unitary propaga-tor U(t) as AdU(t)[SB (0)] = SB (t). The reduced systemand bath states are given via the partial trace operation by

S (t) = TrB[SB (t)] DS and B(t) = TrS[SB (t)] DB,respectively. The standard prescription for the system sub-

dynamics is thus given by the following quantum dynamical

process (QDP), acting on the initial system-bath state SB (0):

S(t) = TrB[U(t)SB (0)U(t)]. (1)

S DSB

TrB S DS

AdU

TrB TrB

SU

FIG. 1. The set of admissible initial system-bath states S DSBmust be chosen such that a well-defined subsystem dynamical map

SU : TrB S DS exists which makes the diagram commute, i.e.,which satisfies SU(TrB) = TrB (UU) for all S. This re-quirement defines the concept of aU-consistent subset.

Consider some designated set of admissible initial states

S DSB . We shall later assume thatS is convex, but forthe time being we do not restrict its generality.

Definition 1 (S-dynamical map). GivenU U(HS HB),a subsystemS-dynamical mapSU : TrB S DS is any mapthat satisfiesSU(TrB) = TrB(U U

)for all S, i.e., thatmakes the diagram in Figure1commute.

Such a map can only be well-defined if, whenever 1, 2S and TrB1 = TrB2, it follows that TrB(U 1U) =

TrB(U 2U). This defines a necessary property ofS that wecallU-consistency.

We typically want to define not just a single dynamical

map, but a family of them based on a subsemigroup of uni-

tary system-bath evolution operators (semi since we will sett0). This gives rise to the notion ofG-consistency.Definition 2 (G-Consistent Subset). LetG U(HS HB)be a subsemigroup of the unitary group acting on the Hilbert

spaceHS HB. A subset of system-bath statesS DSBwill be calledG-consistent if it isU-consistent for allU G,i.e., if, whenever1, 2

S are such thatTrB1 = TrB2,

TrB(U 1U) = TrB(U 2U)for allU G.Grepresents the semigroup of allowed unitary evolutions ofsystem and bath, in other words, the set of unitary evolutions,

closed under compositions, for which the subsystem dynami-

cal maps are of interest. For example, if the system and bath

will only evolve according to a single fixed time-independent

HamiltonianH, thenG is typically the one-parameter sub-semigroupG ={eitH : t 0}. If the system and/orbath are subject to control, thenG may be a larger subsemi-group representing all unitary operators that may be generated

within the control scheme. Generally, the larger the semigroup

G, the more restrictions G-consistency places on the subset ofadmissible initial states

Sand the smaller such a subset must

be. Indeed, whenG = U(HS HB), the condition becomesthatTrB must describe a one-to-one correspondence betweenSandTrB S, i.e., for each system state S DS, there existsat most one stateSB Ssuch thatTrBSB = S.

Not every state in DS needs to be covered by a state in S. Those that are

not covered are inadmissible initial system states, lying outside the domain

ofSU

. See also the paragraph following the proof of Lemma2.
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Example 1. Stelmachovic and Buzek [14] offered an exam-ple which demonstrates that not all subsetsS DSB areU(HS HB)-consistent, i.e., that there exist initial states1, 2 DSB andU U(HS HB) such thatTrB1 =TrB 2, butTrB(U 1U

)= TrB(U 2U). Specifically, theyconsidered a system and bath comprising one qubit each, such

that

1 =||2

|0000| + ||2

|1111| (2a)2 = ( |00 + |11)( 00| + 11|) (2b)U=i(|11| x+ |00| 1)CNOT, (2c)

which satisfy

TrB1 =||2|00| + ||2|11|= TrB2 (3a)TrB(U 1U

) =||2|00| + ||2|11| (3b)TrB(U 2U

) = ( |0 + |1)( 0| + 1|), (3c)

so that TrB(U 1U) = TrB(U 2U). It follows that if

CNOT G, then noG-consistent subsetS DSB may con-tain both1and2.

The specification of a setS DSB of admissible initialstates may be thought of as a promise that the initial system-

bath state will always lie inS. The constraint thatSmust beG-consistent can impose heavy restrictions on this set, oftenforcing Sto be very small relative to DSB . It will be instructiveto consider the consequences of constraining to a G-consistentsubset Sin order to understand the properties of the resultingdynamical map and to place earlier studies in their proper con-

text.

III. LINEAR DYNAMICAL MAPS

A. G-consistent linear subspaces

Properties of the set of admissible initial states Sare closelyrelated to properties of the resulting dynamical mapsSU, andit is reasonable to wonder if there are either necessary or quite

common properties of subsystem dynamics that should be in-

corporated into the framework we are developing. One such

property, nearly ubiquitous in the open quantum system lit-

erature (see [15] for a notable exception we discuss in Sec-

tionV C), is that of convex-linearity ofSU, i.e., the propertythat for anyS, STrB S, and any[0, 1],

S

U(S+ (1 )S ) = S

U(S) + (1 )S

U(S ). (4)

Clearly, this statement aboutSUrequires thatTrB Sbe a con-vex set. We will go a bit further and assume thatSitself is aconvex subset ofB(HS HB). Any subsetS DSB may beuniquely extended to the C-linear subspace

V= SpanC S B(HS HB) (5)and the mapsTrB

S

, andAdUS

may be likewise uniquely

extended by linearity to TrBV

, and AdUV

. By assuming

V

DSB

FIG. 2. Schematic representation of a G-consistent subspaceV andits relationship to DSB , the convex set of all system-bath density ma-trices, represented by the ball. The intersectionV DSB , indicatedby the ruled area, is such that V= Span

C(V DSB ).

thatSis convex andG-consistent, we may similarly extendthe mapsSU : TrB S DS to C-linear mapsVU : TrB V B(HS)for allU G. It should be emphasized that, while SUis defined only on states, the mapVUis defined on the linearoperator space TrB V B(HS ) which contains both states,and non-state operators.

It is readily verified that the subspace

Vconstructed as the

complex span of a convex, G-consistent Sexhibits the follow-ing properties:

1.V B(HSHB)is a C-linear subspace.2.Vis spanned by states, i.e.,SpanC(DSB V) =V.3.V isG -consistent, i.e., if X, Y V are such that

TrBX = TrB Y and ifU G, thenTrB(U XU) =TrB(U Y U).

