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Construction of extremal local positive operator-valued measures under symmetry

S. Virmani1,2 and M.B. Plenio21 Gruppo di Ottica Quantistica, Dip. di Fisica ‘A. Volta’, Universita degli Studi di Pavia, via Bassi 6, I-27100 Pavia, Italy

2 QOLS, Department of Physics, Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BW, UK(February 1, 2008)

We study the local implementation of POVMs when we require only the faithful reproduction of the statisticsof the measurement outcomes for all initial states. We first demonstrate that any POVM with separable elementscan be implemented by a separable super-operator, and develop techniques for calculating the extreme pointsof POVMs under a certain class of constraint that includes separability and PPT-ness. As examples we con-sider measurements that are invariant under various symmetry groups (Werner, Isotropic, Bell-diagonal, LocalOrthogonal), and demonstrate that in these cases separability of the POVM elements is equivalent to imple-mentability via LOCC. We also calculate the extrema of theseclasses of measurement under the groups thatwe consider, and give explicit LOCC protocols for attainingthem. These protocols are hence optimal methodsfor locally discriminating between states of these symmetries. One of many interesting consequences is thatthe best way to locally discriminate Bell diagonal mixed states is to perform a 2-outcome POVM using localvon Neumann projections. This is true regardless of the costfunction, the number of states being discriminated,or the prior probabilities. Our results give the first cases of local mixed state discrimination that can be anal-ysed quantitatively in full, and may have application to other problems such as demonstrations of non-locality,experimental entanglement witnesses, and perhaps even entanglement distillation.

PACS numbers: 03.67.-a, 03.67.Hk

I. INTRODUCTION

The nature of local and global quantum operations playsa key role in Quantum Information theory [1]. However, al-though global quantum operations are well characterised [2],there is still no convenient mathematical classification oftheso called LOCC transformations (operations that can be im-plemented locally with classical communication). In recentyears some progress has been made by trying to understandthe non-local properties of unitaries and by trying to imple-ment operations with a minimum amount of prior entangle-ment [3,4,6]. Nevertheless, a more general answer remainselusive, and so alternative classes of operations have beende-fined that can be more easily characterised. Perhaps the mostinteresting examples are the Separable and PPT operationsintroduced by Rains [7,8]. These classes of operations (seesection II for definitions) are mathematically more tractablethan the LOCC class, and have been particularly useful astools for investigating the entanglement of distillation and lo-cal state discrimination [8,9]. They have also been used toextend the conventional definitions of entanglement measures[10,11], giving indications that quantum state transformationscan adopt a simplified structure under the class of PPT opera-tions [12].

The separable and PPT operations will also play an impor-tant role in this work, where we investigate the local imple-mentation of quantum measurements. This question is signif-icant as in many protocols for quantum communication it isimportant to know how a distantly separated Alice and Bobmay be able to gain information about quantum states usingonly local operations and classical communication. Examplesinclude entanglement distillation and cryptographic scenarios[9,10,13]. The issue of local measurements is also important

for efforts to locally infer some form of entanglement, suchasin developing tests of non-locality, or the local detectionof in-separability [14]. However, there is perhaps a much more fun-damental reason for the investigation of local measurements:in all uses of quantum states, all we ever ‘see’ at the end of theday are measurement outcomes, so one of the most importantthings to know is what kinds of statistics are possible accord-ing to the classes of operations that we are restricted to. AstheLOCC operations are an important paradigm in quantum in-formation, it is important to know what kinds of measurementoutcomes we can obtain with them.

Most of the discussion of local measurements in recentyears has been directed particularly towards local state dis-crimination [15–19]. However, in this work we pay moreattention to the actual implementation of measurements lo-cally. We will be inspired by initial steps in this directiontaken by [9,20], where both the concepts of PPT-operationsand symmetry were used to design a certain type of crypto-graphic scheme for Werner states of dimensions2n×2n (workthat was extended to the multiparty setting in [21]). Unlikethose investigations, however, we will not be directly inter-ested in any particular application or cost functions, as ourmain aim will be to obtain techniques for deciding the localimplementability of measurements, with particular regardtofinding extreme points. Some of the methods are quite gen-eral, and have implications for finding extremal POVMs underconstraints other than local implementability.

This paper is structured as follows. In section II we discussthe connection between the entanglement of POVM elementsand implementability via the different classes of operationsintroduced by Rains [8]. In sections III and IV we use theseideas to derive the entire (convex) set of local measurementsthat satisfy Isotropic and Werner symmetries, also giving the

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extreme points. These particular symmetries are amenableto an elegant form of solution that is not possible for morecomplicated symmetry groups, where we will require strongermethods. So in section V we discuss general techniques forderiving the extrema of POVMs under a quite general classof constraint that includes the constraints of separability andPPT-ness. In section VI we apply these techniques to BellandOO [22] symmetries, deriving the sets of locally imple-mentable measurements and their extreme points. The Bellsymmetries are particularly interesting, as we show that thebest way to locally discriminate Bell diagonal states is just toperform a 2-outcome POVM using local von Neumann pro-jections. This is true regardless of the cost function, the num-ber of states being discriminated, or the prior probabilities.Finally in section VII we summarise and discuss the implica-tions of our results.

II. ENTANGLEMENT CLASSIFICATION OF POVMELEMENTS AND RELATIONSHIP TO CLASSES OF

OPERATIONS

We begin by investigating the relationship between the en-tanglement properties of POVM elements and the resourcesrequired to implement the POVM. The key result of this sec-tion will be the demonstration that the separability (PPT-ness)of the POVM elements is equivalent to the implementabilityof the corresponding POVM by separable (PPT) operations.

Let us begin by reviewing the concepts of PPT and Separa-ble operations. The set of PPT operationsP is the set of com-pletely positive (CP) transformations that remain completelypositive when conjugated with the partial transposition oper-atorΓ, i.e. bothP andΓ ◦ P ◦ Γ are completely positive. Itis irrelevant whether the partial transposition is taken astrans-position of Bob’s or Alice’s system, as long as the choice isfixed. The set of Separable operations is the class of opera-tionsS that can be written as:

S(ρ) =∑

n

An ⊗BnρA†n ⊗B†

n. (1)

It is not too difficult to verify [7] that all LOCC operations areseparable, and all separable operations are PPT. An exampleof a CP operation that is not PPT is the creation of an NPTstate from a PPT one [7]. The PPT and Separable operationscan also be characterised in an interesting way in the light ofthe Jamiolkowsky isomorphism [5–9,20].

This classification of operations has allowed interestingbounds to be derived in the distillation and creation of en-tangled states [8,23], and is also useful for the study of localquantum state discrimination [9,20]. In the following, we willbe interested particularly in the question of when a measure-ment procedure can be implemented by LOCC. We will not beconcerned with the residual states left over after any measure-ment process, so a POVM will be considered to be ‘locallyimplementable’ if we can perform a measurement using only

local operations and classical communication that allows ex-actly the same statistical inferences to be made. It is importantto note the differences between our use of the word ‘local’ andthat of Beckman et. al. [25] and Groisman et. al. [24]. Thoseauthors are motivated by the implications of quantum opera-tions and observations for causality, and hence they do allowthe use of prior entanglement but disallow the use of classicalcommunication.

