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1-13-2017 General Approach to Quantum Channel Impossibility by Local Operations and Classical Communication
Scott M. Cohen Portland State University, [email protected]
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Citation Details Cohen, S. M. (2017). General Approach to Quantum Channel Impossibility by Local Operations and Classical Communication. Physical Review Letters, 118(2), 020501. doi:10.1103/PhysRevLett.118.020501
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General Approach to Quantum Channel Impossibility by Local Operations and Classical Communication
Scott M. Cohen* Department of Physics, Portland State University, Portland, Oregon 97201, USA (Received 16 June 2016; published 9 January 2017) We describe a general approach to proving the impossibility of implementing a quantum channel by local operations and classical communication (LOCC), even with an infinite number of rounds, and find that this can often be demonstrated by solving a set of linear equations. The method also allows one to design a LOCC protocol to implement the channel whenever such a protocol exists in any finite number of rounds. Perhaps surprisingly, the computational expense for analyzing LOCC channels is not much greater than that for LOCC measurements. We apply the method to several examples, two of which provide numerical evidence that the set of quantum channels that are not LOCC is not closed and that there exist channels that can be implemented by LOCC either in one round or in three rounds that are on the boundary of the set of all LOCC channels. Although every LOCC protocol must implement a separable quantum channel, it is a very difficult task to determine whether or not a given channel is separable. Fortunately, prior knowledge that the channel is separable is not required for application of our method.
DOI: 10.1103/PhysRevLett.118.020501
Quantum information theory is concerned with the study single laboratory. This result provides an important con- of the transmission, storage, and manipulation of information nection between the study of LOCC and that of entangle- when that information is encoded in the form of quantum ment. We note that it is always possible to implement any systems. As it is impossible to completely isolate these channel by LOCC when enough entanglement is preshared systems, any description of their evolution must take into between the various parties—by using teleportation [10] or account their interaction with an environment, suggesting an by perhaps more efficient means [19–21]—indicating that analysis utilizing the powerful techniques available for the entanglement is an important resource [22–24] under the study of open quantum systems [1–4]. Interaction with the restriction to LOCC. environment is then considered to introduce noise into The importance of LOCC in quantum information the system’s evolution, and there are many important ques- processing has long been recognized, it playing a key role tions one may ask about the action of noisy quantum in teleportation [10], entanglement distillation [25,26],one- operations, or channels, not the least of which is to find way [27] and distributed [28] quantum computing, local the capacity of a noisy channel to transmit quantum infor- cloning [29], quantum secret sharing [30,31], and beyond. mation [5–7]. When the input to the channel is one part of an While many important results have been obtained concerning entangled state, this capacity also measures its ability to LOCC [32–47], it has, nonetheless, proven difficult to establish entanglement between the sender and receiver [8,9]. characterize in simple terms. Recently [48], we presented For the input to be part of an entangled state, it must have a method of designing LOCC protocols for quantum mea- first undergone an entangling evolution of its own. Hence, surements that succeeds for every measurement that can be one is led to investigate the entangling capabilities of such implemented by finite-round LOCC and for which failure evolutions, be they unitary or general quantum operations. has the immediate implication that the measurement cannot The relatively recent recognition of the practical value of be implemented by LOCC no matter how many rounds of entangled states—examples ranging from teleportation [10] communication are allowed, including when the number of and superdense coding [11] to quantum cryptography rounds is infinite (see, also, Refs. [35,36] for a different [12–15] and quantum computation [16–18]—has generated approach to designing LOCC protocols for measurements, considerable interest in understanding the conditions under an approach which is, in contrast, restricted to a finite