Geometry of two-qubit states with negative conditional entropy

Nicolai Friis,1, Sridhar Bulusu,2 and Reinhold A. Bertlmann2,

1Institute for Theoretical Physics, University of Innsbruck, Technikerstrae 21a, A-6020 Innsbruck, Austria2Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria

We review the geometric features of negative conditional entropy and the properties of the con-ditional amplitude operator proposed by Cerf and Adami for two qubit states in comparison withentanglement and nonlocality of the states. We identify the region of negative conditional entropyin the tetrahedron of locally maximally mixed two-qubit states. Within this set of states, negativeconditional entropy implies nonlocality and entanglement, but not vice versa, and we show thatthe Cerf-Adami conditional amplitude operator provides an entanglement witness equivalent to thePeres-Horodecki criterion. Outside of the tetrahedron this equivalence is generally not true.

PACS numbers: 03.67.-a, 03.65.Ud, 03.67.Hk

I. INTRODUCTION

The feature of entanglement is the basis for many fasci-nating phenomena in quantum information and quantumcommunication, such as quantum teleportation [1, 2] orquantum cryptography [35]. Although the division ofquantum states into entangled and separable states iswell-defined mathematically, checking whether a givenstate is entangled or not often proves to be extraordi-narily difficult. Consequently, a plethora of inequivalentcriteria and measures is available for the detection andclassification of entanglement [68]. The spectrum ofavailable methods ranges from entanglement monotonessuch as the concurrence [911] or negativity [12], and ge-ometric entanglement detection criteria in the form of so-called entanglement witnesses [1316], to measures thatdirectly quantify the utility of a state for specific tasksrequiring entanglement. One of the most prominent suchtasks, which challenges our preconceptions of the realityof nature [17], is the test of a Bell inequality [18, 19], dis-tinguishing between so-called local and nonlocal states,for which the inequality is satisfied or violated, respec-tively. However, while entanglement is required to violatea Bell inequality, entanglement and nonlocality are notthe same concepts. As discovered by Werner [20], certainmixed states, albeit still being entangled, cannot be usedto violate a Bell inequality and hence behave like strictlylocal states.

In contrast to methods that directly relate en-tanglement to physically measurable quantities standinformation-theoretic approaches based on the entropiesof quantum states. In both classical and quantum in-formation theory entropies play a crucial role. Quitegenerally, entropy represents the degree of uncertainty the lack of knowledge about a (quantum) sys-tem. More specifically, the von Neumann entropy of aquantum state can be interpreted [21, 22] as the mini-mal amount of information necessary to fully specify the

nicolai.friis@uibk.ac.at reinhold.bertlmann@univie.ac.at

state, be it separable or entangled. For the quantificationof the correlations between two subsystems A and B, twoparticularly interesting entropies are the mutual entropy(or mutual information) S(A : B) and the conditionalentropy S(A|B). In analogy to the classical case, themutual entropy S(A :B) corresponds to the amount ofinformation contained in the joint state that exceeds theinformation locally available to A and B, i.e., S(A :B)is a measure for the degree of correlation between sub-systems A and B . On the other hand, S(A|B) is theentropy of the state of subsystem A conditioned on theknowledge of the state of subsystem B. In a series ofpapers [2327] investigating the conditional entropy andthe mutual entropy by means of so-called mutual andconditional amplitude operators (CAO), Cerf and Adamiconcluded that the quantum conditional entropy incontrast to its classical counterpart can become nega-tive for entangled states. This provides a connection be-tween quantum nonseparability and conditional entropy,or mutual entropy that we wish to investigate further inthis article.

The purpose of our article is hence to review the geom-etry of quantum states with negative conditional entropyand to compare it with the different regions of nonlo-cality, entanglement and separability. In particular, wewant to focus on the paradigmatic case of two qubits,which is one of the very few examples where the differ-ent methods for detection and quantification of entan-glement and nonlocality described above are practicallycomputable and can be compared both numerically andgeometrically. In this sense, albeit being a system ofcomparatively small complexity, the two-qubit case is ofhigh significance, since it serves as a guiding examplefor developing the geometric understanding and intuitionnecessary to study more complicated systems.

Our investigation confirms that for the interesting classof locally maximally mixed states, the requirement ofnegative conditional entropy is a strictly stronger con-straint than that of nonlocality, i.e., all states withnegative conditional entropy are nonlocal, and there-fore entangled, but the converse statements do nothold. We then consider an entanglement criterion basedon the Cerf-Adami conditional amplitude operator and

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show that it is equivalent to the Peres-Horodecki crite-rion [13, 28] for the set of locally maximally mixed states,but not for all two-qubit states.

This paper is structured as follows. We begin witha pedagogical review of the basic methods in Sect. II,discussing the geometric entanglement and separabilitycharacteristics in Sects. II A and II B, the boundary be-tween local and nonlocal states in Sects. II C and II D,and the entropic correlation measures in Sects. II Eand II F. We then present the results of our investiga-tion in Sect. III, where we discuss the geometric aspectsof the conditional entropy within the set of locally max-imally mixed states in Sect. III A, provide an examplefor the general inequivalence of the Peres-Horodecki andthe CAO criterion in Sect. III B, and discuss extensionsto generalized entropies in Sect. III C. Finally, we drawconclusions in Sect. IV.

II. METHODS

In this section we will provide a pedagogical review ofthe methods for the detection and quantification of en-tanglement and nonlocality relevant to this study. Thereader already well familiar with the geometry of sepa-rable, entangled, and nonlocal states for two-qubits mayskip directly to Sect. III, where we present our results.

A. Entanglement & Separability

Quantum states are described by density operators ,i.e., positive semi-definite ( 0), hermitean ( = ,which of course follows from positivity) operators withunit trace, Tr() = 1. These operators form a convex

subset H L(H) in the Hilbert-Schmidt space L(H) oflinear operators over the Hilbert space H of pure states.Given a bipartition of the Hilbert space into two sub-systems A and B with respect to the tensor product,H = H

AH

B, one may classify the quantum states into

separable and entangled states. The set S of separablestates is defined by the convex (and compact) hull ofproduct states

S ={ =

n

pn An Bn | 0 pn 1 ,n

pn = 1}, (1)

where An and Bn are density operators in HA L(HA)

and HB L(H

B), respectively. In contrast, any state

that is not separable, i.e., which cannot be expressed as aconvex combination of product states, is called entangled.The set Sc of entangled states hence forms the comple-ment of the set of separable states, such that SSc = H.

