A Return to the Optimal Detection of Quantum Information
In 1991, Asher Peres and William Wootters wrote a seminal paper on the nonlocal processing of quantum information [Phys. Rev. Lett. 66 1119 (1991)]. We return to their classic problem and solve it in various contexts. Specifically, for discriminating the “double trine” ensemble with minimum error, we prove that global operations are more powerful than local operations with classical communication (LOCC). Even stronger, there exists a finite gap between the optimal LOCC probability and that obtainable by separable operations (SEP). Additionally we prove that a two-way, adaptive LOCC strategy can always beat a one-way protocol. Our results provide the first known instance of “nonlocality without entanglement” in two qubit pure states.
One physical restriction that naturally emerges in quantum communication scenarios is nonlocality. Here, two or more parties share some multi-part quantum system, but their subsystems remain localized with no “global” quantum interactions occurring between them. Instead, the system is manipulated through local quantum operations and classical communication (LOCC) performed by the parties.
Asher Peres and William Wootters were the first to introduce the LOCC paradigm and study it as a restricted class of operations in their seminal work Peres and Wootters (1991). To gain insight into how the LOCC restriction affects information processing, they considered a seemingly simple problem. Suppose that Alice and Bob each possess a qubit, and with equal probability, their joint system is prepared in one of the states belonging to the set , where and . This highly symmetric ensemble is known as the “double trine,” and we note that lying orthogonal to all three states is the singlet .
Alice and Bob’s goal is to identify which double trine element was prepared only by performing LOCC. Like any quantum operation used for state identification, Alice and Bob’s collective action can be described by some positive-operator valued measure (POVM). While the non-orthogonality of the states prohibits the duo from perfectly identifying their state, there are various ways to measure how well they can do. Peres and Wootters chose the notoriously difficult measure of accessible information Holevo (1973); *Nielsen-2000a, but their paper raises the following two general conjectures concerning the double trine ensemble, which can apply to any measure of distinguishability:
LOCC is strictly sub-optimal compared to global operations,
The optimal LOCC protocol involves two-way communication and adaptive measurements.
The set of global POVMs will be denoted by GLOBAL, and C1 can be symbolized by GLOBAL LOCC. A two-way LOCC protocol with adaptive measurement refers to at least three rounds of measurement, Alice Bob Alice, where the choice of measurement in each round depends on the outcome of the other party’s measurement in the previous round. We symbolize C2 as LOCC LOCC. In Ref. Peres and Wootters (1991) Peres and Wootters obtained numerical data to support both C1 and C2, but these conjectures have never been proven for the double trine.
Before we present our contribution to the problem, we would like to briefly highlight the legacy of the Peres-Wootters paper. Perhaps most notably is that it subsequently led to the discovery of quantum teleportation Bennett et al. (1993). Other celebrated phenomena can also directly trace their roots to Ref. Peres and Wootters (1991) such as so-called nonlocality without entanglement Bennett et al. (1999) and quantum data hiding Terhal et al. (2001); *DiVincenzo-2002a. More generally, Ref. Peres and Wootters (1991) paved the way for future research into LOCC and its fundamental connection to quantum entanglement Horodecki et al. (2009).
We finally note that in a return to Ref. Peres and Wootters (1991) of his own, Wootters constructed a separable POVM that obtains the same information as the best known global measurement Wootters (2005). A POVM belongs to the class of separable operations (SEP) if each POVM element can be decomposed as a tensor product over the two systems. SEP is an important class of operations since every LOCC operation belongs to SEP Bennett et al. (1999).
In this paper, we prove that conjectures C1 and C2 are indeed true when distinguishability success is measured by the minimum error probability, which is defined as follows. For an ensemble , the error probability associated with some identification POVM is given by . Then the minimum error probability of distinguishing with respect to a class of operations (such as LOCC, SEP, GLOBAL, etc.) is given by the infimum of error probabilities taken over all POVMs that can be generated by . Note that we can replace “infimum” by “minimum” only if is a compact set of operations. While GLOBAL, SEP and LOCC all have this property, LOCC does not Chitambar et al. (2012a, b). Hence, to properly discuss the LOCC minimum error, we must consider the class of so-called asymptotic LOCC, which is LOCC plus all its limit operations Chitambar et al. (2012b). We will prove C1 with respect to this more general class of operations.
