A Proof of optimality for the double trine ensemble

# Optimal sequential measurements for bi-partite state discrimination

## Abstract

State discrimination is a useful test problem with which to clarify the power and limitations of different classes of measurement. We consider the problem of discriminating between given states of a bi-partite quantum system via sequential measurement of the subsystems, with classical feed-forward of measurement results. Our aim is to understand when sequential measurements, which are relatively easy to implement experimentally, perform as well, or almost as well as optimal joint measurements, which are in general more technologically challenging. We construct conditions that the optimal sequential measurement must satisfy, analogous to the well-known Helstrom conditions for minimum error discrimination in the unrestricted case. We give several examples and compare the optimal probability of correctly identifying the state via global versus sequential measurement strategies.

## I Introduction

The problem of quantum state discrimination is most naturally thought of as a task in quantum communications, although it also has applications elsewhere in quantum information theory and quantum metrology (1); (2); (3); (4); (5). The communications scenario is as follows: a sender, traditionally called Alice, chooses a quantum state drawn from a given set with associated a priori probabilities , and sends a system prepared in this state to a receiver, Bob. Bob knows the allowed set of states and their associated probabilities, and his task is to determine which state was sent, thereby recovering the message sent by Alice. The task was first considered in the pioneering work of Helstrom, Holevo, and others in the late 60s and 70s (6); (7); (8); (9); (10); (11); (12). Various strategies exist, each optimising a different figure of merit (see e.g. (10); (13); (14); (15); (16); (17); (18); (19); (20)), and for arguably the simplest such figure, minimising the probability of error in identifying the state, necessary and sufficient conditions for a quantum measurement strategy to be optimal are known (8); (9).

More recently, state discrimination has proved a useful test problem with which to clarify the power and limitations of different classes of measurement. For information encoded across multiple quantum systems, the ability to measure jointly is strictly more powerful (but in general technologically more challenging) than the ability to measure each subsystem independently, even if many rounds of classical communication between systems are allowed. Intuitively, one might expect the difference in performance to be more pronounced when information is encoded in entangled states. That this is not necessarily the case was first revealed through two state discrimination problems. The first, so-called “non-locality without entanglement”, gave a set of multi-partite orthogonal product states between which perfect discrimination is not possible using only local measurements and classical communication (21). The second, complementary and no less surprising, showed that any two orthogonal pure states, regardless of entanglement or multi-partite structure, may be perfectly discriminated using only sequential measurement, i.e. local measurement on each system, with classical feed-forward (22). This was later extended to show that any two in general non-orthogonal pure states may be discriminated optimally by sequential measurement of the subsystems, according to the commonly used minimum error (23) and unambiguous discrimination strategies (24); (25); (26).

Beyond the two state examples, the situation becomes much less clear: for the next simplest example of discriminating three possible qubit states given two copies, it was postulated by Peres and Wootters in 1991 that local measurement was strictly weaker than joint measurement on both copies (27), and only twenty years later was it finally proved that such a gap exists for this problem, for the minimum error strategy (28).

In this paper we consider sequential measurements on a bi-partite system; i.e. subsystem A and B are measured in turn, and the choice of measurement performed on subsystem B is allowed to depend in general on the result of measurement of A. This is often a physically relevant class of measurement; for example if A and B are in different labs it is easy to imagine that feedforward of measurement results from lab A to lab B would be practical but many rounds of classical communication could become unfeasible. Alternatively if A and B interact only weakly or not at all (e.g. photons), joint measurements are difficult to perform, while classical feed-forward from one detector to another apparatus is relatively easily achieved with current technology (see e.g. (29) for such an experiment in the state discrimination context). It is natural then to ask how well information can be retrieved with this restriction on the measurement strategy that may be employed. Further, implementations of joint measurement strategies for extracting information may provide applications for small quantum processors (30), and it is useful to understand when the additional experimental challenge of joint measurement may provide a significant advantage over local measurement strategies. For simplicity, we restrict to bipartite instead of the more general multipartite state discrimination.