Here we would like to emphasize that the characteri-zation of a given state as being entangled or separablevery much depends on the choice of factorizing the alge-bra of the corresponding density matrix [29, 30]. From apractical point of view, this choice of bipartition is of-ten suggested by the experimental setup, e.g., by the

spatial separation of the observers Alice and Bob cor-responding to subsystems A and B, respectively. Fromthe perspective of a theorist on the other hand, one hasa freedom to choose the bipartition into two subsystems.While a given density operator may well be separable

with respect to the decomposition H = HA H

B, it

may be entangled with respect to another factorization

HA HB . Since such a different choice of bipartition

corresponds to a change of basis in H, it can be repre-sented by a (global) unitary transformation. As shownin Ref. [29], every separable pure state admits a unitaryoperator transforming it to an entangled state, and viceversa. Interestingly, for mixed states this switch betweenseparability and entanglement is only possible above acertain bound of purity. This implies that there existquantum states which are separable with respect to allpossible factorizations of the composite system into sub-systems. This is the case if UU remains separable forany unitary transformation U . Such states are called ab-solutely separable states [3133]. Geometrically one maythink of the absolutely separable states as a convex andcompact [34] subset A of the separable states S, muchlike S forms a convex subset of all states. In particu-lar, when dim(H

A) = dim(H

B) = d, one may inscribe

a ball of maximal radius rmax = 1/(dd2 1

)into the

set S, where the distance of a state from the centralmaximally mixed state mix = 1d2/d

2 is measured by the

trace distance || mix|| =

Tr( mix)2. All stateswithin this so-called Kus-Zyczkowski ball [31] are separa-ble. Moreover, since the condition r rmax translates tothe purity as Tr(2) 1/

(d2 1

)and (global) unitaries

leave the purity invariant, all states within this maximalball are also absolutely separable. However, note thatnot all absolutely separable states lie within this ball,see, e.g., Refs. [29, 31, 35].

The convex nesting hierarchy A S H L(H)holds for (bipartite) quantum systems of arbitrary di-

mensions dim(HA

) = dA

and dim(HB

) = dB

. The densityoperators of such systems can be written in a generalizedBloch-Fano decomposition [36, 37] as

=1

dAdB

(1

AB+

d2A1

m=1

am A

m 1B +d2B1

n=1

bn 1A Bn

+

d2A1

m=1

d2B1

n=1

tmn A

m Bn), (2)

where the hermitean operators im for i = A,B are thegeneralizations of the Pauli matrices, i.e., they are or-thogonal in the sense that Tr(im

in) = 2mn, and trace-

less, Tr(im) = 0, and they coincide with the Pauli ma-trices for dimension 2. The coefficients am, bn R arethe components of the generalized Bloch vectors ~a and~b of the subsystems A and B, respectively, which com-pletely determine the reduced states

A= Tr

B() and

B

= TrA

(). The real coefficients tmn are the compo-nents of the so-called correlation tensor. Note that am,

3

bn, and tmn cannot be chosen completely independently,but are jointly constrained by the positivity of .

An interesting subset of the state space is given by theset W of locally maximally mixed states or Weyl states,that is, the set of quantum systems with vanishing Blochvectors, am = bn = 0 m,n such that A = 1A/dA andB = 1B/dB. The set W contains all the maximally en-tangled states (for which the marginals A and B aremaximally mixed) and the uncorrelated maximally mixedstate = 1AB/(dAdB), and all states in W are fully de-termined by their correlation tensors t = (tmn). FordA = dB = d, the singular value decomposition of thecorrelation tensor allows bringing t to a diagonal formt = diag{tn} using two orthogonal transformations R1and R2, such that R1tR2 = t. Moreover, these orthog-onal transformations can be realized by local unitariesU1 U2, which do not change the entanglement (or theentropy) of the state. This means that, up to local uni-taries, all Weyl states for d

A= d

B= d can be represented

by vectors in Rd21 with components tn and density op-erators

=1

d2

(1AB +

d21n=1

tn A

n Bn). (3)

The vector components tn are constrained by the pos-itivity of , and the allowed vectors map out a con-

vex set in Rd21. In the case of two qubits, i.e., whend = 2, the Weyl states can hence be nicely illustratedin R3, where 1 tn +1 and (up to local unitaries)the set W forms a tetrahedron, shown in Fig. 1. Thefour Bell states | = 1

2

(|00 |11

)and | =

12

(|01 |10

), where |0 and |1 are the eigenstates

of the third Pauli matrix with eigenvalues +1 and 1,respectively, are located at the four corners of the tetra-hedron at (t1, t2, t3) = (1,1, 1) (|+), (1, 1, 1) (|),(1, 1,1) (|+), and (1,1,1) (|), while themaximally mixed state mix =

1414 is located at the ori-

gin (0, 0, 0).The region of separability is determined by the so-

called positive partial transpose (PPT) criterion estab-lished by Peres [28] and the Horodecki family [13]. Thecriterion allows to identify bipartite quantum states asentangled, if the partial transposition of their density op-erator does not yield a positive operator. Given a densitymatrix in the general Bloch decomposition of Eq. (2),the partial transposition corresponds to the transpositionof the (generalized) Pauli operators in one of the subsys-tems, e.g., (An)kl (An)lk. In 22 and 23 dimensionsthe PPT criterion is necessary and sufficient to detectentanglement, but in higher dimensions entangled statescan have a positive partial transpose. In our exampleof the Weyl states, the positivity constraint of the par-tial transpose identifies the separable Weyl states to liewithin a double pyramid (see Refs. [15, 38, 39]) with cor-ners at (t1, t2, t3) = (1, 0, 0), (0,1, 0) and (0, 0,1),as illustrated in Fig. 1. The maximal Kus-Zyczkowskiball [31] of absolutely separable states lies within the