number which entanglement can be created. Thus, one may ask what of rounds). characterizes a (multipartite) quantum channel’s ability to There are, of course, significant differences between create and/or increase entanglement between the subsystems channels and measurements. A noteworthy example in the upon which it acts. One such significant characterization is context of LOCC is the finding in Ref. [47] that the set of that the entanglement cannot increase when the channel can finite-round LOCC channels has a nonempty interior, be simulated by local quantum operations and classical whereas the interior of the set of finite-round LOCC communication (LOCC), which are the only operations that measurements is, in fact, empty [49].Itisalsoknown[45] can actually be implemented by spatially separated parties that the set of LOCC channels is not closed, even for the who lack the means to bring their subsystems together in a simplest possible case of two qubits [47], but the question of
0031-9007=17=118(2)=020501(6) 020501-1 © 2017 American Physical Society week ending PRL 118, 020501 (2017) PHYSICAL REVIEW LETTERS 13 JANUARY 2017 whether or not LOCC measurements is a closed set appears to describing any given channel is far from unique. For any be as yet unanswered. K0 N0 other set of Kraus operators, f jgj¼1, that describes the same Another key difference between measurements and chan- channel as the original set, let Nˆ ¼ maxðN;N0Þ and pad the nels is that there are many measurements that are all smaller of the two sets with additional zero operators so that associated with any given channel; see Eq. (3) below. the two sets have the same number of members. Then, there Therefore, one can demonstrate that a channel is LOCC by exists a unitary matrix V such that for each j ¼ 1; …; Nˆ , showing that any one of those measurements is itself LOCC. If, on the other hand, a given channel is not LOCC, none of XNˆ 0 the measurements associated with it can be implemented Kj ¼ VjiKi: ð3Þ by LOCC. This latter point may lead one to expect that the i¼1 characterization of LOCC channels may be a far more Furthermore, any other set describing this channel may be difficult task than that for LOCC measurements. It turns expressed in this way [18]. out, however, that this expectation is overly pessimistic. In the A LOCC protocol involves one party making a meas- present Letter, we show that the method of Ref. [48] for urement, informing the other parties of her outcome, after measurements is readily extended to the case of multipartite which according to a preapproved plan, the other parties quantum channels, allowing one to design a LOCC protocol know who is to measure next and what that measurement to implement a given channel whenever this is possible in a should be. Since each local measurement involves a finite number of rounds, with a computational effort for the number of possible outcomes, the entire process is com- case of channels that is never more than a quadratic increase monly represented as a tree, the children of any given node over that needed for measurements. Our method builds a representing the set of outcomes of the measurement made protocol round by round, starting with the first one. If at any at that stage in the protocol. Consider the cumulative action point in this process, the method finds that no measurement at of all parties up to a given node n in the tree, that “ ” the next round is possible, then this implies that no LOCC cumulative action being represented by Kraus operator protocol exists for the desired channel, even with an infinite K, which since the parties each make only local measure- number of rounds [48].Aswewillseebelow,ifno ments, must be of the product form, A ⊗ B ⊗ . We will measurement is possible at the very first round, then the label each such node by the positive operator K†K, method proves LOCC impossibility via the solution of a set of commonly referred to as a POVM element (where POVM linear equations. We will demonstrate that solving this set of stands for positive operator-valued measure), and we will linear equations is often sufficient for proving LOCC impos- say that node n is “equal” to its label K†K. For any such tree, sibility, illustrating this approach with various examples. the root node represents the situation present before any The remainder of the Letter is organized as follows. We party has yet measured and will, therefore, always be equal begin by reviewing the Kraus representation of quantum to the identity operator IH ¼ IA ⊗ IB ⊗ , where Iα is the channels and the concomitant unitary freedom in such identity operator on the local Hilbert space Hα for each party representations. Following this, we give a more detailed α. At the end of