A. Global and Separable Operations: The double trine ensemble has a group-covariant structure which greatly simplifies the analysis. In fact, Ban et al. have proven that the so-called “Pretty Good Measurement” (PGM) 111Recall that the “Pretty Good Measurement” for an ensemble is the POVM with elements , where Hausladen and Wootters (1994). is indeed an optimal global POVM for discriminating ensembles with such symmetries Ban et al. (1997). For the double trine, the PGM consists of simply projecting onto the orthonormal basis , where
The corresponding error probability is
To show that SEP can also obtain this probability, we explicitly construct a separable POVM. The idea is to mix a sufficient amount of the singlet state with each of the PGM POVM elements so to obtain separability (a similar strategy was employed in Ref. Wootters (2005)). The resulting POVM is with . It is fairly straightforward to compute that , where is a product state. This suffices to prove separability of the POVM.
B. LOCC and Asymptotic LOCC: Let us begin with a clear description of asymptotic LOCC discrimination. In general, a sequence of POVMs asymptotically attains an error probability on ensemble if for every we have for sufficiently large . If each POVM in the sequence can be generated by LOCC, then is achievable by asymptotic LOCC.
It is known that for an ensemble of linearly independent pure states, the global POVM attaining minimum error consists of orthonormal, rank one projectors Yuen et al. (1975) (see also Mochon (2006)). We strengthen this result and extend it to the asymptotic setting.
Let be an ensemble of linearly independent states spanning some space . Suppose that is the global minimum error probability of . Then there exists a unique orthonormal basis of such that: (a) A POVM attains an error probability on if and only if it can also distinguish the with no error, and (b) A sequence of POVMs asymptotically attains an error probability on if and only if it contains a subsequence that can asymptotically distinguish the with no error.
The proof is given in the Appendix. Theorem 1 essentially reduces optimal distinguishability of non-orthogonal linearly independent ensembles to perfect discrimination of orthogonal ensembles. Applying part (a) to the double trine ensemble, if an LOCC POVM could attain the error probability of Eq. (2), then it can also perfectly distinguish the states given by (1). However, these are three entangled states which, by a result of Walgate and Hardy, means they cannot be distinguished perfectly by LOCC Walgate and Hardy (2002). Therefore, the global minimum error probability is unattainable by LOCC.
But is the probability attainable by asymptotic LOCC? If it is, then part (b) of Theorem 1 likewise implies that the must be perfectly distinguishable by asymptotic LOCC. While Ref. Walgate and Hardy (2002) provides simple criteria for deciding perfect LOCC distinguishability of two qubit ensembles, no analogous criteria exists for asymptotic LOCC. The only general result for asymptotic discrimination has been recently obtained by Kleinmann et al. Kleinmann et al. (2011). Here we cite their result in its strongest form, adapted specifically for the problem at hand.
Proposition 1 (Kleinmann et al. (2011)).
If the states can be perfectly distinguished by asymptotic LOCC, then for all there is a product operator such that (i) , (ii) , and (iii) the normalized states are perfectly distinguishable by separable operations.
In the appendix we prove that these three conditions cannot be simultaneously satisfied; therefore, GLOBAL LOCC for minimum error discrimination. Here, we provide a little intuition into why Proposition 1 must be true. For every LOCC protocol that correctly identifies the given state with probability , we can think of the success probability as smoothly evolving from complete randomness () to its final average value (). Then for each , the protocol can be halted after some sequence of measurement outcomes (collectively represented by the product operator ) such that given these outcomes: (1) there is one state that can be identified with probability (which by symmetry we can assume is ), and (2) the transformed ensemble can be discriminated by a separable POVM with success probability no less than . By compactness of SEP, we let and replace (2) by the condition that a separable POVM perfectly distinguishes the post-halted ensemble.
C. LOCC LOCC: We will now compute the minimum one-way error probability for the double trine, and then describe an explicit two-way protocol with a smaller error probability. In the one-way task, Alice makes a measurement and communicates her result to Bob. Without loss of generality, we fine-grain Alice’s measurement so that each POVM element is rank one , with . Given outcome , Bob’s task is to optimally discriminate the ensemble , but now with an updated distribution given by
Here, , and we’ve used the covariance . Additionally, we can assume that , since if fails to generate a distribution with this property, by the symmetry we can always rotate such that is indeed the maximum post-measurement probability. This means we can only restrict attention to .