FIG. 1. Tetrahedron of Weyl states. All states in the setW can (up to local unitaries) be geometrically representedas a tetrahedron spanned by the four Bell states |, |.The singular values (t1, t2, t3) of the correlation tensor serveas coordinates. The set S of separable states forms a doublepyramid (blue) and the entangled states are located in theremaining corners outside. The maximally mixed state mixis located at the origin, and the maximal ball of absolutelyseparable states (purple) around mix is contained within S,but touches the double pyramid at the central points of itseight faces. The states located at the tips of the double pyra-mid, for instance, the Narnhofer state N at the coordinates(1, 0, 0), are the separable states with maximal purity. Allstates that cannot violate the CHSH-Bell inequality lie withinthe dark-yellow, curved surfaces drawn outside the doublepyramid, which includes also some entangled states, whereasall states with positive conditional entropy lie within the out-ermost curved surfaces (red), as discussed in Section III A.The solid red line indicates the Werner states W of Eq. (20),while the dashed red line represents the subset of Gisin stateswith = /4, see Eq. (23).

double pyramid and touches the faces of the pyramidsat the points where |t1| = |t2| = |t3| = 13 , four of whichmark the closest separable states to the four Bell states.The entangled Weyl states are located in the four cornersof the tetrahedron outside the double pyramid, extendingfrom the separable states to the maximally entangled Bellstates at the tips.

B. Entanglement Witnesses

The geometric picture that presents itself for the two-qubit Weyl states, i.e., the separation of separable fromentangled states by planes (the faces of the double pyra-mid), can indeed be generalized to arbitrary dimensions.Owing to the convex structure of the set S and the Hahn-Banach theorem (see, e.g., Ref. [40, p. 75]), one maydefine so-called entanglement witness operators via thefollowing theorem [1315].

4

Theorem II.1 (Entanglement Witness Theorem).

A state is entangled if and only if there exists a her-mitian operator W an entanglement witness suchthat

(W |)HS < 0 , (4a)(W |sep)HS 0 sep S, (4b)

where the Hilbert-Schmidt inner product is defined as(A|B)

HS:= Tr

(AB

)for any A,B L(H) and S de-

notes the set of separable states from Eq. (1).

Geometrically, a witness operator for a given state defines a hyperplane in the Hilbert space H that sep-arates the set S from the point representing the state. An entanglement witness Wopt is called optimal if inaddition to the requirements of Eq. (4) there exists a sep-arable state sep S such that (Wopt |sep)HS = 0. Theoperator Wopt defines a tangent plane to the convex setof separable states.

On the other hand, the minimal (trace) distance ofan entangled state from the set S, the Hilbert-Schmidtmeasure D() given by

D() = minsepS

|| sep|| = || 0||

=

( 0 | 0)HS (5)

can be viewed as a measure of entanglement, where thestate 0 is called the nearest separable state to . Aninteresting connection between the Hilbert-Schmidt mea-sure and the entanglement witness inequality arises whenwe define the maximal violation of the entanglement wit-ness inequality as

B() := maxW

[minsepS

(W |sep)HS (W |)HS]. (6)

Here, the minimum is taken over all separable states andthe maximum over all possible entanglement witnessesW = W L(H) that are suitably normalized, i.e.,||W || = 1. With this, we can formulate the followingTheorem [15].

Theorem II.2 (Bertlmann-Narnhofer-Thirring).

The maximal violation of the entanglement witness in-equality is equal to the Hilbert-Schmidt measure, i.e.,

B() = D() , (7)

and is achieved for 0 and W Wopt, where theoptimal entanglement witness is given by

Wopt =(0 | 0)HS ( 0)

|| 0||. (8)

As an example, consider the totally antisymmetric Bellstate

= || = 14(12 12 ~ ~

), (9)

where ~ ~ =3n=1 n n is used as a shorthand

and the n are the usual Pauli matrices. The optimal

entanglement witness W

opt for this state is given by

W

opt =1

23

(12 12 + ~ ~

). (10)

The Hilbert-Schmidt product with the entangled state ofEq. (9) yields

(W

opt |)HS = Tr(W

opt) = 1

3< 0 , (11)

as required for an entanglement witness in inequal-ity (4a). To confirm that also the second inequality (4b)is satisfied, first note that any separable state can bewritten as a convex combination of product states with

local Bloch vectors ~ai and ~bi, that is,

sep =i

pi Ai Bi =i

pi4

(1A + ~ai~

)(1B +

~bi~).

(12)

The correlation tensor of any separable state hence hascomponents tmn =

i pia

mi b

ni and its trace is given by

Tr(t) =i

pi ~ai ~bi =i

pi|~ai| |~bi| cos i , (13)

where i is the angle between the Bloch vectors ~ai and~bi. For two qubits we have |~ai|, |~bi| 1 and hence1 Tr(t) 1. If we then compute the Hilbert-Schmidtproduct of W

opt with an arbitrary separable state wetherefore find

(W

opt |sep)HS = Tr(W

optsep)

=1

2

3

(1 + Tr(t)

) 0 ,

(14)

as required in (4b). The nearest separable state 0, for

which (W

opt |0)HS = 0 is given by

0 =14

(12 12 13~ ~

), (15)

which can be seen to lie on the face (closest to the cornerrepresenting |) of the double pyramid illustrated inFig. 1. The optimal witness W

opt from Eq. (8) hencedefines the plane containing this face of the pyramid.Finally, we can now easily compute the Hilbert-Schmidtmeasure D() from Eq. (5) and compare it to Eq. (11),obtaining

D() = || 0|| =13

= (W

opt |)HS . (16)

Since minsepS (W |sep)HS = (W

opt |0)HS = 0 we cantherefore conclude from Eq. (6) that, indeed, D() =B(), as claimed in Theorem II.2.

5

C. Bell Inequalities & Nonlocality

Let us now turn from the geometric aspects of en-tanglement to the property referred to as nonlocality.A quantum state is said to be nonlocal if it allows forthe violation of a Bell inequality [18, 19]. This termi-nology originates in Bells locality hypothesis for localhidden-variable theories. In such models, the possiblemeasurement outcomes A and B of two (distant) partiesare determined by a hidden parameter . These theoriesare local in the sense that the values A = A(,~a) and

B = B(,~b) depend on their local measurement settings

~a and ~b, respectively, but not on the setting of theother party. As can be shown [18, 19], combinations ofexpectation values of local hidden-variable models areconstrained by Bell inequalities, which may be violatedby certain (entangled) quantum states.