Any C-linear subspaceV B(HS HB)will be called aG-consistent subspaceif it satisfies these three properties. Fig-

ure2illustrates how a G-consistent subspace relates to DSB .Note that it is implied by C-linearity ofV and property2thatV is self-adjoint, i.e., X V implies X V. Note

further that anyC-linear subspaceV B(HS HB) whichis self-adjoint and spanned by states is alwaysG-consistentwhen G is the trivial group G ={1}.

Within anyC-linear subspaceV B(HS HB), we mayidentify a further subspace V0 Vby

V0 := ker

TrBV

={X V : TrB(X) = 0}. (6)

Then property3,G-consistency, is equivalent to the propertythatG V0 ker TrB, i.e., for any X V0 and U G,TrB(U XU

) = 0. In other words, differences in states withthe same reduced state cannot play any role in the final re-

duced state after evolution:TrB() = 0for , VDSBmust implyTrB[U()U] = 0 for all U G . The ker-nel of this idea goes back at least to [11] (section II.C), and

possibly earlier.

B. Assignment maps

Given aG -consistent subspaceV, consider the quotientspaceV/V0 in which each element corresponds to an affine


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subspace of the formX+V0 V for someX V. Be-cause of linearity, property3above is equivalent to the state-

ment that ifY, Z V are such that TrBY = TrBZ, thenTrB(U Y U

) = TrB(U ZU)for allU G. It follows that the

linear mapsTrB :V TrB VandAdU :V B(HS HB)for eachU G uniquely define linear mapsTrB :V/V0TrB VandAdU :V/V0 B(HS HB)/ kerTrB. Moreover,the map TrB :V/V0TrB Vis a one-to-one correspondence,and so may be inverted, yielding the assignment map.

Definition 3 (Assignment Map). The assignment mapAV :TrB V V/V0 is defined by

AV(X) =

TrBV

1(X) ={Y V : TrBY =X}. (7)

for anyX TrB V. For eachX TrB V, the outputAV(X)may be thought of either as an affine subspace ofV, or as apoint in the vector space V/V0.

This definition of the assignment map is a generalization

of the original definition introduced by Pechukas [4]. A

Pechukas-like assignment map (which assigns to each oper-

ator inTrB Va unique operator in V, rather than an affine sub-space) is recovered if and only if the G-consistent subspace VisU(HS HB)-consistent, as demonstrated in the followinglemma.

Lemma 1. LetV B(HS HB) be a C-linear, self-adjointsubspace which is spanned by states, i.e., a{1}-consistentsubspace.V isU(HS HB)-consistent if and only ifTrB

V

is injective, i.e.,V0 := ker

TrBV

={0}. In this case,

V/V0=V, so that the assignment map carries each operatorinTrB Vto a unique operator inV, as in Pechukas originaldefinition of the assignment map [4].

Proof. First, observe that ifV0 ={0}, thenG V0 =V0ker TrB for any subsemigroup G U(HS HB). Therefore VisG-consistent for anyG U(HS HB), and, in particular,isU(HS HB)-consistent. On the other hand, ifV isU(HS HB)-consistent, thenSpanC

U(HS HB) V0

ker TrB isan invariant subspace of the adjoint representation ofU(HS HB) on sl(HS HB) = kerTr [the zero trace operators inB(HS HB)]. Sinceker TrB is a proper subspace ofker Tr(assumingdim HS > 1), this invariant subspace generatedbyV0 cannot be all ofsl(HS HB ). The only if part ofthe lemma then follows from the irreducibility of the adjoint

representation ofU(HS HB)on sl(HS HB)[16,2.4.4]),which implies that the invariant subspace must be{0}, andtherefore V0 ={0}.

C. How constrained are the domains of dynamical maps?

Some authors have expressed a desire to keep TrB V =B(HS), so that the domain of the dynamical maps VU is un-constrained. Lemma1 shows that, whenGis all ofU(HSHB), any G-consistent subspace must be severely constrainedto have dimension no higher thanB(HS ). In Lemma2 wedemonstrate that, ifG contains non-local (entangling) opera-tors, then anyG-consistent subspaceVmust necessarily be a

proper subspace ofB(HS HB), and therefore is constrainedin some way. Specifically, Lemma2says that ifG containsnon-local unitaries, and V=B(HS HB), then V0 = ker TrBis such thatG V0 ker TrB, so thatV is notG-consistent.Thus, the onlyG-consistent subspaces must be proper sub-spaces ofB(HS HB).

Lemma 2. The full kernel

kerTrB :={X B(HS HB) : TrBX= 0} (8)isG-invariant if and only ifGis a subsemigroup of the groupU(HS) U(HB )of local unitary operators.

(Essentially this same question, posed and answered in dif-

ferent language, was the subject of [17].)

Proof. First, observe that if is trivial, i.e., ifU U(HS ) U(HB), thenAdUker TrB ker TrB since, ifU = US UB,thenTrB(U XU) = US[TrBX]U

S = 0 for anyXker TrB.

To prove only if, i.e., ifAdUker TrB

ker TrB, then

U U(HS )U(HB ), note first thatB(HS )1 is the or-thogonal complement to ker TrB in the Hilbert-Schmidt ge-ometry. This may be seen from the fact thatB(HS HB) =B(HS)1ker TrB since any X B(HS HB) may beuniquely decomposed asX= TrB(X) 1/dB+ Xker, whereXker = X TrB(X) 1/dB ker TrB, andB(HS) 1and ker TrB are orthogonal subspaces [ifA B(HS) andX ker TrB, thenA 1, XHS =A, TrBXHS = 0].Now, U U(HS HB) satisfiesAdUker TrB ker TrB ifand only if0 =AdUX, A 1HS =X, AdU(A 1)HSfor all X ker TrB and A B(HS), i.e., if and only ifAdU [B(HS)1] B(HS)1. In other words, the con-dition is that, for any A B(HS), there exists B B(HS)such thatU

(A 1)U = B 1. This condition implies, inparticular, that Ubelongs to the normalizer ofU(HS) 1,which isU(HS) U(HB)[18].