In our context the classification introduced by [7,8] has aparticularly elegant structure, as the PPT and separable def-initions of operations directly map to the definitions of PPTand separable states [26,27]. A given POVM described by aset of elements{Mk|k = 1..N,Mk ≥ 0,

∑

Mk = 11} can beimplemented by a PPT operation iff all the elements (after nor-malisation) correspond to PPT states, and can be implementedby a separable operation iff all the elements correspond to sep-arable states:

can do POVM by Separable ops⇔ elements separable (2)

can do POVM by PPT ops⇔ elements PPT (3)

The fact that all measurements implemented by separable op-erations must involve separable POVM elements was notedin [9]. The fact that a measurement can be implemented bya PPT operation iff the POVM elements are PPT was notedby [20,28]. As far as we are aware, however, the fact thatall POVMs with separable elements can be implemented bya separable operation has not been noted anywhere despiteits relatively simple proof. For completeness we thereforepresent the following proof of correspondence (2):

Observation 1.A given POVM can be implemented by a sep-arable superoperator iff the POVM elements associated withall N outcomes{Mk|k = 1..N,Mk ≥ 0,

∑

Mk = 11} corre-spond to separable states, i.e. the POVM can be implementedby a separable superoperator iff the density matrices definedbyMk/tr{Mk} are separable.

Proof. Any POVM can be realised as a superoperatorwhere the subscriptsn on the Kraus operators have beenrecorded classically. Any separable operation will correspondto POVM elements where the subscriptsn will be groupedinto setsSk corresponding to the different inferencesk. Hencethe expectation values of any POVM elementMk can be writ-ten:

tr{Mkρ} =∑

n∈Sk

tr{An ⊗BnρA†n ⊗B†

n} (4)

=∑

n∈Sk

tr{A†nAn ⊗B†

nBnρ}. (5)

Hence we can write that:

Mk =∑

n∈Sk

A†nAn ⊗B†

nBn (6)

As this sum is completely in terms of positive operators, wehave that tr{Mk} =

∑

n tr{A†nAn}tr{B†

nBn}. This meansthat we can also write:

2

ρk :=Mk

tr{Mk}

=∑

n∈Sk

tr{A†nAn}tr{B†

nBn}tr{Mk}

A†nAn

tr{A†nAn}

⊗ B†nBn

tr{B†nBn}

.

Hence we see that each POVM element from a separable op-eration must correspond to a separable state. To see the con-verse, simply work backwards through the above procedure,using the fact that any positive operatorX has a decomposi-tionX = Q†Q. The generalisation to multi-party systems isstraightforward�.

Observation 1 is extremely powerful, and can be used toprovide simple derivations of many results. For example wecan immediately see that an entangled pure state cannot beperfectly locally discriminated from its orthogonal comple-ment, as this would require a POVM with non-separable el-ements (see also [18]). A similar argument also shows thatwe cannot perfectly locally distinguish an Unextendible Prod-uct Basis (UPB) [29] from an equal mixture of the pure statesin its orthogonal complement, adding to the other intriguingdiscrimination properties of such bases [29].

Is it possible that all separable discrimination protocolscan be implemented locally? Unfortunately this is definitelynot the case, as has been observed from the example ofnon-locality without entanglement presented in [30] (see also[16]). The authors of that paper present sets of orthogonalseparable states that cannot be discriminated perfectly usingonly local operations. If all separable POVMs could be im-plemented locally then those sets of orthogonal states couldbe discriminated perfectly using only local operations. Asthisis not the case, this means that not all separable operationscanbe implemented locally.

Nevertheless, the constraint that any local POVMs must beseparable is still quite a strong one, and in some simple casesof high symmetry we will see that it is exactly equivalent tothe local implementability of POVMs. The results thereby al-low the construction of optimal local discrimination protocolsfor any number of states of such symmetries under any costfunction for any prior probabilities. The ideas may also haveother applications in cryptographic schemes, demonstrationsof non-locality, and the local detection of entanglement. Wewill begin by considering theIsotropicsymmetries.

III. LOCAL OBSERVATIONS WITH ISOTROPICSYMMETRIES

In this section we begin by analysing the so calledIsotropicsymmetries. The class of measurements invariant under thisgroup is actually a subset of theOO symmetric measurementsthat we will consider later in the paper. However, we considerthe Isotropic and Werner cases independently in this sectionand the next, as they afford a particularly elegant form of so-lution and demonstrate the usefulness of the partial transposi-tion mapping between Isotropic and Werner states.

The Isotropic states are those that commute with all localunitaries of the formU ⊗ U∗, where the∗ denotes complexconjugation in a fixed local basis. Any isotropic stateσ(f) onCd ⊗ Cd can be written as:

σ(f) = f |+〉〈+| + (1 − f)11− |+〉〈+|d2 − 1

, (7)

wheref ∈ [0, 1], and |+〉 is a canonical maximally entan-gled state|+〉 =

∑ |ii〉/√d. In the local observation of such

states we can use standard symmetry arguments to restrict ourattention to POVMs with elements of the form:

Mk = ak|+〉〈+| + bk(11− |+〉〈+|). (8)

As we require these elements to form a valid POVM, theymust satisfy the constraints:

ak ≥ 0

bk ≥ 0∑

k

ak = 1

∑

k

bk = 1. (9)

It is well known that isotropic states are separable iff theyarePPT (see e.g. [22]), and so it is easy to compute the additionalconstraint that eachMk must be separable fromMΓ

k ≥ 0:

(d+ 1)bk ≥ ak. (10)

We will now see that there is a local protocol that can beused to attain any Isotropic POVM satisfying equations (9)and (10). The protocol consists of two steps:

• Alice and Bob perform an isotropic twirl [31],T (ρ).This step is superfluous if the states to be observed areisotropic anyway. It is only included to make the totalmeasurement exactly equal to the POVM satisfying (9)and (10).

• Then Alice and Bob perform a measurement with ele-mentsNk described by:

Nk =d

∑

i=1

|i〉〈i| ⊗ [xk|i〉〈i| + yk(11B − |i〉〈i|)] (11)

with

xk = ak

yk =(d+ 1)bk − ak

d, (12)

where|i〉 is a pure state from the computational bases of Alice& Bob, and11B refers to the identity on Bob’s space.

The probability that Alice and Bob will find outcomekfrom the above procedure given an input stateρ is given by:

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tr{NkT (ρ)} = tr{T (Nk)ρ}. (13)

It is not difficult to verify thatMk =T(Nk) for all k, and soour two step protocol gives exactly the same statistics for eachk as the original measurement. Moreover, examination of theform of theNk shows that they correspond to a local mea-surement involving only one way communication, as long asthe condition (10) holds (otherwise theNk will not be posi-tive). Therefore any POVM consisting of PPT Isotropic ele-ments can be attained locally. The transformation above alsogives a direct derivation of the extreme points of this classofPOVMs. The mapping from the PPT (ak, bk) to the (xk, yk)is linear and invertible. Thexk and yk form two indepen-dent probability distributions, and so the extreme points willjust correspond to situations when thexk, yk take on valuesof {0, 1}. This observation allows relatively straightforwardoptimisation of local discrimination of isotropic states underessentially all cost functions.

It is also worthwhile noting that there are many other possi-ble local protocols that can match the PPT isotropic POVMs.This is because the above protocol essentially makes full useof the maximally entangled component of the isotropic statesto leave Bob in a residual mixture of some pure state with theidentity, which can then be used to discriminate between theoriginal isotropic states. Indeed, any measurement that Alicecan perform on a maximally entangled state that would leaveBob with a known pure state can be used to construct a sim-ilar protocol. One example is teleportation. Suppose Aliceand Bob first teleport a known pure state|ψ〉 using whicheverof the isotropic states that they share. Then if Bob performsa POVM defined by elementsxk|ψ〉〈ψ| + yk(11B − |ψ〉〈ψ|)then they will also achieve the same POVMs as theNk definedabove.

IV. LOCAL OBSERVATIONS WITH WERNERSYMMETRIES

Having calculated the set of locally implementableisotropic POVMs, let us see why we can almost immediatelywrite down the set of locally implementable Werner POVMs[26]. TheWernerstates are those that commute with all localunitaries of the formU ⊗ U . Any Werner symmetric POVMcan be written with elements of the form:

Mk = akPA + bkPS , (14)

wherePA andPS are the so calledantisymmetricandsym-metricprojectors respectively [26].