To be more specific, we consider the Clauser-Horne-Shimony-Holt (CHSH) inequality [19, 41] which, in anal-ogy to the entanglement witness inequalities (Theo-rem II.1), can be written as

(21 BCHSH |loc)HS 0 , (17)

where the CHSH-Bell operator BCHSH is given by

BCHSH = ~a ~A (~b~b) ~B + ~a ~A (~b+~b) ~B ,

and the vectors ~a,~b, ~a,~b R3 denote the measurementdirections. All local (and all separable) states loc sat-isfy the inequality (17). On the other hand, for nonlo-cal states, like the maximally entangled Bell state ofEq. (9), the CHSH inequality can be violated for somechoice of measurement directions. That is, there exist

states nonloc and settings ~a,~b, ~a,~b such that

(21 BCHSH |nonloc)HS < 0 , (18)

mirroring Eq. (4a) in Theorem II.1. Since all separablestates are local, the operator (21 BCHSH) can be seenas a witness for nonlocality, and as a (non-optimal)entanglement witness. Unfortunately, this witness is notuseful for arbitrary measurement settings. However, thecumbersome task of explicitly determining the directions

~a,~b, ~a,~b can be circumvented via another powerfultheorem by the Horodecki family [42].

Theorem II.3 (CHSH operator criterion).

Let be the density operator of a two-qubit state withcorrelation tensor t = (tmn), see Eq. (2), and let 1 and2 be the two largest eigenvalues of M = t

T t. The state

is nonlocal if BmaxCHSH, the maximally possible expectationvalue of the Bell-CHSH operator, is larger than 2, i.e., if

BmaxCHSH = max~a,~b,~a,~b

BCHSH = 21 + 2 > 2 . (19)

Using the CHSH operator criterion, it is straightfor-ward to verify that there are quantum states that are en-tangled, but nonetheless local in the sense of the CHSHinequality. This is best exemplified by a certain family ofbipartite mixed states, the so-called Werner states [20],given by

W = + 14 (1 )14 =

14

(12 12 ~ ~

).

(20)

For the parameter range 0 1, the state W()can be viewed as an incoherent mixture of the maxi-mally entangled Bell state | with probability onone hand, and the maximally mixed state 1414 with prob-ability (1 ) on the other. However, W() representsa valid density operator also for the range 13 0.Geometrically this can be understood as parameteriz-ing a straight line in Fig. 1, that connects the corner rep-resenting | (for = 1) with mix = 1414 (for = 0)at the origin, but continues onward until it intersectsthe opposite face of the double pyramid for = 13 .As Werner discovered [20], the state W() is entangledfor half its parameter range, that is, for 13 < 1,the partial transpose of W() has a negative eigenvalue.However, the correlation tensor for this state is found tobe t = 13 and the CHSH operator criterion (Theo-rem II.3) hence informs us that BmaxCHSH = 2

22. Conse-

quently, the Werner state is nonlocal for > 12. This

means that, in the range 13 < 12

the states in

Eq. (20) are entangled but nevertheless cannot violatethe CHSH inequality.

Interestingly, there exist other Bell inequalities thatare more efficient than the CHSH inequality in the sensethat one may find states which violate the former, butnot the latter. For instance, in Ref. [43], a Bell-typeinequality was introduced for which the Werner statesshow nonlocality already when > 0.7056, whichis slightly smaller than 1

2 0.7071. At the same

time, recent improvement [44] of a known bound [45]has revealed that Bell inequalities based on projectivemeasurements cannot be violated by Werner stateswith 0.682, leaving only a small window of uncer-tainty. By employing general positive-operator-valuedmeasurements (POVMs), one may in principle even gobeyond the results for projective measurements and theWerner states may be nonlocal also for values of below0.682. Bounds on the region of nonlocality have alsobeen obtained in this case. In Ref. [44] it was shownthat the correlations of Werner states with < 0.4547can be explained by local hidden-variable models forany measurement (improving on the previously knownbounds 0.416 [46] and 0.4519 [47]).

In general one may in fact even find states withpositive partial transposition that can violate certainBell inequalities [48]. The relationship of nonlocalitywith the PPT criterion, bound entanglement [4951], orsteering criteria [52] is hence complicated. For example,

6

there are states whose entanglement is bound (no pureentangled state may be distilled from any numberof copies of the state), which may yet violate a Bellinequality. Conversely, there are states with non-positivepartial transposition (NPT) that do not violate anyBell inequality. And while all entangled states withpositive partial transpose are bound entangled, it isnot known whether a non-positive partial transpositionimplies distillability. For the remainder of this paper wewill therefore focus on nonlocality in the sense of theviolation of the CHSH inequality.

To incorporate this notion of nonlocality into our geo-metric picture, one can systematically apply the CHSHoperator criterion to all Weyl states, noting that all lo-cally maximally mixed states for which maxi 6=j

[t2i +t

2j

]>

2 are nonlocal. The resulting region of nonlocality is il-lustrated in Fig. 1 where it is situated in the four cornersof the tetrahedron outside the dark-yellow parachutes.The region of local states can be found within theseparachutes and contains all separable but also a numberof (mixed) entangled states [29, 53].

D. Hidden Nonlocality

Since, as Werner demonstrated [20], certain entangledmixed states may satisfy all possible Bell inequalities, lo-cality is not a sufficient criterion for separability. At thispoint it is important to note that the definition of nonlo-cality that we have used here is not the only one possible.Indeed, we call states nonlocal only if they can be directlyused to violate a Bell inequality. However, as shown byGisin [54], for some initially local quantum states theentanglement may be amplified by local filtering opera-tions to allow for the violation of a Bell inequality. Inthis way the nonlocal character of the quantum systemcan be revealed (see also Ref. [55] in this connection).

To understand this phenomenon, we consider a familyof quantum states that arise as mixtures of pure (entan-gled) states = ||, where

| = sin() |01 + cos() |10 , (21)

for 0 < < 2 , with the mixed state given by

top =12

(|0000|+ |1111|

)= 14

(12 12 + z z

).

