When offers no advantage over finite LOCC
We consider bipartite LOCC, the class of operations implementable by local quantum operations and classical communication between two parties. Surprisingly, there are operations that cannot be implemented with finitely many messages but can be approximated to arbitrary precision with more and more messages. This significantly complicates the analysis of what can or cannot be approximated with LOCC. Towards alleviating this problem, we exhibit two scenarios in which allowing vanishing error does not help. The first scenario involves implementation of measurements with projective product measurement operators. The second scenario is the discrimination of unextendible product bases on two -dimensional systems.
We consider bipartite finite dimensional quantum systems, and what state transformations can be achieved given arbitrary quantum operations on each system and classical communication between them. This class of quantum operations is known as LOCC. LOCC arises in many natural settings. For example, it is much easier to transmit classical data than quantum data over long distances. For another example, quantum gates involving multiple registers are much harder to implement, and methods to effect them using entangled states, measurements, and classical feedback hold high promise. The study of LOCC also provides insights on the nature of quantum information, leading to discoveries including teleportation [BBC93], quantum error correcting codes [BDSW96], and security proofs for quantum key exchange [LC99, SP00].
Unfortunately, LOCC, as an operationally defined class, does not have a succinct mathematical description. Traditionally, one turns to relaxations of LOCC such as SEP or PPT instead of analysing LOCC operations. This method has proved fruitful for many problems such as data hiding [DLT02] and state discrimination [Wat05]. Yet this approach fails to answer other interesting questions such as whether more rounds of communication, or infinitely many intermediate measurement outcomes can make a difference, or whether there are operations that can be approximated arbitrarily closely with LOCC but do not belong to LOCC (i.e., whether LOCC is equal to its topological closure or not). Recent investigation of the LOCC class itself has resolved these questions; for example, more communication rounds can be helpful [Chi11] and [Chi11, CCL12b, CCL12a, CLM12].
A common technique to prove that a certain task cannot be accomplished perfectly by a finite LOCC protocol is to start by assuming the contrary. Then the properties of steps taken in any perfect implementation of the task are shown to be incompatible with the structure of an LOCC protocol. However, it could still be possible to accomplish the task with vanishing error using LOCC protocols. Excluding the possibility of approaching perfect implementation is much harder and few results have been established [BDF99, KTYI07, KKB11, CLMO13, CH13b, CH13a].
In this paper, we focus on two problems concerning . Our first problem is, are there natural classes of measurements in LOCC that are closed? In other words, are there sets of measurements such that it does not help to allow vanishing error and more and more rounds of communication? Our second problem concerns the possibility of discriminating sets of orthogonal product states called UPBs (for unextendible product bases though these states are not bases) in .
We first summarize prior works in the two problems of interest. The first example of a task for which gives no advantage over finite LOCC concerns perfect discrimination of the so-called domino states [BDF99]. They cannot be discriminated by either set. Over a decade later a generalization was reached by establishing that the set of full basis measurements implementable by LOCC is closed [KKB11]. For our second question, [Ter99, DMS03] established that UPBs cannot be distinguished with finite LOCC. Reference [Rin04] studied discrimination of UPBs in . However, as pointed out in [KKB11], the proof in [Rin04] is incomplete and a particular claim in the proof contradicts other proven results.
We make partial progress in the two problems posed above. First, we show that the set of LOCC implementable projective measurements with tensor product operators is closed (see Theorem 3). Second, we prove that cannot be used to perfectly discriminate states from a UPB in (see Theorem 9). Both results are applications of a necessary condition from [KKB11].
The rest of the paper is organized as follows. In Section 2 we set up the notation and give basic definitions. In Section 3 we identify a closed class of projective measurements that can be implemented with finite LOCC protocols and provide an application of this result. In Section 4 we establish that two-qutrit UPBs cannot be perfectly discriminated even with . We conclude in Section 5.
Finite and asymptotic LOCC
We say that a measurement can be implemented with finite LOCC, and write , if can be implemented exactly using a finite LOCC protocol (i.e., an LOCC protocol with finitely many communication rounds). We say that can be implemented using asymptotic LOCC, and write , if there exists a sequence of finite LOCC protocols that implement with vanishing error. Note that asymptotic LOCC is the (topological) closure of finite LOCC. It represents the set of operations that can be implemented by LOCC protocols with arbitrary precision. For more detailed explanation of classes and see [CLM12, Man13].