The argument here follows from the partial transpositionmapping between Isotropic and Werner operators that was dis-cussed in [22]. The authors of that paper point out that if acertain class of operators is invariant under a given group oflocal unitary operationsU(γ) ⊗ V (γ), then the partial trans-positions of those operators will be invariant under the ‘par-tially conjugated’ groupU(γ)⊗V ∗(γ) (modulo a problem ofphases, see [22]). This means that if we have a set of PPT

POVM elements that is invariant under one local symmetrygroup,

{Mk|∑

Mk = 11,Mk ≥ 0,MΓk ≥ 0} (15)

then the partial transposition of these elements will form an-other POVM that is a PPT POVM invariant under the partiallyconjugated group, and vice versa. Indeed the extremal PPTmeasurements under one group of local unitaries are in one-to-one correspondence with extremal PPT measurements un-der the ‘partially conjugated’ group:

Theorem 2. If {Mk} forms an extremal PPT POVM for onelocal symmetry group, then the partial transposition{MΓ

k }forms an extremal PPT POVM for the partially conjugatedsymmetry group.

Proof. If a POVM {Mk|k = 1...n} is PPT andnot extremal,then it is possible to write:

{Mk|k = 1...n} = {pM1k + (1 − p)M2

k |k = 1...n} (16)

for some probabilityp, where{M1k |k = 1...n} and{M2

k |k =1...n} are also PPT POVMs. Therefore the partial transposi-tion of this equation is also true:

{(Mk)Γ|k = 1...n} = {p(M1k )Γ + (1 − p)(M2

k )Γ|k = 1...n},(17)

where{(M1k )Γ|k = 1...n} and{(M2

k )Γ|k = 1...n} are alsoPPT POVMs. So the partial tranposition of a PPT POVM isextremal iff the PPT POVM itself is extremal�.

This convenient partial transposition connection is pre-cisely the relationship between PPT POVMs of Werner sym-metry and PPT POVMs of Isotropic symmetries. Hence allimplications discussed above for the local discriminationofisotropic states also immediately apply to local discriminationof Werner states. In particular, the partial transpositionof thelocal protocols given for Isotropic symmetries above will al-low us to obtain any PPT POVM with Werner symmetries.To be explicit, let us denote a Werner-symmetric POVM byelements:

Mk = akPA + bkPS . (18)

The condition that the POVM be PPT forces us to impose therequirement that:

bk

(

d+ 1

d− 1

)

≥ ak (19)

in addition to the other positivity and completeness con-straints. Any such POVM can be attained by the partial trans-position of a PPT isotropic measurement, and so can be at-tained by the ‘partial transposition’ of the local measurementused to attain that particular isotropic POVM, i.e.:

• First Alice and Bob perform a Werner twirl [31],T (ρ).Again, this step is superfluous if the states to be ob-served are Werner anyway.

4

• Then Alice and Bob perform a measurement with ele-mentsNk described by:

Nk =

d∑

i=1

|i〉〈i| ⊗ [xk|i〉〈i| + yk(11B − |i〉〈i|)] (20)

with

xk =(1 − d)ak + (d+ 1)bk

2yk = ak. (21)

It can readily be verified that the protocol matches any PPTWerner POVM.

In the other symmetry groups that we will consider in thispaper, no elegant and simple solution along the lines of theone found for the Isotropic/ Werner states can work. The rea-sons for this will be discussed in more detail in the section onOO symmetries and in the appendix. Consequently, for othersymmetry groups we have to adopt a different approach. Herewe choose to construct the extremal points of the (convex) setof PPT POVMs for the groups that we consider, and then tryto find local protocols that match these extrema. We then needto invoke a theorem of convex analysis which states that anyconvex compact set is the (closed) convex hull of its extremepoints [32], and hence if we can show that the extremal pointsof the PPT measurements can be obtained locally, then by con-vexity so can the whole set (for a finite number of outcomesthe set of PPT POVMS is clearly compact). In the cases thatwe consider here our search for local protocols will alwaysbe successful. In fact, the most difficult exercise is usually theconstruction of the extremal points. The task is made tractableby some general techniques that we will discuss next.

V. GENERAL TECHNIQUES AND NOTATION

In this section we will explain the major tools that we willuse throughout the rest of the paper. Sometimes we will de-scribe these techniques in quite general language, as theremaybe applications to other problems.

The first technique, that we will also use later in this sub-section, is a quite common method for determining constraintson the extrema of convex sets. Suppose that we have a convexsetA, and we are trying to decide whether a candidate ele-ments of the set is extremal. When we write the phrase ‘bythe following perturbation’, this will mean that we considerthe following common approach to deciding extremality. Wecan consider perturbations ofs by a small amountδ. We thentry to decompose the candidate extremums in the followingway:

s =1

2s(+) +

1

2s(−)

s(+) = s+ δ

s(−) = s− δ. (22)

In order thats be extremal, we will require that at least oneof s(+) ands(−) be outside the setA. In our specific ap-plication, we will usually chooseδ such that the separabilityand completeness of the POVM is maintained, and so we willhave to infer that eithers(+) or s(−) be non-positive, andthis will allow us to derive strong constriants of the form ofthe extrema. So when we write ‘by the perturbationδ we cansee that the extremal points must be of the form...’ we are infact referring to the inferences that can be made by the aboveprocess.

Now let us consider how we can use some of the struc-ture of sets of separable POVMs in order to simplify the huntfor extremal points. The most important feature that we willrely on is the fact that the properties of positivity, separabilityand PPT-ness are closed under linear combinations with non-negative scalars. Some of the methods that we will employcan be used to extremise POVMs for any other constraints thathave this property, and not only PPT-ness or separability. Inlight of this we will find it convenient to make the followinggeneral definition:

Definition 3. A constraint on two operatorsM1,M2 will becalledhomogeneousif wheneverM1 andM2 satisfy the con-straint then so doesαM1 + βM2, for all non-negative scalarsα, β. Hence both Separability and PPT-ness are homogeneousconstraints. We will sometimes use the symbolH to denoteanyset of homogeneous constraints under consideration, notnecessarily only PPT-ness or separability.

Definition 4. A POVM {Mk|k = 1...n,Mk ≥ 0,∑

Mk =11} satisfying any extrahomogeneouscontraintsH under con-sideration will be called afeasiblemeasurement.

The set of feasible POVMs is clearly a convex set itself, inthat if {M1

k |k = 1....n} and{M2k |k = 1....n} are feasible

POVMs, then so is{pM1k + (1 − p)M2

k |k = 1....n} wherep ∈ [0, 1]. Hence we can ask questions about what the ex-tremal POVMs are, and how to find them. One particularlyimportant result in this direction is the following:

Theorem 5. In any extremal feasible POVM, the non-zerooperator elements must be linearly independent.

Proof. The proof can be adapted straightforwardly from aproof of this statement for global POVMs by Lo Presti&D’Ariano [33], and indeed their proof is essentially true forany constraints that are closed under multiplication by non-negative scalars�.

This theorem will be particularly useful as it will allow usto restrict the number of non-zero outcomes that we need toconsider. It also provides another reason why in the caseof Isotropic/Werner symmetries the extremal PPT POVMshave essentially two non-zero outcomes, simply because un-der these symmetries there are at most two non-zero linearlyindependent POVM elements.

The consideration of homogeneous constraints [34] will al-low us to use a technique that involves what we call ‘basicvectors’. This should not be confused with the term ‘basic so-lution’ that is often used in linear programming problems. In

5

the rest of this section we will explain exactly what we meanby a ‘basic vector’. Let us suppose that we can find a class ofelements{V (α)} whereα represents one or more parameters(discrete or continuous) with the following properties:

• Each element from{V (α)} is non-negative.

• Each element from{V (α)} satisfies the homogeneousconstraintsH under consideration.