Next, we observe that Bob’s task of distinguishing the ensemble is no easier than distinguishing between the two weighted states and . Indeed, any protocol distinguishing the three can always be converted into a protocol for distinguishing and by simply coarse-graining over all outcomes corresponding to and . The minimum error probability in distinguishing and is readily found to be (see Appendix):
which simplifies to . In the interval , a minimum is obtained at and . This corresponds to and with an error probability of . Now, this probability lower bounds the error probability along each branch of Alice’s measurement, and therefore it places a lower bound on any one-way LOCC measurement scheme. In fact, this lower bound turns out to be tight. When Alice performs the POVM outcome will eliminate but leave the other two states with an equal post-measurement probability. Thus, in each branch we obtain the error probability , and this provides the minimum one-way error probability.
If we allow feedback from Bob, there exists better measurement strategies. The following protocol generalizes the optimal one-way scheme just described. (Round I) Alice performs the measurement with Kraus operators given by with
Here is the state orthogonal to (explicitly ). Note that this is the square-root of the POVM given by Peres and Wootters Peres and Wootters (1991). Without loss of generality, we suppose that Alice obtains outcome “” and communicates the result to Bob. Her (normalized) post-measurement states are , , and . (Round II) From Bob’s perspective, he is still dealing with the original states , but now their prior probabilities have changed to . He now proceeds as if Alice had completely eliminated the state (i.e. if she had chosen as the strength of her measurement). Specifically, he projects onto the eigenbasis of which are the states . A “” outcome is associated with and a “” outcome is associated with ; this is the optimal measurement for distinguishing between two pure states Helstrom (1976). By the symmetry of the states, it is sufficient to only consider the “” outcome, which he communicates to Alice. The conditional probabilities are , , and . These can be inverted to give . (Round III) At this point, Alice still has three distinct states , and . Here, will have the greatest probability while will have the smallest when is close to . Alice then ignores and performs optimal discrimination between just and . Letting , the minimum error probability is given by the well-known Helstrom bound Helstrom (1976) with normalized probabilities:
By symmetry, each sequence of outcomes - with , - occurs with the same probability. Hence, the total error probability across all branches is given by . The plot is given in Fig. 1. It obtains a minimum of approximately , which is smaller than the one-way optimal of . The one-way optimal probability is obtained at the point .
Discussion and Conclusions: Our results for minimum error discrimination of the double trine ensemble can be summarized as:
We thus put substantial closure to a problem first posed over 20 years ago. A primary motivation for studying this problem is to better understand the limitations of processing quantum information by LOCC. Our results complement a series of recent results in this direction Kleinmann et al. (2011); Childs et al. (2012); Chitambar et al. (2012b). In particular is Ref. Kleinmann et al. (2011) which provides a necessary condition for perfect discrimination by asymptotic LOCC discrimination (Prop. 1 above). Theorem 1 of our paper largely extends this result as we reduce asymptotic minimum error discrimination of linearly independent states to asymptotic perfect discrimination.
Our proofs of C1 and C2 are the first of its kind for two qubit ensembles, and we contrast it with previous work on the subject. C1 was first shown by Massar and Popescu for two qubits randomly polarized in the same direction Massar and Popescu (1995). However, a different distinguishability measure was used and the asymptotic case was not considered. Later, Koashi et al. showed an asymptotic form of C1 for two qubit mixed states with respect to the different task of “unambiguous discrimination” Koashi et al. (2007) (the same can also be shown for the double trine ensemble Chitambar and Hsieh (2013)). Finally, C2 has been observed by Owari and Hayashi on mixed states and only for a special sort of distinguishability measure Owari and Hayashi (2008). Our work is distinct from all previous results in that it deals with pure states and minimum error probability, a highly natural measure of distinguishability. The fact that we consider pure ensembles with three states is significant since it is well-known that any two pure states can be distinguished optimally via LOCC (i.e. LOCC GLOBAL) Walgate et al. (2000); Virmani et al. (2001). Thus, with the double trine being a real, symmetric, and pure ensemble of two qubits, we have identified the simplest type of ensemble in which LOCC GLOBAL for state discrimination.