It follows from Lemma2 that anyC-linear subspaceV B(HS HB)which is spanned by states (i.e., a {1}-consistentsubspace) isU(HS ) U(HB )-consistent. This is because, forany suchV,V0 = ker

TrB |V

ker TrB is mapped byU(HS)U(HB )intoker TrB(by Lemma2), which is the con-dition forU(HS ) U(HB)-consistency. In particular, for anyG U(HS)U(HB ), V=B(HSHB)is a valid G-consistentsubspace and always gives rise to completely positive dynam-

ics[19], since for anyU=US

UB G

,VU(X) = USXU

S

for anyXTrB V, which is trivially completely positive.With G-consistency imposing such strong constraints on the

space of admissible system-bath operators, the desire to keep

the domain of the dynamical maps unconstrained seems mis-

placed. If it is reasonable to promise that the initial system-

bath state will neverlie outside a low-dimensional subspace V,it should be reasonable to also promise that the initial system

state will never lie outside a proper subspaceTrB V B(HS).Building on this observation we now proceed to identify a

unique linear dynamical map for the subsystem.


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D. Linear dynamical maps from G-consistent subspaces

Note that the quotient spaceV/V0 admits some additionalstructure that will be useful in characterizing the assignment

map. First, observe that V0is a self-adjointC-linear subspace.An affine subspace X+V0 V/V0contains a Hermitian oper-ator ifX+V0= Y+V0with Y =Y, which holds if and onlyif the affine subspace is self-adjoint, i.e., (X+

V0) =X+

V0

Such an affine subspace will then be spanned by Hermitianoperators, i.e., if(X+ V0) = X+ V0 andHX is the set ofHermitian elements in X+V0, then X+V0is the C-affine hullofHX . We may also identify a closed convex cone (V/V0)+of positive elements in V/V0as the collection of affine sub-spaces in V/V0that each contain at least one positive operator.Thus V/V0 is an ordered vector space with a conjugate-linearinvolution . The partial traceTrB maps each affine subspacein(V/V0)+ to a positive operator inTrB V. Likewise, we callthe assignment mapAV : TrB V V/V0 a positive mapifAVmaps every positive operator in TrB Vto an element of(V/V0)+.Lemma 3. The assignment map

AV : TrB

V V/

V0defined

in Eq. (7) is C-linear,-linear [i.e.,AV(X) =AV(X)],and trace-preserving.

Proof. C-linearity ofAVfollows from the linearity ofTrBV

.

Similarly, for any X B(HS HB),TrB(X+V0) = TrB(X)and (X +V0) = X +V0, so that TrB[(X +V0)] =TrB(X) = TrB(X). And ifTrB(X+ V0) = TrB(Y + V0),then X Y V0, so X +V0 = Y +V0. ThereforeAV(TrB(X))can only be X + V0 =AV(TrB(X)), so thatAV is-linear. Finally, Tr(AV(X)) = Tr(TrB[AV(X)]) =Tr(X) for any XTrB V, sothat AVis trace-preserving.

We are now able to define a uniqueC-linear dynamical mapfor the subsystem, presented in the following lemma.

Lemma 4. LetV B(HS HB)be a G-consistent subspace.For anyU G there is a unique mapVU : TrB V B(HS)such that the diagram in Figure 3 commutes, i.e., such that, for

any operatorX V,VU(TrBX) = TrB(U XU). That mapVU : TrB V B(HS), given byVU = TrB AdUAV is C-linear,-linear, and trace-preserving over the domainTrB V,and acts as the dynamical map for system states inTrB(DSB V) DS.Proof. The uniqueness ofVUis due to the uniqueness of the

assignment mapAVand the mapsAd0U :V/V0 B(HSHB)/ker TrB and Tr0B : B(HS HB)/ kerTrB B(HS).Since these maps are all C-linear and-linear, Uis as well.The-linearity of

V

Uis equivalently expressed asU beingHermiticity-preserving, which is sometimes shortened to justHermitian. For any stateS TrB(DSB V), the affinesubspace AV(S)must contain a state in DSB , so the transfor-mationSVU(S)reflects the unitary evolution of a validsystem-bath state in V, and therefore VU is the dynamicalmapfor such a state.

Definition 4 (Physical Domain). We call the convex set

TrB(DSB V) DS the physical domain because, as de-scribed in Lemma4,these are the system states for which the

V B(HS HB)

V/V0 B(HSHB)kerTrB

TrB

V B(

HS)

AdU

TrB

Ad0U

VU

AV Tr0B

TrBTrB

FIG. 3. For any G-consistent subspace V and unitary evolutionoperator U G U(HS HB), this commutative diagramuniquely defines theC-linear,-linear, trace-preserving map VU =Tr0B Ad

0UAVwhich acts as the time evolution operator for system

states in TrB (DSB V) DS .G-consistency is the condition that, foreveryU G, a mapAd0U : V/V0 B(HS HB)/ker TrB existswhich makes the top rectangle commute.

mapsVUact as the physical dynamical maps. This is calledthe compatibility domain in [6].

We stress a few key points concerning this construction:

1. The linearity of the mapsAV and VU is due to ourchoice to assume that the set of admissible initial

system-bath states belong to a linear subspace. Depend-

ing on the physical processes involved, other choices

are possible, leading to non-linear assignment maps and

non-linear evolution operatorsVU [2,14,20].

2. The mapVUdoes not generally act as the evolution op-erator for all system states, or even for all system states

inDS TrB V. Any state DS TrB Vwhich doesnot lie in the convex physical domain TrB(DSB V)is mapped by the assignment map to an affine subspace

ofB(HS HB) which contains no valid system-bathstates inDSB . Since the transformation byVUof sucha system state is not tied to the unitary evolution of

a valid system-bath state, it is empty of any physical

meaning. Such a mapVUshould never be describedwithout clearly indicating the physical domain of sys-

tem states on which it can meaningfully be applied. To

apply the map outside this domain is to overinterpret the

mathematics.

3. For eachU G , the definition ofVUdepends entirelyupon the G-consistent subspace V. Different consistentsubspaces, even if they include some shared fiducial

stateSB , will yield different mapsVU.

We illustrate these concepts in Fig.4.