State discrimination problem
Let , be a probability distribution, and be a set of bipartite states. In the state discrimination problem an index is chosen with probability and Alice and Bob are given their respective registers of . Their task is to find the index without error. The scenario where they are allowed to err is also of interest but will not be studied in this paper. In the error free case the probability distribution is not relevant and will therefore be chosen to be uniform. We say that the states from can be discriminated with finite LOCC (asymptotic LOCC), if there exists a measurement () that discriminates the states perfectly.
There is a close connection between discrimination of mutually orthogonal states and implementation of projective measurements. In particular, the states from can be discriminated by finite LOCC (asymptotic LOCC) if and only if finite LOCC (asymptotic LOCC) can be used to implement the projective measurement onto the supports of the states [CLMO13].
We now introduce the concept of non-disturbing operators which is central for state discrimination with LOCC.
Let be a set of orthogonal states. We say that is non-disturbing for , if
for all distinct . We say that a measurement is non-disturbing for if each of its POVM elements of is non-disturbing for .
Let denote the support of . Then Condition (1) is equivalent to requiring that for all distinct and all and
Note that any measurement protocol transforms to a new set conditioned on the culmulative measurement outcome. In a perfect discrimination protocol for , at any point, the next measurement applied to this conditioned set must not disturb it. In particular, the protocol must start with a measurement that is non-disturbing for . In an LOCC protocol each measurement must be local. For finite LOCC, each measurement has to be non-trivial. Hence, the states from a set can be perfectly discriminated with finite LOCC only if admits a non-disturbing product operator where exactly one of the matrices is the identity matrix. If such an operator does not exist, the states from cannot be discriminated with finite LOCC. Non-disturbing operators also provide a necessary condition for state discrimination with asymptotic LOCC.
Theorem 1 ([Kkb11]).
Consider a set of states such that does not contain any nonzero product vector. Then can be discriminated with asymptotic LOCC only if for all with there exists a positive semidefinite product operator satisfying all of the following:
is non-disturbing for .
Theorem 1 implies that the set of LOCC implementable full basis measurements is closed.
Corollary 2 ([Kkb11]).
If a full orthogonal basis measurement can be implemented using asymptotic LOCC then it can already be implemented with finite LOCC.
In the next section we generalize Corollary 2 for a larger class of projective measurements.
3 Projective measurements with tensor product operators
In light of the findings of [KKB11] presented in the previous section it is natural to ask whether a similar result holds for the class of all projective measurements, or even for the class of all POVM measurements. To this end, in this section we show that the class of all projective measurements with tensor product operators that can be implemented with finite LOCC is indeed closed.
Let be a projective measurement. Then implies that .
We say that a set of states is a full orthogonal set if the states are mutually orthogonal, and has full rank. We now rephrase Theorem 3 in terms of state discrimination.
Let be a full orthogonal set. If the states from can be discriminated with asymptotic LOCC then they can be discriminated with finite LOCC.
We first prove two lemmas that help establish Theorem 4. The main ingredient for proving Theorem 4 is the construction of an operator such that either or is non-disturbing for . A matrix is non-disturbing for a complete orthogonal set if and only if the row spaces of are -invariant for each . Here, a subspace is said to be -invariant, if for all .
Our first lemma provides useful characterization of -invariance.
Let , be an -dimensional subspace of , be a fixed orthonormal basis of , , and . Then the following are equivalent:
for an matrix ;
has an orthonormal basis consisting of eigenvectors of .
Note that the operator maps to the subspace and the action of on (in the basis ) represents the action of on (in the basis ).
We first prove . If is -invariant, then for all
for some . If we let , then
Next, let us show that . If , then as . Since is Hermitian, is also Hermitian and it has a spectral decomposition . Then for all
Therefore, the vectors are eigenvectors of . Finally, for all we have . So the set
is an orthonormal basis of consisting of eigenvectors of .
Last, we prove that . Let be a set of orthogonal eigenvectors of with corresponding eigenvalues . Then any vector can be expressed as for some . Now we have
as desired. ∎
We now show that whenever is non-disturbing for a full orthogonal set of product states, so are and .
Let and be a projector onto a subspace . If and is -invariant, then is also and -invariant.
Let be the dimensions of and , respectively. Fix some orthonormal basis of and let , where . Define similarly. Then and . If is -invariant, then
for an matrix . Note that is a tensor product, since . Since we also have that . Hence, Equation (8) together with the fact that is a tensor product implies that
Proof of Theorem 4.