• All POVM elementsMk satisfy the homogeneous con-straints under consideration iff they can be written inthe following way:

Mk =∑

i

pki V (αi) (23)

where thepki are non-negative real numbers.

Then the set{V (α)} will be referred to as a set ofbasic vec-tors. The coefficientpk

i will be referred to as theweightofthe basic vectorV (αi) in Mk. Of course this decomposition,and hence the weights, need not be unique. Such sets of basicvectors can be very useful tools for deriving the set of ex-treme points, and in subsequent sections we will prove a fewlemmas that will demonstrate their power. The source of thispower is the fact that in the ‘basic vector’ approach, each fea-sible POVM element is automatically decomposed into a sum(with non-negative weights) of operators that automaticallysatisfy the homogeneous constraints that we are considering.So we are free to try to perturb any non-zero weights by smallamounts without violation of our constraints. This allows usto derive strong limitations on which weights can be non-zeroin an extremal measurement. In particular the following twoLemmas prove to be useful:

Lemma 6. In an extremal feasible POVM, no basic vectorcan have non-zero weight in more than one elementMk ofthe POVM.

Proof. This can be proved along similar lines to the proof ofthe following lemma�.

Lemma 7. If a feasible POVM is extremal, then any set oflinearly dependent basic vectors must have non-zero weightsin at most one elementMk.

Proof. Suppose that we have a set of basic vectors{Vm|m =1...L} such that:

∑

m

λmVm = 0 (24)

for some set of non-zeroλm’s. We can take the Hermitianconjugate of this equation and average it with the original togive:

∑

m

1

2λmVm +

1

2λ∗mVm =

∑

m

Re{λm}Vm = 0. (25)

Hence theλms can be taken to be real without loss of gener-ality (this is unless all the originalλm are purely imaginary,

but in that case we can just divide the equation through by√−1). Suppose that the basic vectorVL has non-zero weight

in elementM2, whereas the other basic vectors of the set havenon-zero weight inM1. Then we have that:

∑

m 6=L

λm

λLVm = −VL (26)

We define two new POVMss(±) with exactly the same ele-ments and basic weights as the original POVM except for thefollowing changes:

p1m 6=L(±) = p1

m ± δλm

λL; p2

L(±) = p2L ± δ (27)

The two new POVMs average to give the original, are com-plete by construction, and if theδ is chosen small enough,will not violate any of the homogeneous constraints or posi-tivity. The same argument can easily be extended to the casewhere more than one of the vectors from the set is placed inM2. Therefore we require that any set of linearly dependentbasic vectors contains non-zero weights in at most one fixedelement of an extremal POVM�.

The exact utility of the basic vectors approach will becomeclearer in the examples that we tackle later, where we will ex-plicitly construct sets of basic vectors. Unfortunately itis notalways easy to do this construction. Nevertheless, there isoneapproach based upon construction of two outcome extremalPOVMs that can allow us to draw some interesting generalconclusions. Let us consider the set of feasible two outcomePOVMs:

{M, 11−M}. (28)

Each such measurement is characterised by just one elementM , as the other is fixed by completeness. Suppose then thatwe can somehow find the extreme points of this set, and theyare given by the set{M(α)}, whereα represents some param-eters (discrete or continuous) that label the extrema. Thenit isclear that{M(α)} will form a set of basic vectors for feasiblePOVMs with any number of outcomes. This is because anyparticular elementMi of anN -outcome POVM can be viewedas a member of a two outcome POVMMi, 11−Mi, where theremaining elements have been grouped together as one out-come, and so any element of a feasible measurement can al-ways be written as a convex combination of the{M(α)}. Thisfeature will be particularly useful to us in the construction ofthe separableOO symmetric extrema, as in that case we willbe able to construct the two outcome extrema more easily.

There are also more general conclusions that can be madefrom this observation. Suppose that each two-outcome fea-sible POVM element can be constrained to live on aK-dimensional vector space. If the elements arem ×m Hermi-tian operators, then there is a natural upper bound toK ofm2,although this can be reduced in cases of symmetry. Then byLemma 7 above, we know that each linearly dependent set of

6

basic vectors can contribute non-zero weights to at most onePOVM element. As anyK+1 elements drawn from{M(α)}are linearly dependent, we must conclude that in anyK non-zero outcome extremal feasible measurement, each elementmust be proportional to one and only one distinct member of{M(α)}. Hence we have that:

Lemma 8. In any feasible extremal measurement ofK non-zero outcomes, each POVM element must be proportional toan element of a 2-outcome extremal feasible POVM.

This result is interesting as it shows how information from2-outcome optimisations may be used to draw conclusionsabout problems with higher numbers of outcomes. Note thatthe possible constants of proportionality will be uniquelyfixedby completeness, as each of theK 2-outcome extrema con-tributing to the POVM are linearly independent. It is impor-tant to note that there cannot be any extremal feasible mea-surements with more thanK non-zero outcomes anyway bythe linear independence requirement.

We are now in a position to apply these tools to the cases ofsymmetry that we will consider in the rest of the paper.

VI. BELL AND OO SYMMETRIC MEASUREMENTS

In the previous section we have discussed a number of use-ful results that we can use calculate the extreme points of sep-arable POVMs. Here we will now use these techniques andnotations to derive the set of local measurements of Bell andOO symmetries, and to characterise the extreme points.

The symmetry groups considered here are all taken from theexamples discussed in [22]. They have the convenient prop-erty that the invariant POVMs have elements that can be writ-ten as a decomposition into a finite numbern of orthogonalprojectorsA,B,C...:

Mi = aiA+ biB + ciC...... (29)

(i.e. thecommutant[35] of the group is abelian). Thereforeeach POVM element can be represented by a vector of co-efficients(ai, bi, ci, ...)

T . As at mostn vectors withn com-ponents can be linearly independent, this means that we willbe able to write the most general extremal PPT POVM for thesymmetry group under consideration as ann×nmatrix, whereeach columnj contains the column vector(aj , bj , cj , ...)

T

corresponding to the POVM elementMj .Another key feature of the symmetries that we consider is

that checking the positivity of the partial transposition of aparticular POVM element is relatively simple, as the partialtransposition of each invariant POVM element can always bewritten in the following way:

MΓi = a′iA

′ + b′iB′ + c′iC

′...... (30)

where the(a′i, b′i, c

′i....) form a vector of real coefficients and

the(A′, B′, C′...) are a set of mutually orthogonal projectors.

Note that this is not always the case, in that there can be lo-cal symmetry groups with an abelian commutant whose par-tial tranposition is no longer abelian. Nevertheless, there aremany interesting symmetries that have this property, includingthose of Bell diagonal states that we consider next.

A. Local observation with Bell symmetries

In this section we will derive the extreme points of the setof Bell Diagonal PPT POVMs, and then see that they can beattained locally. We are forced to adopt this route, as no directand simple solution in the manner of the Isotropic/ Wernercase is possible for the Bell group. A more detailed explana-tion for this will be given in the appendix.

First we will explain our notation. The Bell basis is definedas the orthonormal set of two qubit pure states:

|Φ±〉 =1√2(|00〉 ± |11〉)

|Ψ±〉 =1√2(|01〉 ± |10〉), (31)

and it constitutes a set of projectors that is invariant under thesymmetry group{σx⊗σx, σy ⊗σy, σz⊗σz , 11⊗11} discussedin [22]. Consequently, any POVM element invariant under thesame group can be written as:

Mi = ai|Ψ+〉〈Ψ+| + bi|Ψ−〉〈Ψ−|+ ci|Φ+〉〈Φ+| + di|Φ−〉〈Φ−|. (32)

As we require linear independence between the non-zeroPOVM elements, we can restrict our attention to candi-date extremal measurements represented by a4 × 4 ma-trix, where each columni, i = 1...4 has elements given by{ai, bi, ci, di}T , the quadruple representing the POVM ele-mentMi:

s =

a1 a2 a3 a4

b1 b2 b3 b4c1 c2 c3 c4d1 d2 d3 d4

. (33)

In each column all elements must be non-negative, and eachrow must sum to 1 for completeness.