E. Operator Sum Representations

The mapVUmay be extended to a C-linear, -linear, trace-preserving map VU onB(HS), for example by definingVUto be zero onB(HS)/ TrB V, the orthogonal complement ofTrB V. Such an extension, though arbitrary, may be desirable


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V

TrB

TrB V

AdU

AdUV

DSB

TrB

VU DS

FIG. 4. Schematic representation of the role of states and, in par-

ticular, the physical domainTrB(DSB V)in the descriptions of thespaces and maps involved in the present framework. In the upper two

diagrams, the ball representsDSB , the convex set of all system-bath

density matrices, and the ruled areas are the intersections V DSBand AdUV DSB . In the lower two diagrams, the ball representsthe convex set DS of all system density matrices while the ruledareas represent TrB(V DSB ), which is the physical domain, andTrBAdU(V DSB ), which is the image under

VU of the physical

domain.

because the resultingVUadmits an operator sum representa-tion (OSR) with real-valued coefficients

VU(X) =k

akEkXEk, (9)

forak R [21,22].In the case of an OSR such that ak = 1 for all k , we call

such a representation a Kraus OSR.

Lemma 5. The assignment map AV : TrB V V/V0associ-ated to any G-consistent subspace Vadmits an OSR with real-valued coefficients of the formAV(X) =

kakQkXQ

k +V0.

Proof. This may be shown, for example, by extending AV toa C-linear,-linear mapAV :B(HS) B(HS HB)/V0and applying Chois method toAVas follows. First, chooseorthonormal bases{|i}dSi=1 and{|}dB=1 forHS andHB ,observe thatAV idB(HS) is-linear so that the affine sub-spaceAV idB(HS)(|EE|) is spanned by Hermitian opera-tors, where |E= i |i |i HS HS. Choose any Her-mitian operatorA in this space, choosingA 0 if possible,and eigendecompose

AV idB(HS)(|EE|)A = k

ak|qkqk|. (10)

The operators Qk B(HS; HS HB)defined by

Qk =j,,i

j, , i|qk|j, i| (11)

yield an OSR for the extended assignment map:AV(X) =

kakQkXQk+ V0.

It follows that VU(X) =k, akEkXEk , whereEk =|U Qk B(HS ).

In special cases, this general recipe may not be the most

efficient for obtaining an OSR. For example, in the case of a

Kraus map where AV(X) = X B and B =

p||,an OSR forAV is given by the operators Q =p1|, so that VU(X) =

,E,XE

, where E, =

| U Q. It bears repeating that, regardless of these choicesfor constructing the OSR, the resulting map VUonly acts asthe subsystem dynamical map due to Uon system states in thephysical domainTrB(DSB V) DS .

IV. COMPLETE POSITIVITY

Having described, for each allowed joint system-bath uni-

tary evolution U G, the unique dynamical map U :TrB V B(HS) which is C-linear, -linear, and trace-preserving, we turn to the question of complete positivity.

A. Notions of Complete Positivity

BecauseAV and Uare generally defined on a subspaceofB(HS), rather than the full algebra, we may consider threegenerally nonequivalent definitions of complete positivity of

the subsystem dynamics; one which is essentially the originaldefinition of Stinespring [23], and two more that we introduce.

Definition 5 (Complete Positivity). LetKandHbe Hilbertspaces and letR B(K)be a self-adjointC-linear subspacespanned by positive operators. AC-linear,-linear mapF :R B(H)is

1. Completely Positive (CP) [23] ifFid :RB(HW)B(H HW)is a positive map for all finite dimensionalHilbert spaces HW, i.e., every positive operator in

RB(HW) = SpanC{AB : A R, B B(HW)} (12)

is mapped to a positive operator inB(H HW).HW may be thought of as the state space of a non-interacting, non-evolving witness system;

2. Completely Positively Extensible (CPE) ifF admits acompletely positive extensionF :B(K) B(H), i.e.,if there exists a CP map F :B(K) B(H) such thatFR

= F;

3. Completely Positively Zero Extensible (CPZE) ifF PR :B(K) B(H)is completely positive, where


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7

PR :B(K) R is the orthogonal projection ontoRwith respect to the Hilbert-Schmidt inner product.

It should be noted that a CPE map exhibits the property

that the CP extension, being a CP map defined on the entire

algebraB(HS ), admits a Kraus OSR [22,24]. As such, theexistence of a Kraus OSR for the map F :R B(H)is analternate characterization of CPE-ness.

It is straightforward to see that if a mapF :R B(H)is CPZE, then it is CPE, since the zero extension is just one

possible extension of F. Likewise if a map F : R B(H) is CPE, it is also CP, since restricting the CP exten-sion F :B(K) B(H) to the subspaceR B(K) doesnot break complete positivity. The following two theorems,

translated into this terminology of consistent subspaces and

CP/CPE/CPZE maps, may be considered partial converses of

these statements.

Theorem 1(Arveson[25]). If1TrB V, then every CP mapwith domainTrB Vcan be extended to a CP map onB(HS),i.e., every CP map onTrB Vis CPE.

Theorem 2 (Choi & Effros [26]). IfTrB V is a unital C

-subalgebra ofB(HS), then the orthogonal projectionPTrB V iscompletely positive, and therefore every CP map with domain

TrB Vis CPZE.That the property of being CPE does not necessarily imply

CPZE-ness is demonstrated by the following counterexample.

Example 2. SupposeHS is 3-dimensional with orthonormalbasis {|i}2i=0. Fix some DB and let

V= SpanC{|ij| : (0, 1)= (i, j)= (1, 0)}. (13)The assignment map AVassociated with thisVis CPE but notCPZE, demonstrating that the hypothesis of2thatTrB Vbe asubalgebra ofB(HS)is necessary.Proof. TrB Vis given by

TrB V= SpanC{|ij| : (0, 1)= (i, j)= (1, 0)}. (14)The orthogonal projection ontoTrB Vhas Choi matrix

Choi(PTrB V) =

0i,j2(0,1)=(i,j)=(1,0)

|ij| |ij|. (15)

This matrix has eigenvector |00 |11 + 2 |22 witheigenvalue 1

2 < 0, so it is not positive, and there-

forePTrB Vis not CP. This demonstrates that there exist self-adjoint subspacesTrB V B(HS)containing the identity andwhich are not C subalgebras for which the orthogonal pro-jectionPTrB V is not CP. The zero-extended assignment mapA=AV PTrB Vis simply A() = (PTrB V()) . There-fore, for anyX B(HS HW),

A idB(HW)(X) = (PTrB V id(X)) , (16)

so that A is not CP becausePTrB Vis not CP, and therefore

AVis not CPZE. However, for anyXTrB V B(HW),AV id(X) = X , (17)

which is a positive map for all finite-dimensional witnesses, so

that AVis CP. Moreover, since 1TrB V, Theorem 1impliesthat AVis CPE.