We prove by induction on . Clearly the states in can be discriminated with both finite and asymptotic LOCC if . We assume that the theorem statement holds for all values for some .
Suppose and the states in can be discriminated with asymptotic LOCC. Then for every there exists a product operator satisfying the three conditions in Theorem 1. Our goal is to choose appropriate value of and use Lemma 6 to conclude that both and are non-disturbing for .
Pick any and let be the corresponding operator. Let us now check that is nontrivial and satisfies the hypothesis of Lemma 6. The range of is chosen so that Conditions (1) and (2) together imply that
cannot be proportional to the identity matrix (from now on we assume that is not proportional to the identity matrix as the other case is similar);
for all we have .
For each , let be the column space of and be the projector onto . Then the last item implies that . Since is non-disturbing for , the subspace is -invariant. Due to Lemma 6, is -invariant.
Let be the projector onto the -eigenspace of . Due to the equivalence of (1) and (3) in Lemma 5, the subspaces are -invariant for all . So if is the identity measurement on Bob, the nontrivial local projective measurement
is non-disturbing for .
Suppose we measure the states in using and obtain outcome . If we restrict the unnormalized post-measurement states to the column space of , we have the following set:
, and is some orthonormal basis of the -eigenspace of . We now want to use the induction hypothesis to conclude that the states in can be discriminated with finite LOCC. To do so, we have to check that is a set of mutually orthogonal states that can be discriminated with asymptotic LOCC and that has full rank.
First, since is positive semidefinite and has full rank and has full column rank, the matrix
has full rank. Suppose that a sequence of finite LOCC protocols can be used to certify that the states in can be discriminated with asymptotic LOCC. Let be the finite LOCC protocol in which Alice first embeds her input space in by applying the isometry and then the two parties proceed with the protocol . After the embedding, Alice and Bob have the states up to a normalization. Since the column space, , of is -invariant, the column space of is contained in . Therefore, the sequence can be used to certify the asymptotic distinguishability of the states from .
Since is non-disturbing for , the states in are mutually orthogonal. Finally, as , the states from can be discriminated by a finite LOCC protocol by induction hypothesis. Combining the measurement with the finite LOCC protocol gives a finite LOCC protocol for discriminating the states in (See Figure 1). ∎
We can lift the tensor product requirement for one of the states in Theorem 4.
Let be a full orthogonal set and assume that all but one can be expressed as . If the states from can be discriminated with asymptotic LOCC then they can be discriminated with finite LOCC.
and , where . Let be the space onto which projects. Although is -invariant, it is neither - nor -invariant, since and .
Therefore, the general question of whether the set of POVM measurements implementable by is closed remains open, despite partial progress presented by Theorem 3.
Although Corollary 7 is only a slight generalization of Theorem 3, it provides answers to natural questions. For example, let us consider the following orthonormal product basis is known as the domino basis [BDF99] :
where . It is known that the domino states cannot be discriminated by asymptotic LOCC [BDF99]. However, as soon as we modify the problem slightly the answer becomes unclear. For example, it is not known whether the states from
can be discriminated with asymptotic LOCC. Questions about asymptotic LOCC can be difficult to answer. In some settings, Theorem 3 and Corollary 7 allows us to reduce such questions to those about finite LOCC which usually are more tractable, as demonstrated in the following example.
Let be such that or and is the uniform mixture of the remaining domino states. Then the states from cannot be discriminated with .
Because of Corollary 7, it suffices to prove that cannot be discriminated with . To do so, we only need to disprove the existence of nontrivial (i.e., not proportional to the identity) positive semidefinite operators of the form and .
Let and assume is non-disturbing for the set of states . Then
since both for and and is Hermitian. Similarly, from we obtain . Either or belongs to the support of a different state from than . Hence, either or . In both cases we obtain . To see that all the diagonal elements have to be equal, note that and . Thus, is proportional to the identity matrix. Via similar analysis, one can reach the same conclusion for . Therefore, all the non-disturbing operators for are proportional to the identity matrix and our lemma follows. ∎
Building on [KKB11], [CH13a] presents a necessary condition for discriminating two states with asymptotic LOCC. When the support of these two states cover the whole space this condition yields a simple criterion. This allows the authors to show that
cannot be discriminated with asymptotic LOCC. Since our set is a refinement of the impossibility to distinguish the states from with asymptotic LOCC also follows from the result of [CH13a]. We illustrate Corollary 7 on the domino states because they are well-known. There are cases where Corollary 7 applies but the criterion from [CH13a] cannot be used to conclude that a set of states cannot be discriminated with asymptotic LOCC.