(22)

The Gisin states [54] G are hence given by

G(, ) = + (1 ) top (23)

= 14[12 12 cos(2)

(z 12 12 z

)+ sin(2)

(x x + y y

)+ (1 2)z z

],

for real probability weights 0 1. Note that theGisin states are in general not locally maximally mixed,i.e., the local Bloch vectors do not vanish for the whole

parameter range. Only the subset for which = /4can be represented in the tetrahedron of Weyl states as aline connecting the state top at the upper corner of theseparable double pyramid with the maximally entangledstate |+, as shown in Fig. 1.

With the help of the PPT criterion one immediatelyfinds that the Gisin state is entangled if and only if1 < sin(2). We can furthermore quantify the en-tanglement of G using an entanglement monotone calledconcurrence [911]. For an arbitrary two-qubit densityoperator , the concurrence C[] is given by

C[] = max{0,1

2

3

4} , (24)

where the i (i = 1, 2, 3, 4) are the (nonnegative) eigen-values of y yy y in decreasing order (1 2 3 4), and is the complex conjugate of with respect to the computational basis. For the Gisinstate, a simple calculation reveals that

C[G] = max{0, sin(2) + 1} , (25)

which is illustrated in Fig. 2 for the allowed range of and. In contrast, we can determine the parameter range forwhich G(, ) is nonlocal using the CHSH operator cri-terion of Theorem II.3. Reading off the matrix elementsof the correlation tensor from the Bloch decomposition

FIG. 2. Gisin states. The parameter regions of entangle-ment and nonlocality are shown for the family of Gisin statesfrom Eq. (23). Parameters (, ) lying in the blue region onthe left-hand side describe separable states. For the remainingentangled region (green) on the right-hand side the contourlines of the concurrence C[G] from Eq. (25) are drawn forvalues 0.1 to 0.9 in steps of 0.1. The orange lines on theright-hand side delimit the region of nonlocal Gisin states inthe sense that BmaxCHSH(G) > 2. The horizontal dashed redline indicates those Gisin states that are also Weyl states,corresponding to the dashed red line in Fig. 1.

7

in Eq. (23), one finds the maximally possible expectationvalue of the Bell-CHSH operator to be

BmaxCHSH(G) = 2 max{

2 sin2(2)+(12)2,

2 sin(2)}.

(26)

The parameter region for which the Gisin states are non-local is indicated in Fig. 2. Similar to the Weyl states inFig. 1, some of the Gisin states may be local althoughbeing more entangled (as measured by the concurrence)than some of the nonlocal Gisin states.

However, the most interesting feature of the Gisinstates is revealed by applying a local filtering procedure.That is, suppose that after sharing the state G(, ) forsome between 0 and /4, Alice and Bob locally amplifytheir qubit states |0

Aand |1

B, respectively. Since in

that case sin() < cos(), this increases the component of|01 with respect to that of |10 in |, which effectivelymoves the state closer to the maximally entangled state|+. Likewise, if /4 < < /2, amplifying |1

Aand

|0B

, respectively, will have the same effect. Mathemati-cally, these filtering operations are represented by a fam-ily of local, completely positive, and trace-nonincreasingmaps F, parameterized by , and given by

F : 7 F() = F F . (27)

Here, we choose Kraus operators satisfying F F 1which are given by

F =

{F0() F1() if 0 < 4F11 () F

10 () if

4 < 2. The dashed orangeline delimits the parameter region for which the unfilteredGisin states G(, ) are nonlocal, see Fig. 2.

8

the maximally possible expectation value of the CHSHinequality from Theorem II.3, which yields

BmaxCHSH(F) =2

sin(2) + 1 (31)

max{

2 sin2(2) +(1 sin(2)

)2,

2 sin(2)}.

Focussing on the parameter region where F is entangled,

i.e., for >(1+ sin(2)

)1, we find the condition for the

filtered Gisin state to be nonlocal as

BmaxCHSH(F) > 2 >1

(

2 1) sin(2) + 1. (32)

As illustrated in Fig. 3, the nonlocal parameter regionfor F(, ) includes the entire region of nonlocalityof the unfiltered state, but is also strictly larger.Some previously local (entangled) states become morestrongly entangled and even nonlocal due to the filter-ing. The amplification of entanglement hence revealsthe hidden nonlocality of some of the Gisin states,while others remain local. Although this separationmay attributed to the choice of filtering operation, itshould be remarked here that not every entangled statecan become nonlocal under local filtering operations [56].

Further note that, in contrast to Gisins nonunitarybut local filtering operations, one may instead use a uni-tary but nonlocal operation to increase the entanglementof the Gisin state. This simply corresponds to anotherchoice of factorizing the algebra of a density matrix [29].In this case, the mixedness of the state would remainunchanged. For instance, consider the unitary transfor-mation given by

U =12

(f+()12 12 if()x y

), (33)

where f() = cos() sin(). Since from Eq. (21)is transformed to the maximally entangled state +, i.e.,

UU =

+, and top = UtopU is left invariant by

the unitary transformation, the Gisin states become

U () = UG(, )U =

+ + (1 ) top . (34)

The unitarily transformed Gisin states are independentof , and more specifically, U () = G(, /4). Theunitary hence corresponds to vertically moving states inFig. 2 towards the dashed red line of = /4 while keep-ing fixed. It can easily be seen that this allows forseparable states to become entangled, and even nonlocalwith respect to the new factorization.