One observation to make about

G-consistency, that will be

useful in applying these notions of complete positivity, is that,when a G-consistent subspace Vis tensored with the operatoralgebra of a witness system as in the definition of complete

positivity above, the resulting operator space is still appropri-

ately consistent. In other words, G-consistency is stable in thefollowing sense:

Lemma 6. IfG U(HS HB) andV B(HS HB) is aG-consistent subspace, then for any finite-dimensional Hilbertspace HW, V B(HW) B(HS HB HW)is a GU(HW)-consistent subspace. In particular, it is a G 1HW -consistentsubspace.

Proof. First, observe that, ifV B(HSHB) is a G-consistentsubspace, then it is C-linear and spanned by states. It followsthat V B(HW)is C-linear. Furthermore, since Vand B(HW)are each spanned by states,V B(HW) is spanned by statesof the form with V DSB and DW.

It remains only to proveG U(HW)-consistency. To thatend, let X ker

TrB

VB(HW)

=V0 B(HW), U G ,

andV U(HW). Xmay be expanded as X =

i Yi Ziwhere {Yi} V0 and {Zi} B(HW). Then

TrB

(U V)X(U V) = i

TrB(U YiU) (V ZiV)

= 0 (18)

sinceYi V0 andV isG-consistent. Then(UV)X(UV) ker

TrB

B(HSHBHW)

, soV B(HW) isG

U(HW)-consistent. It follows trivially thatV B(HW) isG 1HW -consistent.

B. Defining Completely Positive Assignment Maps

LetV be aG-consistent subspace, letV0 = ker

TrBV

,

and letHW be a finite-dimensional Hilbert space. We wouldlike to apply the above notions of complete positivity not only

to the dynamical maps U, but also to the assignment map

AV : TrB V V/V0. To do so, we will need to identify the

positive elements withinV/V0 B(HW). To that end, webegin by establishing a natural isomorphism between V/V0 B(HW)and a space in which the positive elements are readilyidentifiable.

Lemma 7. The space V/V0B(HW)is naturally isomorphicto V B(HW)/V0 B(HW).Proof. We may construct a natural isomorphism

h:V B(HW)/V0 B(HW) V/V0 B(HW) (19)


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9

for anyY B(HSHW),

h1 A id(Y) = TrS[(

S1)Y(S 1)]Tr(2S )

+ V0 B(HW), (22)whereh :B(HS HB) B(HW)/V0 B(HW) B(HS

HB)/

V0

B(

HW)is the isomorphism constructed analogously

to that in Lemma 7. It follows that A idB(HW) is a posi-tive map for any finite-dimensional witness HW. Thus,AV isCPZE.

It is an open problem to find some generalizations of these

results to arbitraryG-consistent subspaces. In particular, Ex-ample2shows that Lemma 8does not extend to higher di-

mensionalTrB Vwithout modification.

C. Why Complete Positivity?

Definition5advocates different notions of complete posi-

tivity. In order to properly address the question of complete

positivity of system dynamics, we must first ask ourselves:

what aspects of complete positivity make it a physically inter-

esting property? There may be several valid answers to this

question and it is possible that there will be no general con-

sensus as to which is most compelling. Here are two:

1. The existence of a non-interacting, non-evolving wit-

ness (for which the initial system-witness state can be

entangled) should not be enough to break the positivity

of the evolution. This is frequently claimed as a man-

date for complete positivity.

2. As mentioned above, CP maps are sometimes repre-

sentable by Kraus OSRs() = EiEi , and KrausOSRs are ubiquitous in quantum information theory.For example, such an OSR can be thought of as a

convex combination () =

pii, where pi =

Tr(EiEi ) and i = EiE

i /pi. As such, it offers a

convenient interpretation as a non-selective measure-

ment of the system by the environment, yielding out-

come statei with probabilitypi.

1. Witnessed Positivity

If the importance of complete positivity in quantum dynam-

ics is to do with maintaining positivity in the presence of a

witness, i.e., point1above, then the key question, as we nowshow, is whether the assignment map is CP. If we consider

some dynamical mapVU : TrB V B(HS)forU G and afinite-dimensional witness Hilbert space HW , then

VU idB(HW) = TrB AdU1HW (AV idB(HW)) (23)

is the dynamical mapVB(HW)U1 for theG 1HW -consistent

subspace V B(HW)(see Lemma6). IfU idB(HW)is pos-itive for all finite-dimensionalHW, then, mathematically (by

V/V0 B(HW) B(HSHB)kerTrB B(HW)

TrB V B(HW) B(HS ) B(HW)

Ad0U1

0

VU id

0

AV id 0 Tr0B id 0

FIG. 7. It is possible for non-positivity in the assignment map (pos-

sibly tensored with the identity on a witness matrix algebra) to be

hidden by the positivity ofAdU and TrB for all U G, resultingin dynamical maps which appear positive or completely positive ac-

cording to the mathematical definitions. However, strictly speaking,

these dynamical maps do not satisfy the first answer to the question

why complete positivity?. An example is considered in Example3.

definition5), it is considered completely positive. However, it

is possible that there exists a witness state space HW such thatU idB(HW)is positive, while AV idB(HW)is non-positive,as illustrated in Fig.7.

Observation 1. In such a situation (whereU idB(HW)0but

AV

idB

(H

W)

0), the positivity of U

idB(H

W)should be considered non-physical as it arises from assign-

ing some states inTrB V B(HW) to non-positive operatorsinV B(HW) and evolving those non-positive operators.If the desire was to show that all system-witness states in

TrB V B(HW) DSW evolve to states inDSW, then we havenot fulfilled this goal. It is only satisfied if the assignment map

AVidB(HW) is itself a positive map. This obviously holdsfor all finite dimensional witnessesHW if and only ifAV iscompletely positive (CP).

We now give an explicit example of such non-physicalcom-

plete positivity.