Definition 9. A largest element in a given column will be re-ferred to as the amaximalelement of the column. In manycases the maximal element of a column will not be unique, inwhich case we are free to choose among the possibilities.

The condition that each column corresponds to a PPT mea-surement is that a maximal element of a column must be lessthan or equal to the sum of the other three elements of thesame column.

Definition 10. We will refer to a column as beingtight if it hasa maximal elementequalto the sum of the remaining matrixelements in its column.

Lemma 11.In an extremal Bell POVM with a finite integerNof non-zero outcomes, at leastN − 1 columns must be tight.

7

Proof. Suppose that we consider two columns, say with-out loss of generality the first two. Suppose that there aretwo rows such that both columns contain non-zero matrix ele-ments. Suppose w.l.o.g. these two rows are the top two rows.Then we can perturb the POVM as follows with a small posi-tive δ:

s(±) = s±

+δ −δ 0 0+δ −δ 0 00 0 0 00 0 0 0

. (34)

If δ is small enough, this perturbation maintains positiv-ity and completeness. Therefore PPT-ness must be violated.Computing the partial transpose ofMi in eq. 32 the positivityof MΓ

i is equivalent to

ai + bi ≥ |ci − di| (35)

ci + di ≥ |ai − bi|. (36)

If PPT-ness should be violated by arbitrarily small perturba-tionsδ this implies that at least one of these conditions has tobe an equality. Together with the positivity of theai, bi, ci anddi, this implies the tightness of at least one column.

Alternatively, it could be the case that there are no two rowsin which both columns contain non-zero matrix elements.This can only be the case if one of the columns contains onlytwo non-zero matrix elements. In that case both non-zero ma-trix elements of the column in question must be maximal inorder for it to be PPT, and so that column must be tight.

We can take any two columns in this argument, and so atleastN − 1 of the non-zero columns must be tight�.

Lemma 12. Any tight columnj can itself be written in thefollowing way:

∑

i

pjiPi

1100

+ qj

0000

, (37)

where the summation runs over all permutation matricesPi

and the coefficients{pji , q

j} form a probability distribution.

Proof. Let the sum of the matrix elements in the column be2σj . Then as the column is tight, we must have that its maxi-mal element is equal toσj . Therefore the column is majorised[36] by the column vector:

σj

σj

00

. (38)

It is a well known result that if a column vector~g majorises acolumn vector~h then~h can be written as a convex combina-tion of permutations of~g [36]. Hence we can write that:

σj

∑

i

xjiPi

1100

+ (1 − σj)

0000

, (39)

where the{xji} form a probability distribution. As no max-

imal element can be greater than 1 (from completeness), theσj , 1 − σj also form a probability distribution, and so by set-ting pj

i = σjxji andqj = σj we have a decomposition of the

column as stated in the lemma. Note that the decompositionneed not be unique.

Lemma 13. All columnsj in a Bell PPT POVM can be ex-pressed as

∑

i

pjiPi

1100

+ qj

0000

+ rj

1111

(40)

where the{pji , q

j , rj}i form a probability distribution.

Proof. Let us for now just consider extremal measurementsof two non-zero outcomes. One of the two non-zero columnsof any such measurement must be tight, let this w.l.o.g. becolumn 1. Then from completeness and Lemma 12 we canexpand column 2 as:

a2

b2c2d2

=

1111

−

∑

i

p1iPi

1100

+ q1

0000

=∑

i

p1iPi

0011

+ q1

1111

:=∑

i

p2iPi

0011

+ r2

1111

,

where as before{p2i , r

2} also forms a probability distribu-tion. Hence the theorem holds for all 2 non-zero outcomeextremal PPT Bell measurements, and by convexity holds forthe non-extremal ones as well. But then any column from anN outcome PPT measurement can be viewed as a member of atwo outcome measurement where the remaining columns havebeen summed. Therefore the result is true for any column inany Bell PPT POVM.�.

A consequence of this lemma is that the set of column vec-tors of the form:

Pi

1100

or

1111

(41)

forms a set of basic vectors for the Bell Diagonal PPTPOVMs. For a columnj, the probabilitypj

i or rj will hence

8

be taken as the weights of the correponding basic vector. Wewill refer to as(1, 1, 1, 1)T as theidentity basic vectoras itcorresponds to the identity POVM.

Lemma 14. There are no 4 non-zero outcome extremal BellPOVMs.

Proof. In a 4 non-zero outcome extremal POVM, each non-zero column must have weight from only one basic vector dis-tinct from the basic vectors contributing to the other columns.Otherwise the linear independence condition in Lemma 7 willbe violated (as any five 4-component column vectors must belinearly dependent).

If one column contains weight from the identity, then thePOVM cannot be extremal, as it can be expressed as the con-vex sum of the identity and a renormalised POVM that con-sists of only the remainder from the original.

For four outcomes where no column contains weight fromthe identity, a little thought shows that the only possibility thatcan maintain completeness is a POVM given by a row and/orcolumn permutations of the following:

w 0 0 zw x 0 00 x y 00 0 y z

, (42)

However, this measurement cannot be extremal as thecolumns are linearly dependent To see this, note that we canobtain the zero vector by (a) multiplying each column by anon-zero factor such that each non-zero matrix element hasthe same value, and then (b) subtracting the 2nd and 4thcolumns from the sum of the 1st and 3rd. Therefore, thereare no four non-zero outcome extremal PPT Bell POVMs.

Lemma 15. There are no 3 non-zero outcome extremal BellPOVMs.

Proof. As with the proof for four non-zero outcomes, therecan be no column with weight from the identity. Thereforewe require that each column can be decomposed into con-vex sums of basic vectors other than the identity, such that nomore than four basic vectors are involved in total. Thereforetwo columns must be directly proportional to a basic vector,as each basic vector can have non-zero weight in at most onecolumn. Let these two columns be 1 and 2 w.l.o.g.. Thereare essentially only two possibilities upto row and columnpermutations - the first two columns have a common row inwhich they both have non-zero matrix elements, or they haveno such common row. The third column is fixed by complete-ness. Therefore up to row and column permutations we have:

x 0 1 − x 0x y 1 − x− y 00 y 1 − y 00 0 1 0

or

x 0 1 − x 0x 0 1 − x 00 y 1 − y 00 y 1 − y 0

, (43)

but these POVMs cannot be extremal unless they have onefurther zero column. For the first matrix we can see this fromthe following. Consider the perturbation

±δ

+1 0 −1 0+1 −1 0 00 −1 +1 00 0 0 0

. (44)

Unless eitherx or y are zero, this perturbation maintains com-pleteness, positivity and PPT-ness. Hence the first matrix can-not be extremal. For the second matrix we can consider thefollowing perturbation:

±δ

+1 0 −1 0+1 0 −1 00 −1 +1 00 −1 +1 0

. (45)

This perturbation implies that to be extremal eitherx or y arezero, or that the third column is tight. We cannot have threenon-zero columns ifx or y are zero, so let us deal with thepossibility that the third column is tight. The third columncanonly be tight if one ofx or y is 1. Suppose w.l.o.g. thatx = 0,then the second and third columns become proportional, andso it cannot be extremal. Hence neither of the possibilitiesin(43) can be extremal�.

Therefore there the only extremal POVMs of Bell diagonalform are those of at most two non-zero outcomes.