We begin with the case where the bipartite state is simply a two-copy state. We construct necessary conditions that a given sequential measurement must satisfy to be optimal in the sense of minimising the error in determining the state, analogous to the well-known Helstrom conditions (8); (9). We further find a condition which is both necessary and sufficient, but which requires optimisation over an arbitrary measurement on one subsystem. We illustrate the two-copy case through the example of the trine states considered in (27); (28), and give the probabilities of correctly identifying the state for sequential and global strategies, as well as discussing features of the optimal measurements in each case.

We extend the discussion to arbitrary bi-partite states, and as an example give the optimal sequential strategies for discriminating three Bell states, and for discriminating the so-called domino states introduced by Bennett et al. in (21). Finally we discuss an interpretation for our necessary and sufficient condition in terms of a related discrimination problem.

## Ii Review: Helstrom conditions

We first recall the minimum error problem, where there are no restrictions on the allowed measurement: a quantum system is prepared in one of a known set of states with associated probabilities . Any physically allowed measurement may be represented by a POVM (positive operator-valued measure) (31), also referred to as a POM (probability operator measure) (11), that is, a set of Hermitian operators satisfying:

 πi ≥ 0, ∑iπi = \mathbbmss1.

For a measurement described by operators , if outcome is taken to indicate state , the probability of correctly identifying the state is given by:

 Pcorr=∑ipiTr(ρiπi). (1)

The operators describing the optimal measurement satisfy the following conditions (5); (8); (9); (32):

 ∑ipiρiπi−pjρj ≥ 0,∀j (2) πi(piρi−pjρj)πj = 0,∀i,j. (3)

It is worth noting that the conditions are not independent, as the second follows from the first. Condition (3) may be thought of as analogous to the condition in an optimisation problem that the first derivative vanish at a stationary point, while condition (2) is analogous to the second derivative condition: it is the sign of the second derivative which determines whether the corresponding point is a local maximum or local minumum. Condition (3) is therefore necessary but not sufficient for to be an optimal measurement, however (2) is both necessary and sufficient. We give here a sketch of the proof, following the treatment of (32), which is extended to the sequential case in the rest of the paper.

If is optimal then for all other physically allowed measurements we require

 Pcorr({πi})≥Pcorr({π′i}).

From this we obtain

 ∑ipiTr(ρiπi)−∑jpjTr(ρjπ′j) ≥ 0 ∑jTr[(∑ipiρiπi−pjρj)π′j] ≥ 0 (4)

Note that for positive operators , it is always true that , which may be seen by evaluating the trace in the eigenbasis of :

 Tr(AB)=∑i⟨ai|AB|ai⟩=∑iai⟨ai|B|ai⟩≥0, (5)

where are the eigenvalues of , are the eigenkets of , and the inequality follows from the positivity of . As is a positive operator it is therefore clear that condition (2) is sufficient in order for the inequality (4) to be satisfied. That this condition is also necessary may be shown by introducing the Hermitian operators

 Gj=∑ipi12{ρi,πi}−pjρj.

Now if such that , the variation

 π′i=(\mathbbmss1−ϵ|λ⟩⟨λ|)πi(\mathbbmss1−ϵ|λ⟩⟨λ|)+ϵ(2+ϵ)|λ⟩⟨λ|δij (6)

results in a measurement with higher probability of success than , which therefore cannot be an optimal measurement. Finally it is possible to show (32) that

 ∑jGjπj=0

and thus

 ∑ipi12{ρi,πi}=∑jpjρjπj.

Thus the requirement reduces to condition (2), which is therefore both necessary and sufficient.

It is useful to denote . We finish by noting that for an optimal measurement , we require

 Pcorr=Tr(Γ)=∑jpjTr(ρjπ′j),

and therefore

 ∑jTr((Γ−pjρj)π′j)=0.

As discussed above, for positive operators , , , and it is clear from eqn (5) that equality holds if and only if . Thus we require that each term in the sum be identically zero, which further requires

 (Γ−pjρj)π′j=0,∀j (7)

for any optimal measurement . This is an alternative necessary (but not sufficient) condition, and is sometimes useful for finding optimal measurements. It also implies, on summing over , that is unique, for any optimal (see also (33)).

## Iii Two copy state discrimination with sequential measurement

### iii.1 Necessary conditions

Now let us consider the two copy case, with sequential measurement. Suppose therefore we are provided with two copies of a state drawn from a known set with associated probabilities . The allowed measurement procedures are as follows: make a measurement described by some POVM on system ; given outcome make a measurement on system , as shown in the tree in Figure 1.