E. Classical and Quantum Entropy Measures

Let us now turn to another major category of quanti-ties used for the characterization of correlations. Manyfundamental features of multi-party (quantum) systems

can be captured by entropy, a key concept in both classi-cal and quantum physics. In classical information theory,the basic quantity is the Shannon entropy. For a randomvariable A whose possible values a are encountered withprobability p(a), the Shannon entropy is given by

H(A) = a

p(a) log p(a) , (35)

where the logarithm is understood to be to base 2. TheShannon entropy H(A) represents the uncertainty for theoccurrence of the values a in the sense that it quantifiesthe amount of information (in bits) that is gained onaverage by sampling the random variable once. For a bi-partite system with independent random variables A andB with values a and b, respectively, the joint probabilitydistribution factorizes p(a, b) = p(a)p(b). In this case thejoint entropy H(A,B) is additive, i.e.,

H(A,B) := a,b

p(a, b) log p(a, b) = H(A) + H(B) ,

and any information gained about b does not reveal anyinformation about a, or the other way around. In gen-eral, however, the joint entropy is subadditive, that is,H(A,B) H(A) + H(B). The strict inequality holdswhen the random variables A and B are correlated, suchthat information about the occurrence of b gives us in-formation about the occurrence of a, and vice versa. Foran arbitrary joint probability distribution p(a, b), the en-tropy can hence be written as

H(A,B) = H(A|B) +H(B) = H(B|A) +H(A), (36)

where H(A|B) is the (classical) conditional entropy de-fined as

H(A|B) = a,b

p(a, b) log p(a|b) (37)

= H(A,B) H(B) ,

where p(a|b) = p(a, b)/p(b) is the conditional probability,i.e., the probability of the occurrence of a conditionalon the occurrence of b . In other words, the conditionalentropy of Eq. (37) characterizes the uncertainty aboutthe value a when the value b is already known. From theabove definitions it immediately follows that

0 H(A|B) H(A) . (38)

If one wishes to define a measure for the correlationsbetween A and B, the (classical) mutual informationH(A :B) readily presents itself. It can be defined as theamount of information that is encoded in the joint dis-tribution p(a, b) but which is not contained in the localdistributions p(a) and p(b), i.e.,

H(A :B) := H(A) + H(B) H(A,B) . (39)

9

Another possible definition for the mutual information isas the difference between the local uncertainty H(A) andthe conditional uncertainty H(A|B), that is,

H(A :B) := H(A) H(A|B) . (40)

As can be easily seen from Eq. (36), these definitions forthe mutual information are equivalent. Moreover, sinceH(A,B) H(A) +H(B), and the conditional entropiesare nonnegative, H(A|B), H(B|A) 0, the two defini-tions in Eqs. (39) and (40) further imply the bound

0 H(A :B) min{H(A), H(B)}. (41)

However, when we extend these entropic measures toquantum systems, we will encounter some interesting dif-ferences to the classical case, especially when entangledsystems are considered.

The quantum analogue to the classical entropy ofEq. (35) is the von Neumann entropy S(), defined asthe Shannon entropy of the spectrum of the density op-erator representing the quantum state, that is,

S() := Tr( log

)=

n

pn log pn , (42)

where =n pn |nn| for some orthonormal basis

{|n}. Similar to the Shannon entropy, the von Neu-mann entropy represents the uncertainty the lack ofinformation we have about the state represented by. This definition naturally applies to bipartite systemswith density operators AB, such that the joint entropyis

S(A,B) S(AB) = Tr(AB log AB

). (43)

Since the von Neumann entropy of pure states vanishes,one may quantify the entanglement of bipartite purestates |

ABby the entropy of the reduced states, i.e.,

one can define the entropy of entanglement E(|AB

) as

E(|AB

) = S(A) = S(B) , (44)

where S(A) S(A

) and S(B) S(B

). However, whengeneralizing this concept to mixed states, it becomesproblematic to distinguish the contributions of the jointstate entropy and entanglement to the entropy of the sub-systems. This necessitates the introduction of a compli-cated optimization procedure when defining the so-calledentanglement of formation EoF of a mixed state as

EoF() = min{(pn,|n)}

n

pn E(|n) , (45)

where the minimization is carried out over all pure-state ensembles realizing the density operator =n pn |nn|. It is not known how to practically carry

out this optimization in general, but for some specialcases, EoF() can be computed explicitly. Amongst these,the most prominent is the case of two qubits, where the

entanglement of formation is found to be a monotonouslyincreasing function of the concurrence [10] of Eq. (24),i.e.,

EoF() = h(1 +1 C2[]

2

), (46)

where h(p) = p log(p)(1p) log(1p) is the Shannonentropy of the Bernouli distribution {p, 1 p}.

In contrast, the straightforward generalization of themutual information from Eq. (39) to the quantum case,given by

S(A :B) := S(A) + S(B) S(A,B) , (47)

does not separate genuine quantum correlations (i.e., en-tanglement) from purely classical correlations. Instead,as emphasized by Cerf and Adami [23], the quantum mu-tual information S(A : B) is a measure of the overallcorrelations. Moreover, S(A : B) has an interesting in-terpretation in the context of quantum thermodynamics.The quantum mutual information can be shown to beproportional to the work cost of its creation from an ini-tial thermal bath [57]. That is, the maximal amount ofcorrelation as measured by the mutual information thatcan be created between two initially thermal, noninter-acting systems at temperature T at the expense of thework W is S(A :B) = W/T (in units where ~ = k

B= 1).

And while the quantum mutual information also re-mains positive, S(A :B) 0, just as the classical mutualinformation in Eq. (39), it can exceed the classical upperbound from Eq. (41) by a factor of 2 such that we have

0 S(A :B) 2 min{S(A), S(B)} . (48)

The quantum information bound of Eq. (48) follows di-rectly from the definition in Eq. (47) using the Araki-Liebinequality S(A) S(B) S(A,B) . (49)As we shall see in the next section, when one also intro-duces the generalization of the conditional entropy to thequantum regime one encounters some more surprises.

F. Conditional Entropy and ConditionalAmplitude Operator

A straightforward generalization1 of the conditionalentropy of Eq. (37) to bipartite density operators

AB

on a joint Hilbert space is

S(A|B) := S(A,B) S(B) . (50)

1 Note that other generalizations for the quantum conditional en-tropy are possible, for which the equality in Eq. (51) does nothold [5860].