Theorem 16. The extremal points of the PPT Bell diago-nal POVMs can be obtained by local protocols. The extremalpoints are given by the possible row and column permutationsof the following POVMs:

1 0 0 01 0 0 01 0 0 01 0 0 0

;

1 0 0 01 0 0 00 1 0 00 1 0 0

. (46)

Proof. The fact that these are the only extremal points can beproven using similar arguments to the ones above. They arealso locally attainable, as the the first matrix correspondstothe ‘do nothing’ measurement, and the other POVMs corre-spond to POVMs of the form:

M = σk ⊗ σl[|00〉〈00|+ |11〉〈11|]σ†k ⊗ σ†

l ; 11−M (47)

where theσi are drawn from the four Pauli matrices. ThesePOVMs are trivially locally attainable. They simply cor-respond to measurements in a product basis, and can beachieved using one way communication�.

Despite some prior results showing that the four orthogonalBell states cannot be perfectly discriminated locally [19,16],this result is interesting as it shows that there can be a hugedifference between locally and globally obtainable informa-tion underall figures of merit, even when we are only deal-ing with two qubits. Indeed, as the extremal local measure-ments involve only 2 non-zero outcomes, there can be at most1 classical bit of information obtained by local measurementsof Bell diagonal states. However, in the global case there can

9

be 4 orthogonal Bell diagonal states that can be perfectly dis-criminated, thereby allowing a full 2 classical bits of informa-tion to be obtained globally. It might be tempting to proposea bit hiding scheme on the basis of this observation. How-ever, the scheme would be susceptible to cheating with a priorshared singlet state (as then Alice could teleport her particle toBob), and hence the technology required to break the schemeis equivalent to that required to implement it.

B. Local Observations withOO Symmetries

In the notation of [22], the OO group is the set of unitariesof the formO ⊗ O, whereO is an orthogonal transformation(i.e. OOT = IA/B) in some fixed local bases. Any POVMelement invariant under this group can be written in the form:

a [|+〉〈+|] + b

[

(11− F )

2

]

+ c

[

(11 + F )

2− |+〉〈+|

]

,

whereF =∑ |ij〉〈ji| is the swap operator,a, b, c are real co-

efficients, and the three terms in square brackets are mutuallyorthogonal projections. Hence a full measurement obeyingthe OO symmetry withN outcomes is characterised com-pletely by the triples(ak, bk, ck), corresponding to each ofthe POVM elements indexed byk = 1, ..., N . One could tryto look for local protocols to match these triples in a simi-lar spirit to the solution presented for the Isotropic case.Tobe specific, it might be hoped that a linear invertible transfor-mation could be found taking the(ak, bk, ck) to new triples(xk, yk, zk), such that the(xk, yk, zk) are the positive coeffi-cients of some local protocol iff the original(ak, bk, ck) cor-respond to a PPT POVM. However, we can show that no suchnaive transformation exists for theOO symmetries (see ap-pendix 1 for details). Consequently, in the following we willinstead restrict our attention to theextreme pointsof the con-vex set ofOO symmetric PPT POVMs. We will present localprotocols that attain these extreme points, and thereby arguethat allOO symmetric PPT POVMS can be attained locally.First we need to show how to construct these extreme points.

1. Extreme points of theOO symmetric POVMs

First let us discuss the partial transposition ofOO symmet-ric operators. Partial transposition takesM to an operator ofthe same form, with new coefficients given by:

a′

b′

c′

= R

abc

. (48)

where

R =1

2d

2 d(1 − d) (d+ 2)(d− 1)−2 d (d+ 2)2 d (d− 2)

. (49)

In order to satisfy the constraints of being a PPT POVM,all the triples(ak, bk, ck) corresponding to each elementMk

must satisfy:

ak, bk, ck ≥ 0 (positivity)

a′k, b′k, c

′k ≥ 0 (PPT-ness)

∑

k

ak =∑

k

bk =∑

k

ck = 1 (normalisation). (50)

We would like to construct a set of basic vectors for theOOsymmetries. One way to do this is to first construct the setof 2-outcome PPT extrema under these symmetries. In suchsituations given one elementM the other is fixed as11 −Mby completeness. We hence require:

11 ≥M ≥ 0

11 ≥MΓ ≥ 0.

Denoting the only free element by a column vector of its tripleM = (a, b, c)T , the constraints will result in a polyhedron inR3, for which the extreme points are the vertices (see figure1).

D2

D1

C2

C1 B2

B1

A2

A10.2

0.4

0.6

0.8

1

0.20.4

0.60.8

1

0.20.4

0.60.8

1

FIG. 1. The polyhedron of allowed PPT 2-outcome POVMs un-der OO symmetry, sketched ford = 3. The vertices are given inequation (51).

It is trivial but tedious to calculate these vertices. They aregiven by the following POVM elements:

A1 :

000

A2 :

111

B1 :

00

2d/[d+ 2][d− 1]

B2 :

11

(d+ 1)(d− 2)/[d+ 2][d− 1]

10

C1 :

11/[d− 1]

(d− 2)/[d+ 2][d− 1]

C2 :

0(d− 2)/[d− 1]d2/[d+ 2][d− 1]

D1 :

10

2/[d+ 2]

D2 :

01

d/[d+ 2]

(51)

Those POVM elements labelled with the same upper case let-ter arecomplementary, in the sense that they sum to the iden-tity. Hence each letter labels a complete extremal PPTOOsymmetric POVM of 2 non-zero outcomes. The entire set ofvectors in equation (51) hence also forms a set of basic vec-tors forOO symmetries. We would like to use this set to con-struct the entire set of extremal points. We require any candi-date extremal POVM to consist of measurement elements thatare linearly independent. Therefore we need only restrict ourattention to POVMs with at most 3 non-zero outcomes, i.e.those measurements that can be represented by a3×3 matrix:

s =

a1 a2 a3

b1 b2 b3c1 c2 c3

, (52)

where each column characterises one POVM element. Asa consequence of Lemma 7, we can only use three of ourbasic vectors to construct the POVM, and as we require allthree columns to be non-zero, each column of the extremal3-outcome POVMs must beproportionalto one and only onedistinct basic vector. As there are essentially only 7 importantbasic vectors to choose from in the set above (A1 is trivial),we are already constrained as to the possible form. However,there is another observation that will provide much tighterconstraints:

Lemma 17. It is not possible to have 2 basic vectors from thesame complementary pair contributing non-zero weights inthe same extremal POVM. Consequently, no extremal POVMcan have any column with non-zero weight from the identitybasic vector, unless it is the trivial POVM with only one non-zero outcome.

Proof. Let us suppose that the two complementary basic vec-tors areV1 andV2 respectively. Then suppose that two POVMelementsj, k (where we can havej = k) contain weightspj

1

andpk2 . Suppose thatpj

1 ≤ pk2 . Then we can write the whole

POVM as a convex combination with probabilitypj1 of a two-

outcome POVM (M1 = V1,M2 = V2) and1−pj1 of the rest of

the POVM rescaled to be complete. Hence the POVM cannotbe extremal unless one of thepj

1, pk2 is zero�.

An immediate consequence of Lemma 17 is that we cannotuse either of the basic vectorsA1, A2 to construct our POVM.Another consequence is that from the remaining possible ba-sic vectors, we must utilise exactly one vector from each ofthe pairs{B1, B2}, {C1, C2} and{D1, D2}. We also re-quire that in choosing each basic vector, and its weight in a

column, we must respect completeness. Therefore it is notpossible to choose all three selected basic vectors with a 1 inthe top component. This leaves only six possible choices forthe basic vectors with non-zero weight:

{B1, C1, D1}{B1, C1, D2}{B1, C2, D1}{B2, C1, D2}{B2, C2, D1}{B2, C2, D2} (53)

It is not difficult, although tedious, to verify that the onlycombination that can be made complete with positive weightsis the combination{B1, C1, D2}. We recommend the useof a standard computational package for algebraic manipu-lation, with which it can readily be verified that the selec-tion {B2, C1, D2} is linearly dependent, and that the remain-ing possibilities other than{B1, C1, D2} all require neg-ative weights in order to satisfy completeness [37]. For{B1, C1, D2} the weights are also fixed uniquely, as the solu-tion involves a set of 3 linearly independent equations in 3 un-knowns. It turns out that two of the three weights are 1 (thoseof B1 andC1), and as these elements are also 2-outcome ex-trema, they cannot be perturbed in any way while keeping thewhole POVM feasible. Hence this whole 3-outcome POVMis extremal, and we have the following:

Lemma 18. The only genuine 3-outcome extremal POVMforOO symmetries, upto relabelling the outcomes, is given bythe POVM with elements represented by the following triples:

M1 = (0, 0,2d

(d+ 2)(d− 1))

M2 = (0,d− 2

d− 1,

d(d− 2)

(d+ 2)(d− 1))

M3 = (1,1

d− 1,

d− 2

(d+ 2)(d− 1)), (54)

where we have made the slight abuse of notation that a POVMelement will be set equal to the triple representing it.