Even more, since the double trine ensemble consists of product states (i.e. no entanglement), we have shown that “nonlocality without entanglement” can exist in even the simplest types of ensembles with more than two states. This distinction is further sharpened by considering that LOCC SEP for the optimal discrimination of the double trine. Separable operations are interesting since, like LOCC operations, they lack the ability to create entanglement. Nevertheless, SEP evidently possesses some nonlocal power as it can outperform LOCC in discriminating the double trine. Thus, entanglement and nonlocality can truly be regarded as two distinct resources, even when dealing with two qubit pure states.
Acknowledgements.We would like to thank Runyao Duan, Debbie Leung, and Laura Mančinska for helpful discussions on the topic of LOCC distinguishability.
Appendix A Appendix
a.1 Proof of Theorem 1
(a) We first recall a few general facts about minimum error discrimination. A POVM is optimal on if and only if for all , in which the operator is hermitian Holevo (1973); *Yuen-1975a; *Barnett-2009a. Since , we have
Then as and , we must have that
Our argument now proceeds analogously to the one given in Ref. Mochon (2006). Let be the projector onto , and for some POVM that obtains on , define . As the are linearly independent, there exists a set of dual states such that . We first note that implies , where . Thus, the POVM also obtains on . We next note that for all . For if this were not true for some , then we could contract with to obtain .
Next, since is an optimal POVM, the corresponding equality of Eq. (5) is . Applying to the RHS yields
Thus, (which is nonzero) lies in the kernel of for , while is an eigenvector of with eigenvalue +1 when . Hence, and , with . We obviously have , which means the original POVM can perfectly distinguish the . Conversely, any POVM that perfectly distinguishes the will satisfy , and will therefore obtain on .
Finally, let be the compact, convex set of POVMs with elements, each having support on . We have just shown that the continuous linear function given by can be maximized only by an extreme point of (rank one projectors). Convexity of implies that this extreme point must be unique.
(b) For the asymptotic statement, we will need to endow with a metric. For two POVMs and in , we can define a distance measure by , where 222A perhaps more natural distance measure between two POVMs is the difference in measurement probabilities, maximized over all trace one, non-negative inputs: . As we are only concerned with issues of convergence, it suffices to consider the equivalent metric introduced above.. Note that when is pure, we have Nielsen and Chuang (2000).
For any , define . Suppose there exists a sequence of POVMs such that for any , for sufficiently large . As is a sequence in the compact metric space , by the Weierstrass Theorem from analysis, there will exist some convergent subsequence . Continuity of implies that (recall ). However, by part (a), the POVM in obtaining is unique and so . Thus, , and so the error on of each subsequence satisfies
Conversely, if , then Nielsen and Chuang (2000), which means . By continuity of , we have .
a.2 All Conditions of Proposition 1 Cannot be Simultaneously Satisfied
Condition (iii) requires orthogonality for , and so in the basis , must take the form
where , , and denotes the complex conjugate. If is a product operator across Alice and Bob’s system, then must commute with . Here we are taking partial contractions on Alice’s system so that and are operators acting on Bob’s system. By directly computing the commutator using Eqs. (1) and (7), the condition becomes
With (condition (ii)), it is clear that . However, if , then this equation cannot hold for any . Thus, the product form constraint on requires .
Next, we focus on the range , which because of (iii), guarantees that is full rank. It is known that the can be perfectly distinguished by separable operations if and only if , where is the concurrence of the state and (see Thm. 2 of Duan et al. (2009)). We combine this with the fact that for a general two qubit state , its concurrence is given by is Verstraete et al. (2001). Therefore after noting that and writing , condition (iii) of Proposition 1 can be satisfied if and only if
To compute , we use Cramer’s Rule which says , where denotes the adjugate matrix. From (7), we have that . Substituting this into the above equation gives
where we have used (7) and Hadamard’s inequality: . It is a straightforward optimization calculation to see that under the constraint , the RHS of (A.2) obtains a minimum of 1 if and only if . This proves that condition (iii) is impossible whenever .