As the choice of measurement on system can in general depend on the outcome of measurement on , we denote the associated POVM , where for all and , and for each

 ∑iNBi|j=\mathbbmss1B.

The measurement on the joint system is thus of the form , with the probability of correctly identifying the state given by:

 Pcorr = ∑ijpiTrAB(ρAi⊗ρBiMAj⊗NBi|j) (8) = ∑ijpiTrA(ρAiMAj)TrB(ρBiNBi|j).

In the following we drop the superscripts , , whenever it is not confusing to do so. We begin by pointing out that each of , may be interpreted as an optimal measurement for an appropriately defined discrimination problem, as follows. We first note that, given measurement result on system , we can update the probabilities as follows, using Bayes’ rule:

 P(i|Mj)=P(i,Mj)P(Mj)=piTrA(ρiMj)∑kpkTrA(ρkMj)=pi|j. (9)

Thus given result on system , the possible states of system occur with probabilities . Clearly should thus be optimal for discriminating the states with the updated priors , and thus a necessary condition is

 ∑ipi|jρiNi|j−pk|jρk≥0,∀k,

or equivalently, using eqn. (9):

 ∑ipiTrA(ρiMj)ρiNi|j−pkTrA(ρkMj)ρk≥0,∀k, (10)

which must hold for each . This set of conditions is necessary, but not sufficient (we haven’t done any optimisation over ). Finally, summing over gives:

 TrA(∑i,jpi(ρi⊗ρi)(Mj⊗Ni|j)−pkρk⊗ρk)≥0,∀k, (11)

which is rather similar to the Helstrom condition (2), but with a partial trace over system .

Conversely, we can re-write eqn (8) as follows:

 Pcorr = ∑jTrA(∑ipiTrB(ρBiNBi|j)ρAiMAj) = ∑jcjTrA(σAjMAj),

where we have defined:

 σAj = ∑ipiTrB(ρBiNBi|j)ρAi∑kpkTrB(ρBkNBk|j), (12) cj = ∑ipiTrB(ρBiNBi|j). (13)

We can interpret the trace one operators as density operators, and if we further define probabilities , it follows that must be optimal for discriminating the states with probabilities . The Helstrom condition (2) then gives:

 ∑jqjσAjMAj−qkσAk≥0,

which may be re-written as:

 ∑j(∑ipiTrB(ρBiNBi|j)ρAi)MAj −∑ipiTrB(ρBiNBi|k)ρAi ≥ 0. (14)

Finally we obtain

 TrB(∑ijpi(ρAi⊗ρBi)(MAj⊗NBi|j) −∑ipi(ρAi⊗ρBi)(\mathbbmss1A⊗NBi|k)) ≥ 0. (15)

Again, this is necessary, but not sufficient (this time we haven’t done any optimisation over ). One might hope that the conditions (10,15) when taken together are also sufficient, and could then imagine that it may be possible to construct an iterative procedure for numerical solution of the optimization problem. However, this turns out not to be the case; we will return to this point later. Each of conditions (10,15) however have a clear interpretation; note that it might have been expected that should be optimal for the updated priors given measurement of ; that plays a complementary role for a different discrimination problem is less obvious a priori. We return to give an interpretation of this discrimination problem later.

### iii.2 A necessary and sufficient condition

We now turn to the problem of simultaneously optimising both the measurement on and that on system . We find that the condition

 ∑i,jpiTrB(ρiNi|j)ρiMj−∑kpkTrB(ρk˜Nk)ρk≥0 (16)

where is any physically allowed measurement on system is both necessary and sufficient for optimality of . Unfortunately this still contains an arbitrary measurement on system , and thus is not as readily applicable as the original Helstrom conditions to verify optimality of a candidate measurement. Nevertheless we will give examples in which it can be used to prove optimality analytically. We also note that the inclusion of an arbitrary measurement on one subsystem means that analysis beyond the bipartite case becomes complicated and our method is not readily extended to multipartite discrimination.