Example 3. LetdS = dim HS

,dB = dim HB

, =

1

dSdB1and

=|| || for any states| HS and| HB.Then V= SpanC{, } is a U(HSHB)-consistent subspace.The corresponding assignment mapAV is not positive (seealso Fig.8), butU = TrB AdUAVis CPZE for allUU(HS HB).Proof. First, note thatAV is not positive, because for a 0and a

dS b


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10

0 dB1dSdB1

1

0

dSdBdBdSdB1

1

||

1

||

dS

1

FIG. 8. The system statesD S TrB Vfor the consistent subspaceV described in Lemma 3 are convex combinations of the states|| and (1 ||)/(dS 1), indicated by the thin blue linein this figure. The physical domain TrB (DSB V), indicated by

the thick red line, comprises convex combinations of || and(dB1 ||)/(dSdB 1). In particular,AV

(1 ||)/(dS

1)

= (dS )/(dS 1)is a unit trace,non-positiveoperator in Vand therefore not a state.

2. Kraus OSR for Dynamical Maps

On the other hand, if the existence of a Kraus OSR for the

dynamical maps (point 2 above) is of primary importance,

then the key question is not whether VU is CP, but ratherwhether VU is completely positively extensible (CPE) foreachU G. This is because

Lemma 9. For any G-consistent subspace Vand any U G,the dynamical mapVU admits a Kraus OSR if and only if

VU

is CPE.

Proof. IfVU is CPE, the CP extensionU is defined over

the entire algebra B(HS), so that Chois theorem [22] may beinvoked to conclude that Uadmits a Kraus OSR. Likewise,ifUmay be written in terms of a Kraus OSR, then that OSR

defines a CP extension U :B(HS) B(HS)so thatU isCPE.

It should be noted that, while the map Uis uniquely de-fined by the choice ofG-consistent subspace Vand the choiceofU

G, neither its CP extension U(if one exists) nor the

Kraus OSR thereof is uniquely defined. It should also be notedthat any Kraus OSR obtained in this way is still subject to the

restrictions mentioned earlier: it cannot be meaningfully ap-

plied to any state outside of the physical domain TrB (DSB V).

V. EXAMPLES

In this section we apply the general framework developed

above to earlier work on the topic of complete positivity. In

many cases we reinterpret this earlier work in light of the no-

tion ofG-consistency, CPE, and CPZE maps.

A. Kraus (1971)

The standard Kraus map [24] is built from aU(

HS

HB)-

consistent subspace of the formV =B(HS)B for somefixed bath stateB DB, so that the admissible initial statesSB {S B : S DS}=DSB Vare uncorrelated. Aswas mentioned at the end of SectionIIIE,the assignment map

A : TrB V V/V0 associated with this subspaceVadmitsan OSR A(X) = QXQwith operators Q= 1| where B =|| is the eigendecomposition ofB. This OSR is evidently completely positive. It followsthat, for any joint unitary evolution U U(HS HB), thesubdynamical mapVUis completely positive and admits theOSR

S(t) = VU[S(0)] = ,

ES(0)E (25a)

forE=| U Q=

| U| , (25b)

where the operation elementsEare the Kraus operators. Itis worth noting that, in this case, the physical domain in-

cludes all system states, i.e., TrB(DSB V) =DS. A the-orem of Pechukas [4], the proof of which was supplied by

Pechukas in the case of a 2-dimensional system, and by Jor-

dan, Shaji, and Sudarshan [6] for general finite-dimensional

systems, shows that the only U(HS HB)-consistent sub-spaces exhibiting this property (thatTrB(DSB V) =DS) arethose of the form V=B(HS) B for afixedB DB.

B. Pechukas (1994)

To illustrate the fact that subsystem dynamics need not be

completely positive, Pechukas offered an example involving a

two-state system and bath and aU(HS HB)-consistent sub-space defined by a C-linear,-linear assignment map of theform

S

S

eqS1

eqSB + eqSB

eqS1

S

/2 (26)

where eqS = TrB(eqSB ) are fixed equilibrium states and

eq

S

> 0 (not just

0). IfeqSB is nota tensor product state,and the domain of this map is taken to be all ofB(HS), thenthe image will be a well-defined U(HSHB)-consistent sub-space ofB(HS HB). However, the assignment map cannotbe positive under these assumptions, let alone completely pos-

itive, because pure statesS are not mapped to product statesS B as a positive assignment map must do (this last pointmay be seen as a consequence of the fact that a bipartite state

SB with a pure reduced state S possesses zero mutual in-formation, i.e., saturates subadditivity of the von Neumann

entropy [27], and must therefore be a product state [28]).


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11

C. Alicki (1995)

Commenting on Pechukas theorem concerning complete

positivity, Alicki [15] suggested abandoning either consis-

tency (i.e. TrB AV = id) or positivity of the assignmentmap, rather than giving up complete positivity of the dynam-

ical maps. To illustrate the possible loss of consistency, he

describes a scheme for turning a CP map T :

B(

HS)

B(HS) into a CPE assignment map with domain given bythe fixed points ofT. SupposeT is represented asT(S) =

n VnSVn and fix some

eqSB DSB . Then one can postulate

a C-linear assignment map

AV(S) =n

VnSVn

TrS[(VnVn1)eqSB ]

Tr[(VnVn1)eqSB ](27a)

=ijn

QijnSQijn (27b)

whereQijn B(HS; HS HB)is the operator

Qijn = Vni|(Vn

1)eqSB |j

Tr[(VnVn1)eqSB ](28)

so thatAVis CPE. SinceTrB AV = T, the correct domainof definition of this assignment map is the C-linear space offixed points ofT inB(HS). LettingV be the image of thisdomain through AV, we may define aU(HS HB)-consistentsubspace byV = SpanC(DSB V). Since the assignmentmapAV is CPE, it is clear that the dynamical maps VU :TrB V B(HS) are also CPE for allU U(HS HB). Itis also clear that, although this construction is described by

Alicki as involving a loss of consistency, it is entirely valid

within the framework ofG-consistent subspaces. It shouldbe noted, however, that this construction depends not only onthe map Tand the equilibrium state eqSB , but also on therepresentationofT. In general, two equivalent OSRs T =

Vn Vn =

Rm Rm will not yield the same consistentsubspace V, thus will not yield the same assignment map AVor dynamical mapsVU.