Appendix B Eq. (4) and the Minimum Error for one Mixed and one Pure State
We compute an analytic formula for the minimum error probability in distinguishing weighted qubit states and . The minimum error probability is given by , where is the trace norm. Since is hermitian with eigenvalues , we have . Thus, it is just a matter of computing the eigenvalues of . Taking , we write in coordinates:
For a matrix, , its eigenvalues are given by the expression . Thus, have that
Letting , we can compute that and
Therefore, we arrive at the following
For the weighted states and , the minimum error probability is
if , and if .
- Peres and Wootters (1991) A. Peres and W. K. Wootters, Phys. Rev. Lett. 66, 1119 (1991).
- Holevo (1973) A. Holevo, J. Multivar. Anal. 3, 337 (1973).
- Nielsen and Chuang (2000) M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
- Bennett et al. (1993) C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
- Bennett et al. (1999) C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 59, 1070 (1999), quant-ph/9804053 .
- Terhal et al. (2001) B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, Phys. Rev. Lett. 86, 5807 (2001).
- DiVincenzo et al. (2002) D. P. DiVincenzo, D. W. Leung, and B. M. Terhal, IEEE Trans. Inf. Theory 48, 580 (2002), quant-ph/0103098 .
- Horodecki et al. (2009) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
- Wootters (2005) W. Wootters, (2005), quant-ph/0506149 .
- Chitambar et al. (2012a) E. Chitambar, W. Cui, and H.-K. Lo, Phys. Rev. Lett. 108, 240504 (2012a).
- Chitambar et al. (2012b) E. Chitambar, D. Leung, L. Mančinska, M. Ozols, and A. Winter, (2012b), arXiv:1210.4583 .
- (12) Recall that the “Pretty Good Measurement” for an ensemble is the POVM with elements , where Hausladen and Wootters (1994).
- Ban et al. (1997) M. Ban, K. Kurokawa, R. Momose, and O. Hirota, Int. J. Theor. Phys. 36, 1269 (1997).
- Yuen et al. (1975) H. Yuen, R. Kennedy, and M. Lax, IEEE Trans. Inf. Theory 21, 125 (1975).
- Mochon (2006) C. Mochon, Phys. Rev. A 73, 032328 (2006).
- Walgate and Hardy (2002) J. Walgate and L. Hardy, Phys. Rev. Lett. 89, 147901 (2002).
- Kleinmann et al. (2011) M. Kleinmann, H. Kampermann, and D. Bruß, Phys. Rev. A 84, 042326 (2011), 1105.5132 .
- Helstrom (1976) C. W. Helstrom, Quantum detection and estimation theory (Academic Press, New York, 1976).
- Childs et al. (2012) A. M. Childs, D. Leung, L. Mančinska, and M. Ozols, (2012), arXiv:1206.5822 .
- Massar and Popescu (1995) S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 (1995).
- Koashi et al. (2007) M. Koashi, F. Takenaga, T. Yamamoto, and N. Imoto, (2007), 0709.3196 .
- Chitambar and Hsieh (2013) E. Chitambar and M.-H. Hsieh, (2013), manuscript in Preparation.
- Owari and Hayashi (2008) M. Owari and M. Hayashi, New J. Phys. 10, 013006 (2008), 0708.3154 .
- Walgate et al. (2000) J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Phys. Rev. Lett. 85, 4972 (2000).
- Virmani et al. (2001) S. Virmani, M. F. Sacchi, M. B. Plenio, and D. Markham, Phys. Lett. A 288, 62 (2001).
- Barnett and Croke (2009) S. M. Barnett and S. Croke, J. Phys. A 42, 062001 (2009).
- (27) A perhaps more natural distance measure between two POVMs is the difference in measurement probabilities, maximized over all trace one, non-negative inputs: . As we are only concerned with issues of convergence, it suffices to consider the equivalent metric introduced above.
- Duan et al. (2009) R. Duan, Y. Feng, Y. Xin, and M. Ying, IEEE Trans. Inf. Theory 55, 1320 (2009), 0705.0795 .
- Verstraete et al. (2001) F. Verstraete, J. Dehaene, and B. DeMoor, Phys. Rev. A 64, 010101 (2001).
- Hausladen and Wootters (1994) P. Hausladen and W. K. Wootters, J. Mod. Opt. 41, 2385 (1994).