We begin by proving sufficiency of condition (16). If is optimal among sequential measurements, we require

 TrAB(∑i,jpi(ρi⊗ρi)(Mj⊗Ni|j))≥ TrAB(∑k,lpk(ρk⊗ρk)(M′l⊗N′k|l)),

for all . Inserting the identity and re-arranging gives

 ∑lTrAB[(∑i,jpi(ρi⊗ρi)(Mj⊗Ni|j) −∑kpk(ρk⊗ρk)(\mathbbmss1⊗N′k|l))M′l] ≥ 0, ∑lTrA[(∑i,jpiTrB(ρiNi|j)ρiMj −∑kpkTrB(ρkN′k|l)ρk)M′l] ≥ 0.

Condition (16) is therefore sufficient, if is any allowed measurement on .

That condition (16) is also necessary may be seen as follows: as in the unrestricted case, we introduce the manifestly Hermitian operator:

 ΓAsym=∑i,jpiTrB(ρiNi|j)12{ρi,Mj}.

Suppose now that there exists some and some such that

 ⟨λ|ΓAsym−∑kpkTrB(ρk˜Nk)ρk|λ⟩<0.

We can construct a variation of as follows:

 M′j = (\mathbbmss1−ϵ|λ⟩⟨λ|)Mj(\mathbbmss1−ϵ|λ⟩⟨λ|),0≤j

where . Note that if has outcomes, the primed measurement on system has outcomes. Now note that

 Pcorr({M′j⊗N′i|j})=Pcorr({Mj⊗Ni|j}) −ϵTrAB(∑i,jpiρi⊗ρi(|λ⟩⟨λ|Mj+Mj|λ⟩⟨λ|)⊗Ni|j) +2ϵTrAB(∑ipiρi⊗ρi(|λ⟩⟨λ|⊗˜Ni))+O(ϵ2) =Pcorr({Mj⊗Ni|j}) −2ϵ⟨λ|ΓAsym−∑ipiTrB(ρi˜Ni)ρi|λ⟩+O(ϵ2) >Pcorr({Mj⊗Ni|j}).

Finally we note that, by virtue of the fact that is an optimal measurement for discriminating the states , it follows that , where is defined as:

 ΓA=∑i,jpiTrB(ρiNi|j)ρiMj.

Thus we require

 ΓA−∑kpkTrB(ρk˜Nk)ρk≥0,

which completes our proof.

## Iv Example: The double trine ensemble

As an example we consider the so-called double trine ensemble: two copies of the trine states, for which , and

 |ψj⟩=1√2(|0⟩+e2πji/3|1⟩).

These each occur with prior probabilities , and have the symmetry property

 |ψj⟩=Uj|ψ0⟩

where is a rotation of around the -axis in the Bloch sphere.

### iv.1 Optimal sequential measurement

For the two-copy case, Chitambar and Hsieh (28) showed that the optimal sequential measurement rules out one state of the three in the first step, and corresponds to the Helstrom measurement to distinguish between the remaining two states in the second step. We first briefly present this optimal measurement, and then use it to demonstrate our conditions.

The optimal sequential measurement thus makes the measurement on the first copy, where the states form the so-called anti-trine ensemble:

 |ψ⊥j⟩=1√2(|0⟩−e2πji/3|1⟩).

Following this measurement, the updated priors become , and is then the optimal measurement to distinguish the two remaining equiprobable pure states . This is a case of the well-known Helstrom measurement, and is a projective measurement in a basis located symmetrically around the signal states (see e.g. (1)). Thus for , , and for we denote , where

 |ϕ1|0⟩ = 1√2(|0⟩+i|1⟩), |ϕ2|0⟩ = 1√2(|0⟩−i|1⟩), |ϕ0|1⟩ = 1√2(|0⟩+eiπ/6|1⟩)=U|ϕ2|0⟩, |ϕ2|1⟩ = 1√2(|0⟩−eiπ/6|1⟩)=U|ϕ1|0⟩, |ϕ0|2⟩ = 1√2(|0⟩+e−iπ/6|1⟩)=U2|ϕ1|0⟩, |ϕ1|2⟩ = 1√2(|0⟩−e−iπ/6|1⟩)=U2|ϕ2|0⟩.

These states, along with the trine and anti-trine states are shown in the Bloch sphere picture in Figure 2.