Alicki goes on to consider a generally nonlinear assignment

map

A(S ) =

nPnTrS [eqSB (Pn1)]

Tr[eqSB (Pn1)] (29)

where again eqSB DSB is a fixed equilibrium state, andS =nPn is the spectral decomposition of S. ThenTrB A = id on all ofDS , butA :DS DSB is convex-linear if and only ifeqSB is a tensor product state. In eithercase, the image of this assignment map S=A(DS) DSB isa validU(HSHB)-consistent subset as described in SectionII, contained within the subset of zero-discord states in DSB [8]since any state in the image ofA is of the form nPn nfor orthogonal projectors{Pn} and states{n} DSB . Assuch,eqSB Sonly if it has this very special form. It is clearthat the resulting dynamical maps SUwill be positive, but theywill also generally notbe convex-linear. However, the phys-

ical meaning ofeqSB is unclear wheneqSB / S, and moreover

SU id andA id have no meaning when SU andA arenonlinear, so that complete positivity has no meaning either.

D. Stelmachovic and Buzek (2001)

In addition to Example 1, Stelmachovic and Buzek [14]

offer a second example which demonstrates that, if we fix

any S(0), S (T) DS, then it is possible to chooseHB, aU(HS HB)-consistent subspaceV B(HS HB), and aunitary transformationU U(HS HB) such that the cor-responding dynamical mapVUtakesS (0)toS(T). Indeed,they show this can be done with a Kraus-like consistent sub-

spaceV =B(HS)B, whereHB HS andB = S(T),and where the unitary transformation U=

|ij| |ji| isthe swap operator U| |=| |.

E. Jordan, Shaji, and Sudarshan (2004)

Jordan, Shaji, and Sudarshan[6] develop a two qubit ex-

ample (one system qubit and one bath qubit) in detail in

order to examine quantum dynamics in the presence of ini-

tial entanglement. Their example can be expressed in terms

of aG-consistent subspace, whereG U(HS HB ) is theone-parameter subsemigroup generated by the Hamiltonian

H = (/2)Z X where Z and X are Pauli operators.TheG-consistent subspaceV B(HS HB) is then the14-dimensional orthogonal complement (with respect to the

Hilbert-Schmidt inner product) ofSpanC{1 + Y

X, 1

X X} for some fixed choice of, (1, 1). The kernelV0 = ker

TrB

V

is then the 10-dimensional subspace

V0 = SpanC{1 X, 1 Y, 1 Z, X Y, X Z, Y Y,Y Z, Z X, Z Y, Z Z}. (30)

It is straightforward to check thatG V0 ker TrB as re-quired. The quotient spaceV/V0 is then 4-dimensional, sothatTrB V =B(HS). However, as pointed out in [6], unless = = 0, the assignment mapAV : TrB V V/V0 isnot positive. If it were positive, thenAV(|00|)would be anaffine subspace ofB(HS HB)containing a state of the form

|0

0

| B for some B

DB. But unless = = 0, no state

of that form lies inVsince such states are not orthogonal to1 + Y Xand1 X X. It follows that the affine sub-space AV(|00|)contains only unit trace, non-positive opera-tors, so that AVis non-positive. Moreover it is shown that theVU(|00|) need not be positive forU G and thereforeVUis not completely positive for all U G. On the other hand,if = = 0, thenV contains theU(HS HB)-consistentsubspace V =B(HS)1 for whichV = V V0. In thatcase, the assignment map is not only positive, but completely

positive.


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12

F. Carteret, Terno, and Zyczkowski (2008)

Carteretet al. [7], consider an example with a one qubit

system and bath and a U(HS HB)-consistent subspace ofthe form

V= SpanC

11, 2 1, 31, 1 + a3

i=1i i

,

(31)

where{i} are the Pauli operators and1 < a < 1/3 is afixed constant. Fora= 0, the associated assignment map AVis non-positive, admitting a physical domain TrB(DSB V)which, within the Bloch sphere, is the ball centered at 1/2

with radius

(1 + a)(1 3a)for a0 and with radius1 +a for a 0 (this latter point about a < 0 was missed in[7]). The authors note that the dynamical map VUis triviallycompletely positive for anyU U(HS HB)that commuteswith

i i. They then consider unitary transformations

of the form

U= 1 0 0 00 cos sin 00 sin cos 00 0 0 1

, (32)pointing out that the = /4 case yields a CP dynamicalmapVUand incorrectly claiming that = yields a non-CPdynamical map (this case also yields a CP VU, because for = ,U = z z commutes with

i i). However,

it may be shown that the Choi matrix for the dynamical map

VUassociated with such aUis given by

Choi(VU) =1

2

1 + cos2()z z+ a sin(2)1 z

+ cos()x x y y (33)so that other values for and a can yield non-CP maps, forexample the case = /6anda >1/2

3.

G. Rodrguez-Rosario, Modi, Kuah, Shaji, and Sudarshan

(2008)

Rodrguez-Rosario et al. [9], consideredU(HS HB)-consistent subspaces of the formV = SpanC{|ii| i},where{|i} HS is an orthonormal basis for the systemand{i} DB are bath states. ThenV DSB comprisesonly zero-discord states, i.e., those that exhibit only classi-

cal correlations with respect to some measurement basis [8].They showed that, for such aVand anyU U(HS HB),the corresponding dynamical mapVUis always CPE. In fact,more is true: the assignment mapAV : TrB V V isCPZE since AVPTrB V(X) =

i, EiXE

i,is CP, where

PTrB V :B(HS ) TrB Vis the orthogonal projection ontoTrB Vand Ei,=|ii|i | for some orthonormal basis{|} HB .

It should be noted that, for these dim(HS )-dimensionalconsistent subspaces V, the linear domain of definition, TrB V,

of the dynamical maps consists only of operators which are

diagonal in the fixed basis{|i}. As a result, they violate theauthors own requirement that the dynamical map is well-

defined if it is positive on a large enough set of states such

that it can be extended by linearity to all states of the system.