2. Locally attaining the extreme points of theOO symmetricPOVMs

In trying to attain the extreme points derived above, wewill first consider only the 2-outcome extrema, given in equa-tion (51).The important question is how to realise the extremepoints of this polyhedron using LOCC measurements. ThemeasurementA and its complement are trivial, they corre-spond to the do nothing POVM. The measurementD and itscomplement are also not too difficult to obtain. It is easy toconfirm that if Alice and Bob carry out an ‘orthogonal twirl’[31,38] followed by the locally implementable projection:

11

d−1∑

i=0

|ii〉〈ii| (55)

then they will achieve the POVMD, and so bothD and itscomplement can be attained.

The pointsB,C, and complements are a little more subtle.However, by performing an orthogonal twirl [38] it is not dif-ficult to show that if we can find a set of overcomplete purestates on Alice’s particle{|q〉}, q = 1...L, L ≥ d with theproperties that:

d

L

L∑

q=1

|q〉〈q| = IA (56)

∀ q tr{|q〉〈q|(|q〉〈q|)T } = 0 (57)

then the following local measurements:

d

L

L∑

q=1

|q〉〈q| ⊗ |q〉〈q| (58)

and

d

L

L∑

q=1

|q〉〈q| ⊗ (|q〉〈q|)T (59)

will attain pointsB andC respectively when preceded by theorthogonal twirl. It remains to show that a set of pure statessatisfying equations (56) and (57) exists. Two pure states withproperty (57) are:

|u〉 := 1/√

2(|r〉 + i|s〉)|v〉 := 1/

√2(|r〉 − i|s〉) (60)

for any two computational basis states|r〉, |s〉. Suppose thatdis even. Then Alice can obtain a complete set of{|q〉} satisfy-ing the properties by partitioning her space in 2-level orthog-onal subspaces, and constructing a|u〉, |v〉 for each subspace.If d is odd, then she can do the same until she is left with a3-level subspace. We therefore need to construct such a set ofpure states for this 3-level subspace. One route to a solution isas follows. Without loss of generality assume that this residualsubspace is spanned by computational basis states|1〉, |2〉, |3〉.Then Alice can pick any irreducible representation of a finitegroup in3 × 3 real orthogonal matrices, i.e.OOT = I. Sup-pose this set is{Om|m = 1...L}. Then the set of pure states:

Om[1/√

2(|1〉 + i|2〉)] (61)

will satisfy the requirements (56) and (57). It seems likelythat this solution for oddd is overly complicated, however,we were not able to find any significant simplifications.

Therefore we see that all extreme points of the set of 2-outcome PPTOO symmetric measurements can be obtainedby local protocols involving only one way communication.Now we need to turn to the 3-outcome element presented

in equation (54). Given the existence of a set of pure states{|q〉|q = 1...L, L ≥ d} satisfying the conditions (56) and(57), it is not too difficult to verify that the local POVM de-ifened by the following elements:

N1 =d

L

L∑

q=1

|q〉〈q| ⊗ |q〉〈q|

N2 =d

L

L∑

q=1

|q〉〈q| ⊗ (|q〉〈q|)T

N3 =d

L

L∑

q=1

|q〉〈q| ⊗ [11B − |q〉〈q| − (|q〉〈q|)T ] (62)

achieves theM1,M2,M3 of eq.(54) respectively when pre-ceded by the orthogonal twirl.

Therefore all extreme points for N-outcome OO symmetricPOVMs can be attained by local protocols.

VII. SUMMARY & DISCUSSION

The classification of Separable and PPT operations seemsto be particularly suited to the question of measurements,where we find that a POVM can be implemented by a separa-ble operation iff its elements are separable, and by a PPT oper-ation iff its elements are PPT. As the Separable/PPT POVMsare mathematically more tractable than the LOCC measure-ments, and as we have the (strict) inclusions LOCC⊂ Sepa-rable⊂ PPT⊂ Global, these classes of operation are usefulfor deriving bounds on what we can achieve with LOCC ob-servations. For this reason it is useful to have techniques forderiving extremal measurements under the constraints of sep-arability and PPT-ness. In this context we have discussed anumber of useful tools. The partial transposition isomorphismis particularly useful for PPT measurements under partiallyconjugated symmetries, whereas the linear independence re-quirements and basic vectors approach are more generallyuseful for extremising POVMs under any homogeneous con-straints. Applying these techniques we were able to show thatfor Isotropic, Werner, Bell andOO symmetries, the separa-bility of POVM elements is necessary and sufficient for localimplementability. There is no obvious reason why this is thecase, as it is clear from the results of [30] that not all separablePOVMs can be implemented locally.

Perhaps the most immediate consequences of our resultsare for local state discrimination. Imagine that we have beengiven one ofi statesρi with probabilitypi. What is the bestway to tell which state we have been given? Questions suchas this are significant for entanglement distillation protocols,where ‘target’ states are often locally measured to give in-formation about ‘source’ states [10,13], and also for crypto-graphic schemes. In such scenarios one expects the extremallocal POVMs to be the optimal possible measurements formost reasonable cost functions, regardless of the number of

12

states or prior probabilities. Therefore our results also essen-tially give optimal protocols for the local discriminationofIsotropic, Werner, Bell andOO states. In the context of statediscrimination the Bell symmetries give the largest (in gen-eral) separation between local and global state discrimination.If we have global access to one of the four orthogonal Bellpure states, then we can discriminate them perfectly, poten-tially obtaining 2 classical bits of information. However,ifwe are restricted to acting locally, the best measurement wecan do has at most 2 non-zero outcomes, showing that we canlocally obtain at most 1 bit of information. Unfortunately thisdoes not supply an interesting bit hiding scheme in the mannerof [9,21], as the technology required to implement the schemeis the same as the technology required to break it - Alice andBob can share prior Bell states and use them to teleport Al-ice’s state to Bob, thereby allowing Bob full access to the statewithout having to meet Alice. Nevertheless, our results givethe first examples of mixed states for which a full quantitativeanalysis of optimal state discrimination can be performed.

We hope to be able to apply the techniques from this pa-per to more ambitious symmetries, perhaps considering morespecific cost functions or prior probabilities in order to makethe calculations more tractable. A fuller understanding oflocal measurements could have applications not only for lo-cal state discrimination and entanglement distillation, but alsofor any situation in which local observations are required todemonstrate a phenomenon, such as in demonstrations of non-locality, and also the recent program of research on the exper-imental implementation of entanglement witnesses [14].

VIII. ACKNOWLEDGEMENTS

SV thanks Oliver Rudolph, Adam Brazier and TiloEggeling for sharing their insights, and Paolo Lo Presti& Mauro D’Ariano for discussions about global extremalPOVMs. We acknowledge financial support from EC projectEQUIP (IST-1999-11053), EC project ATESIT (contract IST-2000-29681), EC project ATESIT (contract IST-2000-29681),INFM PRA 2001 CLON, US Army Grant DAAD 19-02-0161,the UK EPSRC and the European Science Foundation Pro-gramme on Quantum Information Theory and Quantum Com-puting.