### iv.2 Necessary and sufficient conditions

We now use this strategy to illustrate the conditions presented in the previous section. From the symmetry we find that , for all , where is the probability of success of the Helstrom measurement distinguishing between two equiprobable states with overlap , i.e. from (11):

 pH=12(1+√1−|⟨ψ0|ψ1⟩|2)=12(1+√32).

By construction, this measurement strategy satisfies condition (10). To evaluate (15) and the necessary and sufficient condition (16), we first calculate :

 ΓA = ∑i,jpiTr(ρiNi|j)ρiMj = ∑i,j13pH(1−δij)(|ψi⟩⟨ψi|)(23|ψ⊥j⟩⟨ψ⊥j|) = 13pH(∑i|ψi⟩⟨ψi|)(∑j23|ψ⊥j⟩⟨ψ⊥j|) = 12pH\mathbbmss1=14(1+√32)\mathbbmss1,

where in the last line we have used . We first show that the strategy satisfies condition (15). We obtain

 ∑ipiTr(ρiNi|j)ρAi = 13∑ipH(1−δij)ρi = 12pH(\mathbbmss1−23ρj)

from which it is clear that condition (15) is satisfied for each . Finally, to prove that this is indeed the optimal strategy, we must show that it satisfies the necessary and sufficient condition (16). As we have shown that is proportional to the identity, this amounts to showing that for any allowed measurement on system , the largest eigenvalue of the operator

 ∑kpkTr(ρk˜Nk)ρk

is bounded by . The proof that this holds is straight-forward but needs a few steps, and the details are given in appendix A.

The probability of correctly identifying the state using the optimal sequential measurement is given by

 Pseqcorr=Tr(ΓA)=pH=12(1+√32)≃0.933.

### iv.3 Comparison of global and sequential schemes

For comparison we recall the globally optimal measurement strategy, also discussed in (28). Recall the double trine ensemble satisfies , where is a rotation of around the -axis in the Bloch sphere. For sets with such symmetry the optimal measurement was shown by Ban et al to be given by the so-called square-root measurement (34) (also known as the “pretty-good measurement” (35)). In this case, the optimal measurement corresponds to a projective measurement, with operators , where

 |Φj⟩ = 1√3(|0⟩|0⟩+e2πji/31√2(|01⟩+|10⟩) (17) +e4πji/3|1⟩|1⟩).

The probability of correctly identifying the state is

 Pglobcorr=12+√23≃0.971.

Note that the probability of identifying the state correctly achieved by the optimal sequential measurement is greater than of that achieved by the optimal global measurement. In systems where joint measurement is technologically challenging it is thus perhaps difficult to argue that the additional experimental effort is merited by the improvement in performance in this case.

We comment finally on the optimal sequential measurement as an approximation to the optimal global measurement. For the optimal sequential measurement, given above, we obtain

 π0 = 23(|ψ⊥1⟩⟨ψ⊥1|⊗|ϕ0|1⟩⟨ϕ0|1| +|ψ⊥2⟩⟨ψ⊥2|⊗|ϕ0|2⟩⟨ϕ0|2|) π1 = (U⊗U)π0(U⊗U)† (18) π2 = (U⊗U)2π0((U⊗U)†)2

Considering , after a little algebra we find

 |ψ⊥1⟩⊗|ϕ0|1⟩ = 12e−πi/12[√1+2cos2π12|α0⟩ +i√1+2sin2π12|β0⟩] |ψ⊥2⟩⊗|ϕ0|2⟩ = 12eπi/12[√1+2cos2π12|α0⟩ −i√1+2sin2π12|β0⟩]

where

 |α0⟩ = (1+2cos2π12)−1/2(cosπ12|00⟩ +1√2(|01⟩+|10⟩)+cosπ12|11⟩) |β0⟩ = (1+2sin2π12)−1/2(sinπ12|00⟩ +1√2(|01⟩−|10⟩)−sinπ12|11⟩).

Thus we can write

 π0 = 13(1+2cos2π12)|α0⟩⟨α0| +13(1+2sin2π12)|β0⟩⟨β0| = 13(2+√32)|α0⟩⟨α0| +13(2−√32)|β0⟩⟨β0|.