H. Shabani and Lidar (2009)

Shabani and Lidar [10] expanded the class of consistent

subspaces from the zero-discord subspaces considered in [9],

to the class of all validU(HS HB )-consistent subspaces ofthe form

V= SpanC{|ij| ij}, (34)where{|i} is an orthonormal basis forHS and{ij} B(HB) are bath operators. Ref. [10] showed that, within thisclass of consistent subspaces, the dynamical maps VU areCPE for all U U(HS HB)if and only ifVDSB comprisesonly zero-discord states.

In light of the framework developed here, it is clear thatthe results of [10] do notamount in general to necessary and

sufficient conditions for complete positivity, but are restricted

to the class of consistent subspaces of the form (34). This lack

of generality is a point that has been largely overlooked by the

community, including by two of the present authors. Thus the

present work amounts to a retraction and correction of some

of the claims made in[10]. The issue is that, while the form

(34)is general enough to describe any given stateSB , it is notgeneral enough to describe everyG-consistent subspace. Asa simple example, the2-dimensionalU(HS HB)-consistentsubspace described in Example3 is not of the form (34), nor

is it contained in any subspace of this form.

Let us also comment on [11], which does not address the

question of whether the maps it constructs are CP or not, but

states in its Theorem 3, that the most general form of a quan-

tum dynamical process irrespective of the initial system-bath

state (in particular arbitrarily entangled initial states are pos-

sible) is always reducible to a Hermitian map from the ini-

tial system to the final system state. We note that this only

holds forG-consistent subspaces; when the set of admissibleinitial statesS is notG-consistent, there is no subsystemS-dynamical map (for someU G).

I. Brodutch, Datta, Modi, Rivas, and Rodrguez-Rosario

(2013)

To demonstrate that vanishing discord is not necessary for

complete positivity, Brodutchet al.[12] showed thatU(HSHB)-consistent subspacesVexist for which almost all statesinV DSB are discordant, but the dynamical maps VU arenonetheless CPZE (and therefore CPE) for allU U(HSHB). The counterexamples described in [12] are subspacesof the form V= SpanC{01}{ |ii| i}ni=2 where 01 =|00| 0+|11|1+|++| +, {|i} is an orthornomalbasis for HS , and {+, 0, . . . , n} DB are bath states.


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13

It should be mentioned that, while Brodutch, et al. were

correct to criticize the generality of the conclusion concerning

the necessity of vanishing discord for complete positivity in

[10], their claim of nonlinearity appearing in [10] is not valid.

By expressing the construction of [10] in the language ofG-consistent subspaces as in(34) it should be clear that, while

the initial system-bath states lie in a constrained subspace

as they always must forG-consistency, i.e., to obtain well-defined dynamical maps (see Lemma2) all spaces and mapsin the Shabani-Lidar framework are C-linear.

J. Buscemi (2013)

Buscemi [13] has devised a general method for construct-

ingU(HS HB)-consistent subspaces Vwhich yield CPE dy-namical mapsVUfor all U U(HS HB). The technique in-volves first introducing a reference subsystem R and choos-ing a fixed tripartite stateRSB that saturates the strong subad-ditivity inequality for von Neumann entropy; in other words

the conditional mutual information,I(R : B|S), is zero. Asubspace V

B(HS HB)is then formed asV ={TrR[(L 1SB )RSB ] : L B(HR)} (35)

and the corresponding U(HS HB)-consistent subspace isV = SpanC(DSB V). Buscemi proves that such a con-struction always leads to CPE dynamical maps. While there

is no claim that allU(HS HB)-consistent subspaces yieldingCPE dynamical maps must be formed in this way, the tech-

nique does yield a rich set of examples, including the zero-

discord subspaces of Rodrguez-Rosario et al.[9] and the ex-

amples of Brodutchet al. [12], as well as examples featuring

highly entangled states. It should also be noted that, although

Buscemis method predicts that the dynamical maps will be

CPE in these examples, it does not appear to predict the factthat the assignment maps are CP (indeed, CPZE) in the exam-

ples of [9] and[12].

VI. SUMMARY & OPEN QUESTIONS

We have formulated a general framework for open quan-

tum system dynamics in the presence of initial correlations

between system and bath. It is based around the notion of

G-consistent operator spacesV B(HS HB) representingthe set of admissible initial system-bath states.G-consistencyis necessary to ensure that initial system-bath states with the

same reduced state on the system evolve under all admissible

unitary operatorsU G to system-bath states with the samereduced state on the system, ensuring the dynamical maps

VU are well-defined. Once such aG-consistent subspace ischosen, related concepts like the assignment map and the dy-

namical maps are uniquely defined. In general, the dynamical

maps may not be applied to arbitrary system states, but only

to those in the physical domainTrB(DSB V). Interestingly,non-positive assignment maps can give rise to completely pos-itive dynamical maps, but this is not a situation that is physi-

cally acceptable since it involves the propagation of unphysi-

cal system-bath states.

TheseG-consistent subspaces may be highly constrained,with the result that complete positivity of the assignment and

dynamicalmaps may be defined in several distinct ways, relat-

ing to different interpretations and desirable features of com-

plete positivity for open system dynamics. Indeed, we have

identified two new types of completely positive maps, which

we have called CPE and CPZE. It is important to distinguish

between these flavors of complete positivity and whether one

is considering complete positivity of the assignment map or

of the dynamical maps. We have shown that various spe-

cial cases considered in earlier work can be unified within the

framework ofG-consistency, and that earlier discussions ofcomplete positivity are better understood using CPE or CPZE

maps.

Having laid out this framework ofG-consistent subspaces,we have left many important questions open. The formalism

is general enough to encompass other proposed frameworks,

most notably the assignment map framework introduced by

Pechukas [4], and various worked examples. These various

results and special cases illustrate perhaps the biggest open

question: what structural features characterize the subclass of

G-consistent subspaces which give rise to completely positivedynamics (in any reasonable flavor)? Recent work [9, 10]

suggested that the answer might lie with quantum discord.However, more recently it became apparent that the focus on

quantum discord is too restrictive[12,13], and here we have

shown that earlier work has illuminated some small corners of

the space ofG-consistent subspaces, but a complete analysisremains to be seen.

ACKNOWLEDGMENTS

This research was supported by the ARO MURI grant

W911NF-11-1-0268 and by NSF grant numbers PHY-969969

and PHY-803304.

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a general framework for complete positivity - jason m. dominy, alireza shabani, daniel a. lidar

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