10

With this definition, one recovers the same relation tothe mutual information as in the classical case, i.e.,

S(A :B) = S(A) S(A|B) (51)

as in Eq. (40), and the upper bound for the conditionalentropy remains as in the classical case. That is, sinceS(A,B) S(A) + S(B), one finds S(A|B) S(A),in analogy to the right-hand side of Eq. (38). How-ever, the lower bound is altered. In the quantum caseone can encounter negative conditional entropy. For in-stance, when we consider a pure, maximally entangledstate such as = ||, the joint entropy van-ishes, S() = 0, while the local entropy is maximal,S(Tr

A()

)= log(2), and the conditional entropy hence

is S(A|B) = log(2) < 0. In general, the conditionalentropy is thus bounded by the marginal entropies, i.e.,

S(B) S(A|B) S(A) . (52)

A physical interpretation for the negative quantum con-ditional entropy was given in the context of state merg-ing protocols between two observers [61]. There it wasfound that positive values of S(A :B) quantify the par-tial information in qubits that need to be sent from Ato B, whereas a negative conditional entropy indicatesthat, in addition to successfully running the protocol, asurplus of qubits remains for potential future communi-cation. Moreover, a classical analogue of negative partialinformation can also be given [62]. Other physical inter-pretations of negative conditional entropy arise in quan-tum thermodynamics [63], and when considering mea-surements of quantum systems, where the negative con-ditional entropy quantifies the amount of information inthe post-selected ensembles [64]. These interesting inter-pretations motivate considering entropic Bell inequali-ties whose violation implies a negative conditional en-tropy [65].

Here, we want to better understand the relationship ofnegative conditional entropy and entanglement. In orderto do so, let us first discuss a different way to extend theclassical conditional entropy of Eq. (37) to the quantumcase. That is, we consider the conditional amplitude op-erator

A|B proposed by Cerf and Adami [23, 24], whichis given by

A|B := exp

(log

AB log(1

A

B)), (53)

where the exponential map is understood to be to base2. The conditional amplitude operator is a positive semi-definite hermitian operator defined on the support of ABthat takes over the role of the classical conditional prob-ability p(a|b) in the sense that one can now define theconditional entropy as

S(A|B) = Tr(AB log A|B

)(54)

in analogy to Eq. (37). To see that this definition isequivalent to Eq. (50), simply note that

Tr(AB

log(1A

B))

= Tr(AB1A log

B

)(55)

= TrA

(A1A

)Tr

B

(B

log B

)= S(B) .

Despite this close analogy between the conditional prob-ability distribution and the conditional amplitude oper-ator there are some fundamental differences. Whereasp(a|b) is a probability distribution satisfying 0 p(a|b) 1, its quantum analogue

A|B is not a density matrixin general. While

A|B is hermitian and positive semi-definite, it can have eigenvalues larger than one, andhence A|B 1 . Ultimately, this is what can lead tothe negativity of the conditional entropy. As we haveseen, a state for which this occurs is the maximally en-tangled Bell state. Moreover, we can immediately notethat the spectrum of

A|B (and thus the conditional en-tropy) is invariant under any local unitary transformationof the form U

AU

B, which also leaves entanglement un-

changed. This already suggests that the spectrum of theCerf-Adami operator

A|B is related to the separabilityof quantum states. Indeed, the following theorem due toCerf and Adami [24] can be formulated.

Theorem II.4 (Cerf-Adami Theorem).

The operator

AB

:= log A|B = log(1A B) log AB (56)

is positive semi-definite if the bipartite quantum statescharacterized by AB are separable.

Theorem II.4 implies that any separable bipartite statesatisfies the condition

A|B 1 . In turn, this means thatthe conditional entropy is non-negative, S(A|B) 0, forany separable state. States with negative conditional en-tropy must hence necessarily be entangled. Here, it isimportant to note that the negativity of the conditionalentropy implies that (some of) the eigenvalues of

A|B

exceed the physical boundary of unity, but the converseis not true as we demonstrate by several examples inSect. III A.

Moreover, the condition A|B 1 and the positivity

of S(A|B) are only necessary for the separability of thequantum states, but are in general not sufficient. As re-alized in Ref. [24], there exist entangled quantum statesAB

for which the operator AB

from Eq. (56) is pos-itive semi-definite,

AB 0, and hence

A|B 1 andS(A|B) 0. Such cases are of interest to the presentwork when it comes to detecting entanglement and non-locality. The results of our investigation in 2 2 dimen-sions are presented in Sect. III A. Before we finally turnto these results, also note that an operator analogous tothat of Eq. (53) can be defined for the mutual informa-tion [23]. The mutual amplitude operator

A:Bdefined

as

A:B

:= exp(log(

A

B) log

AB

), (57)

gives rise to the mutual information of Eq. (47) via

S(A:B) = Tr(AB

log A:B

). (58)

11

III. RESULTS

A. Geometry of Two Qubit States with NegativeConditional Entropy

We now wish to incorporate the negativity of theconditional entropy and the conditional amplitudeoperator bound into the geometric picture of two-qubitentanglement. To this end, we first consider againthe Werner states W() from Eq. (20). As we havepreviously argued, these locally maximally mixed statesform a line in the tetrahedron of Weyl states, reachingfrom the maximally entangled state | at = 1,through the maximally mixed state at the origin for = 0 to the opposite side of the separable doublepyramid until = 1/3, see Fig. 1. The Wernerstates are entangled for > 1/3 and violate the CHSH

inequality for > 1/

2.

When we now compute the conditional entropy for theWerner state, we note that the eigenvalues of W are14 (thrice degenerate) and

1+34 . With this we find

that the boundary between negative and non-negativeconditional entropy is given by the state W(0), where

0 0.7476 > 1/

2 is the solution of the transcendentalequation

3(1 ) log(1 ) + (1 + 3) log(1 + 3) = 4 log(2) .

The condition S(A|B) < 0 is hence a strictly strongercondition than nonlocality for the family of Wernerstates, as illustrated in Fig. 4. Indeed, a numerical anal-ysis shows that this is the case for all Weyl states, i.e.,the curved surfaces beyond which the conditional entropybecomes negative lie strictly outside of the local regionwithin the orange parachutes in the tetrahedron of locallymaximally mixed states, see Fig. 1.

- 13 0 13 12 0 1

absolutelyseparable & A B 1 entangled

nonlocal (CHSH)S(A|B) < 0

FIG. 4. Werner states. The parameter regions for theWerner state W() from Eq. (20) are shown.