Note that , and hence this is the eigendecomposition of the operator. We further note that is orthogonal to the signal state and thus does not contribute to the probability of identifying the state. The remaining eigenvector is an approximation to , the state onto which the optimal global measurement projects; an amazingly good one in fact: it turns out . Due to the weighting factor, the overlap between and is given by .

The state is thus very close to a superposition of and , with appropriate normalisation:

 |Φ0⟩≃|α0⟩ = (1+2cos2π12)−1/2(eπi/12|ψ⊥1⟩⊗|ϕ0|1⟩ +e−πi/12|ψ⊥2⟩⊗|ϕ0|2⟩).

The optimal sequential measurement, on the other hand, is formed from a mixture of projectors onto these same states. It gives additional information – one state is ruled out with certainty – at the expense of a slightly lower probability of success.

### iv.4 A non-optimal sequential measurement

The example of the trine states is further illuminating, as there exists another measurement strategy which satisfies both necessary conditions (10) and (15), but which is not an optimal strategy, thus demonstrating that these two conditions, when taken together, are not sufficient to define the optimal measurement. This strategy is to perform the optimal minimum error measurement at each step, with Bayesian update of the probabilities in between measurements. Note that such a strategy is known to be optimal (in fact performs as well as the best joint measurement) for a different set of states - the case of just two pure states (36); (37). For the trine states, the measurement is as follows: is the optimal one-copy minimum error measurement, which consists of weighted projectors onto the trine states themselves (11); (34), . Note that for the trine states , and thus the updated priors upon obtaining outcome are, using equation (9):

 pi|j=23|⟨ψi|ψj⟩|223∑k|⟨ψk|ψj⟩|2=16+12δij

For each , the states with these probabilities have so-called mirror symmetry – the set is invariant under reflection about . For such a set, the minimum error problem was considered by Andersson et al (38): using their results we find for the optimal measurement is of the form:

 N0|0 = (1−a2)|ψ0⟩⟨ψ0|, N1|0 = 12(a|ψ0⟩−i|ψ⊥0⟩)(a⟨ψ0|+i⟨ψ⊥0|), N2|0 = 12(a|ψ0⟩+i|ψ⊥0⟩)(a⟨ψ0|−i⟨ψ⊥0|),

where depends on the geometry of the set and the prior probabilities (38): for our case we find . The optimal measurements for are obtained by symmetry . Note that condition (10) is satisfied by construction. Turning to condition (15), we find

 Tr(ρ0N0|0) = 7475, Tr(ρ1N1|0) = Tr(ρ2N2|0)=3275,

with analogous results for . Concisely, . Finally, we can calculate :

 ΓA = ∑i,jpiTr(ρiNi|j)ρiMj = ∑i,j13(3275+4275δij)(|ψi⟩⟨ψi|)(23|ψj⟩⟨ψj|) = 3075\mathbbmss1=25\mathbbmss1.

For we obtain:

 ∑ipiTr(ρiNi|j)ρAi = 13∑i(3275+4275δij)ρi = 1675\mathbbmss1+1475ρj = 25|ψj⟩⟨ψj|+1675|ψ⊥j⟩⟨ψ⊥j|

from which it is clear that condition (15) is satisfied for each .

An analogous situation arises in state discrimination maximising the mutual information between sender and receiver - a necessary but not sufficient condition is known, and for the example of the trine states, is satisfied by both the trine measurement, which is not optimal (34), and the anti-trine measurement, which is optimal (14). We finally note that the probability of correctly identifying the state using this scheme, , is considerably worse than that given by the optimal sequential measurement from above.

## V General bi-partite case

### v.1 Necessary and sufficient conditions

Above, for simplicity, we confined our discussion of optimal sequential measurement strategies to the case of two-copy state discrimination. The conditions obtained however are easily extended to the general bi-partite case. Suppose therefore we are provided with a bi-partite state drawn from a known set , with known a priori probabilitites . If our measurement strategy is restricted to sequential measurements on each subsystem, with feed-forward, what is the best measurement to make? The allowed measurements on the joint system are again described by POVMs of the form , and the probability of correctly identifying the state is expressed:

 Pcorr=∑ijpiTrAB(ρABiMAj⊗NBi|j).

Following the same reasoning as in Section III, the necessary conditions eqns (10), (15) become:

 ∑ipiTrA(ρABiMj<