4 UBP’s in cannot be discriminated by
In this section, we turn our attention to the problem of discrimination of unextendible product bases (UPBs). It has been shown in [Ter99, DMS03] that UPBs cannot be perfectly discriminated by finite LOCC. Here, we show that any UPB in cannot be discriminated by asymptotic LOCC.
Reference [DMS03] establishes that any UPB in has exactly states of the form which, up to local unitary transformations, can be parametrized by six angles , , , , , :
where , and unextendibility implies that none of , , , vanishes.
The set of states cannot be discriminated by asymptotic LOCC.
First, we show why it suffices to prove the case for . We can replace by in the choice of the local basis on Alice’s system. Then, only appears in . Apply a similar change of basis on Bob’s system. Clearly, replacing and by does not affect distinguishability.
We now proceed to prove the theorem by contradiction. Suppose can be discriminated by asymptotic LOCC.
Then, Theorem 1 applies to since the kernel condition holds automatically when is a UPB. Here, and we take (any will do). The theorem states that that is non-disturbing for . Furthermore, (else ), (else, ), and . From these we derive constraints for and . We have , cannot have two nonzero eigenvalues of opposite signs (else the same holds for ) and similarly for , so, without loss of generality, , . Finally, and .
We will show that the non-disturbing property of is inconsistent with the conditions on .
Our main tool in the analysis is an extension of the orthogonality graph for UPBs defined in [DMS03]. Given , define two graphs as follows. They both have vertex set . () has an edge whenever (). Since is non-disturbing for , the pair is an edge in or for all distinct . Since any three span , no vertex in has degree more than , and similarly for . So, the only possible are -cycles with complementary sets of edges.
Denote the possible -cycles as . We analyse the possible cases for which ( is then fixed) in detail in the appendix. We will see that in one case, which is a contradiction. For all other cases, which is also a contradiction.
We note as a side remark that, using Theorems 2 and 3 in [DMS03], any bipartite UPB with states can be perfectly discriminated by a separable measurement. Theorem 9 thus provides another example of the phenomenon nonlocality without entanglement, in which a set of unentangled states cannot be discriminated by asymptotic LOCC, but can be discriminated by separable operations.
To ease the analysis of asymptotic LOCC we have introduced two scenarios in which no new task can be accomplished (information theoretically) collapse. by allowing vanishing error. The first scenario is the implementation of projective measurements with tensor product operators. The second is the discrimination of the states from an unextendible product basis in . On the first subject, an obvious next step is to investigate whether asymptotic LOCC can be helpful for implementing any projective or general measurement. A second question is whether asymptotic LOCC can help for perfect state discrimination. On the second subject, it is likely that a general UPB cannot be discriminated by asymptotic LOCC. It will be nice to obtain a rigorous proof.
Another very poorly understood subject in LOCC is round complexity. Almost nothing is known about how many messages the parties need to exchange in order to accomplish a task in the LOCC setting, especially when a small probability of error is allowed.
DL is supported by NSERC, NSERC DAS, CRC, and CIFAR. LM is funded by the Ministry of Education (MOE) and National Research Foundation Singapore, as well as MOE Tier 3 Grant “Random numbers from quantum processes” (MOE2012-T3-1-009).
Appendix A Case analysis for Theorem 9
For concreteness, we denote the possible -cycles as follows:
Omitting the subscript (which is irrelevant here), and using the shorthands for , , for , , we have
where , and we rephrase in terms of , and , two states that appear frequently in the analysis.
If we swap with , Eq. (19) becomes
where , , , , and . So the local change of basis swaps with and swaps with with modified angles. Thus the analysis for applies to the case for .
Here, we summarize the methodology in the analysis. In each case, lives in a one-dimensional subspace because it is orthogonal to two (linearly independent) ’s where is adjacent to in . Similarly for . We thus obtain the first two columns of in terms of and two scalar multipliers for . We often use the original orthogonality conditions between the ’s to deduce the form of . With the first two columns of fixed, we use hermiticity of to fix all but one entry of (which we call ). We use the remaining orthogonality conditions to relate until we either obtain (in which case we show as well and thus ) or use to force or thereby contradicting .
Case I: If , then
, so, .
, so, .
By hermiticity, . So, .
Imposing gives , and gives , so .
When , . Up to different choices for the angles, the states are the same as , so, the above analysis applies to and . Thus a contradiction.
Case II: If , then
, so, .
, so, .