Examining, on the other hand, the conditionalamplitude operator for the Werner states, one findsA|B = 2W, since B = TrA(W) =

121 such that

log1A

Bcommutes with log(W). Therefore, the

condition A|B 1 is met as long as 1/3, i.e., as

long as W is separable, whereas A|B has an eigenvaluelarger than 1 for > 1/3. For the Werner states theCerf-Adami condition

A|B 1 is thus equivalent to thePPT criterion [13, 28], a fact already noticed by Cerfand Adami [24].

Indeed, the observations we have made for the Wernerstates also hold for all other Weyl states in additionto this one-parameter subfamily. That is, a numericalevaluation of the conditional entropy of the locallymaximally mixed states of Eq. (3) presented in Fig. 1shows that the negativity of S(A|B) is a strictly strongercondition than nonlocality for these states. That is, thered, curved surfaces indicating where the conditionalentropy changes sign lie outside the orange parachutesurfaces marking the boundary of nonlocality for allWeyl states. Moreover, we can also formulate the fol-lowing conditional amplitude operator (CAO) criterion.

Theorem III.1 (CAO Criterion).

For every locally maximally mixed state W, the crite-rion A|B 1 for the Cerf-Adami conditional amplitudeoperator A|B given by Eq. (53) is equivalent to the PPTcriterion, i.e.,

A|B 1 if and only if W is separable,A|B 1 if and only if W is entangled.

Proof. For the proof of Theorem III.1 we recall Woottersconcurrence [911] from Eq. (24). For calculating C weneed the spin-flipped state = yyyy whichis equal to the density operator for all Weyl states, = . The square roots of the eigenvalues of neededfor the concurrence are hence just the eigenvalues pn (n =1, 2, 3, 4) of , which satisfy

n pn = 1. Consequently,

the concurrence of all Weyl states can be written as

C[] = max{0, p1 p2 p3 p4} = max{0, 2p1 1},(59)

where the largest eigenvalue p1 must exceed the value of1/2 for to be entangled.

Next, recall that for all Weyl states we have A

= B

=121 and the Cerf-Adami conditional amplitude operatoris hence given by

A|B = exp

(log log(1

A

B))

= 2 . (60)

Consequently, we have A|B 1 when the largest eigen-value of A|B = 2 exceeds 1, i.e., when the largest eigen-value of exceeds 1/2. By virtue of Eq. (59) this meansthat the state is entangled. Conversely, all entangledWeyl states must have an eigenvalue larger than 1/2 suchthat A|B 1. The fact that all entangled two-qubitstates have nonzero concurrence and non-positive partialtransposition concludes the proof.

B. Inequivalence of the CAO and PPT Criteria

Having established the significance of the conditionalamplitude operator and the relationship of entanglement,negative conditional entropy, and nonlocality for theWeyl states, we are curious whether the observations we

12

have made also hold for other states. We therefore con-sider the unitary orbit of one of the Weyl states thattakes us outside this set. Starting from the Narnhoferstate N =

14 (12 12 + x x), situated at the corner

of the double pyramid of separable states half-way onthe line connecting |+ and |+ in the tetrahedron ofFig. 1, we apply the unitary transformation

V = 14[(2 +

2)12 12 + i

2 (x y + y x)

(2

2)z z]. (61)

The resulting state, given by

V = V NV = 14

[12 12 + 12 (z 12 + 12 z)

+ 12 (x x + y y)], (62)

lies outside of the setW due to the occurrence of the term12 (z12 +12z). The purity Tr(

2N) = Tr(

2V ) =

12 of

the state is left unchanged by the unitary transformationbut the final state V is entangled. In fact, the con-currence takes the maximally possible value at this fixedpurity, C[V ] =

12 , i.e., the state V belongs to the class

of maximally entangled mixed states (MEMS) [66, 67].In other words, no global unitary may entangle this stateany further.

With this in mind, we now consider a family of statesin the two-qubit Hilbert space along the line from V totop from Eq. (22), i.e., we define

V () := V + (1 ) top , (63)

where 0 1. The eigenvalues of the partial trans-pose of V () are

4 (twice degenerate) and

14

(2 (1

2)). The states along the line are hence entangled if

> 2(

2 1). Now, if we consider the CAO criterion,we first compute the reduced state

V,B() = TrA(V ()

)= 14

(2 + 0

0 2

), (64)

and the spectrum of V (), given by

spectr(V ()

)= { 12 , 0,

12 ,

2} . (65)

To compute the spectrum of A|B, note that V () has(at least) one vanishing eigenvalue [see Eq. (65)], whichis problematic when evaluating log V (). However, asimple work-around is to replace the vanishing eigenvalueby > 0 throughout the computation and take the limit 0 at the end. With this procedure we obtain theeigenvalues of A|B as

{0, 22+ ,222 ,

242 } . (66)

The first three eigenvalues are always smaller than 1,but the last eigenvalue becomes larger than one when > 2

5> 2(

21). We thus see that the PPT criterion

and the CAO criterion are inequivalent in general.Nonetheless, the conditional entropy of the state V ()remains nonnegative for all values , and none of thesestates allows for a violation of the CHSH inequalityeither.

To incorporate also negative conditional entropy andnonlocality into the picture, we hence turn again to theGisin states G(, ) from Eq. (23). The spectrum of thedensity operator G(, ) is given by

spectr(G) = {0, 12 ,12 , } , (67)

while the reduced states A and B are already diagonaland have eigenvalues 12

(1 cos(2)

). The graphical

analysis of the parameter region for and for whichthe conditional entropy is negative reveals an interestingfeature. As can be seen in Fig. 5 (a), while some Gisinstates are both nonlocal and have negative conditionalentropy, some only have one of these properties, but notthe other. That is, contrary to what was found for theWeyl states, in general not all states for which S(A|B) 0 in the computation and consider thelimit 0 at the end. With this method, the eigenvaluesi of A|B are found to be

1 = 0, 2 =1

1 cos(2), 3 =

1 1 + cos(2)

,

and 4 =2 exp

( cos(2) artanh[ cos(2)]

)1 2 cos2(2)

. (68)

While 2 and 3 are smaller than 1 for all values of and , the fourth eigenvalue 4 can become larger than 1.The corresponding region, delimited by the purple linesin Fig. 5 (a), is contained within the region of entangledstates, but there is a region of entanglement where 4