IX. APPENDIX 1: NO ‘NAIVE’ SOLUTION POSSIBLE FOROO SYMMETRIES.

In this appendix we will see that no ‘naive’ solution can befound forOO symmetric POVMs with the same simplicity asthe solution for the Isotropic/Werner measurements. Firstweneed to clarify what we mean by ‘naive’. We will consider lo-cal measurement protocols consisting of POVM elementsNk,each of which can be written in the following form:

Nk = xkX + ykY + zkZ (63)

where the{X,Y, Z} are orthogonal projectors that sum to theidentity, and the (xk, yk, zk) are triples of real coefficients.Then the requirements of completeness and positivity meanthat the sets of{xk}, {yk} and{zk} must form probabilitydistributions. We will also assert thatany such valid POVMis also local. Under these assertions we will also impose thefollowing requirement:

• Any PPT POVM ofOO symmetry can be attained byperforming an orthogonal twirl followed by a (local)measurement of the stipulated form (63).

We will show that no such solution is possible for theOOcase, even though a solution with essentially the same featureswas possible for the Isotropic case.

Let us represent each POVM elementNk by the columnvector ~vk = (xk, yk, zk)T . Then under twirling,Nk will betaken to anOO symmetric POVM element. Let this elementbe represented by the column vector~wk = (ak, bk, ck)T , us-ing the same notation as the main text. There will be a lineartransformationL, such thatL · ~vk = ~wk represents the effectof the twirling.

As stated in the main text, the the only non-zero elements inextremal (PPT) POVMs must be linearly independent (Theo-rem 5). Let us represent the set of general extremal POVMs ofthe stipulated form (63) by the set{ri}, and the extrema of thePPTOO symmetric form by the set{si}. As a consequenceof the linear independence requirement, every one of these ex-trema can be expressed as3 × 3 matrices, where the columnsare linearly independent and represent each of three non-zeroPOVM elements. Hence two general membersr ∈ {ri} ands ∈ {si} of these sets can be written:

r =

x1 x2 x3

y1 y2 y3z1 z2 z3

, s =

a1 a2 a3

b1 b2 b3c1 c2 c3

. (64)

As the columns of these matrices are linearly independent, thematrices are invertible. This will be important in the subse-quent discussion.

Now let us see the implications of the above assertions forthe form of the transformationL. We are forced to have thefollowing properties:

• L must have non-negative matrix elements, as its ele-ments are calculated from the traces of products of non-negative operators.

• Each row ofL must sum to 1. This is from the fact thattwirling retains completeness of the POVM.

• L must be invertible. As we wishall PPTOO POVMsto be obtainable from the stipulated protocol, the ex-treme points{si} of the PPTOO symmetric POVMsmust be contained within set obtained fromL acting onthe extrema{ri} of the stipulated measurement. Con-sequently for somer, s we haveL · r = s, and henceL−1 = r · s−1.

13

Let us now consider the transformationR of equation (48).The vector:

R · L · ~vk = R · ~wk (65)

gives the partial transpose of the twirled POVM element cor-responding toNk. As we wishanyPPT POVM be attainableby a local protocol of our postulated form, and as we requirethatL be invertible, we also require that both:

L · ~vk ≥ 0 ∀ k (66)

and

R · L · ~vk ≥ 0 ∀ k (67)

hold if and only if the sets{xk}, {yk} and{zk} form proba-bility distributions. We also must require the following prop-erties for the matrix productR · L:

• R · L must contain only non-negative elements. Thiscan be seen as follows. If we pick a three element localmeasurement characterised by:

~v1 = (1, 0, 0)T ; ~v2 = (0, 1, 0)T ; ~v3 = (0, 0, 1)T

(68)

then this must produce vectors~wk with non-negativepartial transposition, and it can readily be verified thatthis implies that the elements ofR · L must be non-negative.

• The rows ofR · L must sum to 1. This can be shownfrom the fact that the same property holds for bothRandL.

Now consider the following disallowed choice for the vectors~vk, whereǫ is small and positive:

~v1 = (1/3, 1/3,−ǫ)T ; ~v2 = (1/3, 1/3, (1 + ǫ)/2)T ;

~v3 = (1/3, 1/3, (1 + ǫ)/2)T . (69)

This choice of vectors does not give a valid choice of measure-ment vectors, as the first one contains a negative component.However, the vectors~v2 and ~v3 can in principle come fromvalid probability distributions by themselves, and so it mustbe the case that eitherL · ~v1 orR · L · ~v1 contains a negativecomponent. It can be shown that this, together with the factthat bothL andR · L must be row stochastic, implies that atleast one row of eitherL orR · L must be(0, 0, 1). Similarlywe can ‘place theǫ’ in different rows in the above vectors toshow that in fact we require that at least one row of eitherL orR ·L must be(0, 1, 0) and at least one row should be(1, 0, 0).Regardless of where these rows appear, asR = R−1 containssome negative matrix elements, no such solution is possible,as bothL andR · L are required to have non-negative matrixelements. Although the proof that we have given is tailored toOO symmetries, it can be modified relatively easily for othersituations, including bases of Bell diagonal states.

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14

[21] T. Eggeling and R.F. Werner, Phys. Rev. Lett.89, 097905(2002).

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Rev. A64, 052309 (2001).[26] R.F. Werner, Phys. Rev. A40, 4277 (1989).[27] A. Peres, Phys. Rev. Lett.77 1413-1415 (1996).[28] In [20] this observation was credited to unpublished work by

Eric Rains. Hence here we supply a brief indication of a possi-ble proof. Given a PPT POVM{Mi} define the CP map:

ρ →∑

i

tr(Miρ)|i〉〈i| ⊗ |i〉〈i| (70)

where{|i〉} is a basis of pure orthogonal states. This is a PPToperation, and leads to the implementation of{Mi} if it is fol-lowed by a local von Neumann measurement by Alice and Bobin the basis{|i〉}.

[29] C.H. Bennett, D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolinand B.M. Terhal, Phys. Rev. Lett.82 (1999) 5385.

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[31] TheIsotropic twirlingoperationT in d× d dimensions,d > 2,is defined asT (Q) =

∫

dµU (U ⊗ U∗)Q(U ⊗ U∗)†, wherethe integral is performed with respect to the Haar measure,

U ∈ U(d). The analogous operations for the Bell, Werner andOO symmetries will be referred to as theBell twirl, Wernertwirl andOrthogonal twirlrespectively.

[32] See entry for theKrein-Milman theorem, inCollins Dictionaryof Mathematics, Collins (1989). For more detail see Section 18,Convex Analysis, R. T. Rockafellar, Princeton (1970).

[33] G. M. D’Ariano and P. Lo Presti, quant-ph/0301110.[34] There is at least one other computable separability criterion

known to be independent of PPT that also gives homogeneousconstraints, namely the criterion first proposed by Rudolph(O.Rudolph, quant-ph/0202121). Hence the techniques discussedmay yield further bounds on LOCC observations using that cri-terion instead of PPT.

[35] The commutant of a groupg of unitaries is defined as the set ofall operatorsA for which [U, A] = 0 for all U ∈ g.

[36] Theorem4.3.33, Matrix Analysis, R.A. Horn and C.A. Johnson,Cambridge University Press (1985).

[37] A Maple 7script for this task is available by emailing a requestto s.virmani@ic.ac.uk.

[38] Under orthogonal twirling the product operatorA⊗B becomesan OO symmetric operator with the following values for thetriple (a, b, c):

a =tr{ABT }

d

b =1

d(d − 1)[tr{A}tr{B} − tr{AB}]

c =2

(d + 2)(d − 1)

[

tr{A}tr{B} + tr{AB}

2−

tr{ABT }

d

]

15