any finite branch of the tree is a leaf node explanation of what exactly LOCC means and how it can be (leaf nodes being those that do not themselves have represented. Then, we recall a lemma from Ref. [48] and children), which, since it is terminal, must be labeled by show how the method presented there can be fully adapted to a POVM element associated with a Kraus representation of the present case of quantum channels. These ideas are then the desired channel. Similarly, any infinite branch has a used to analyze certain example channels which are, with sequence of nodes that approach such a POVM element in relative ease, found to be impossible by LOCC. Finally, we the limit. When we say that a finite-round LOCC protocol offer our conclusions. implements Kraus operators fKjg, this means that each leaf Each quantum channel may be described by a set of † K N is proportional to one of the Kj Kj, with positive constant of Kraus operators f igi¼1 [3], which indicate how that channel transforms the state of the quantum system upon proportionality, and the sum of all leaves proportional to † † which it acts. If that system described by Hilbert space H Kj Kj is exactly Kj Kj. For infinite protocols, it means the starts out in state ρ0, then its final state will be given by same but in the limit as the number of rounds approaches XN infinity. Note that the class of infinite-round protocols can be † ρf ¼ Kiρ0Ki ; ð1Þ divided into two distinct subclasses [47]. The first subclass, i¼1 which is a subset of what we will refer to as “LOCC” [47], where with IH the identity operator on H, involves sequences of protocols more and more closely XN approaching a given channel simply by adding more and † Ki Ki ¼ IH ð2Þ more rounds of communication but without changing the i¼1 local measurements implemented in earlier rounds. The so that the channel is trace preserving, TrðρfÞ¼Trðρ0Þ.Of second subclass LOCC in Ref. [47] includes limits of course, as is well known, the set of Kraus operators sequences in which measurements made at the earlier rounds
020501-2 week ending PRL 118, 020501 (2017) PHYSICAL REVIEW LETTERS 13 JANUARY 2017 are changed from one protocol in the sequence to the next. Let fΛμg and fΓνg be orthonormal bases of the space of † As indicated by the notation, LOCC is the topological operators acting on HA and HA¯ , respectively, TrðΛμΛμ0 Þ¼ † closure of LOCC, and as mentioned above, when consid- δμμ0 and TrðΓνΓν0 Þ¼δνν0 , with Λ0 ¼ IA and Γ0 ¼ IA¯ . ering quantum channels, LOCC ≠ LOCC [45]. We note that † Choose any μ and any ν ≠ 0 and multiply Eq. (5) by Λμ ⊗ the results of the present Letter apply to the entire class † Γν to obtain LOCC, including those that involve an infinite number of X X rounds, but not to LOCC. 0 C~ Λ† ⊗ Γ† K†K C~ QðμνÞ; ¼ ii0 Trð½ μ ν i i0 Þ¼ ii0 i0i We will need the following lemma proved in Ref. [48] ði;i0Þ∈S ði;i0Þ∈S (but reworded here to conform to the present context). μ 0; …;d2 − 1; ν 1; …;d2 − 1; Lemma 1. Suppose the tree L represents a LOCC ¼ A ¼ A¯ ð6Þ protocol (finite or infinite), which implements the set of d H d ¯ H ¯ 0 where A is the dimension of A, A is that of A, and Kraus operators fKjg. Then the POVM element E repre- QðμνÞ ≔ Λ† ⊗ Γ† K†K : senting the accumulated action of all parties up to any given i0i Trð½ μ ν i i0 Þ ð7Þ L node in is equal to a positive linearP combination of the set of ~ 0† 0 0† 0 The next step is to form the coefficients Cii0 into an operators, fKj Kjg. That is, E ¼ jcjKj Kj,withcj ≥ 0. jSj-dimensional column vector c~ (by, say, stacking each Without loss of generality, let us assume that Alice QðμνÞ measures first to initiate a LOCC protocol that implements column one below the next) and also the i0i into a row quantum channel E. More specifically, assume the protocol vector for each μ and ν ≠ 0, collecting all the latter row 0 Q implements any set of Kraus operators fKjg that represents vectors into a matrix A. Then, we have from Eq. (6) E. Since Alice has measured first, the other parties have as 0 ¼ QAc~ ð8Þ yet done nothing, which implies that the outcome of Alice’s initial measurement will correspond to a (multipartite) Kraus showing that any initial local measurement by Alice in a A ⊗ I I operator of the form A¯ , where A¯ is the identity LOCC protocol exactly implementing the desired quantum H ¯ operator on A, the Hilbert space describing all parties channel E is determined by the null space of matrix QA. other than Alice. The associated POVM element is then [Note that once the index set S and the orthonormal bases † E A ⊗ I ¯ A A A ¼ A, with ¼ , which, according to Lemma 1, fΛμg and fΓνg are chosen, matrix QA is completely defined c ≥ 0 implies that with j , by the channel itself through the Kraus representation Ki . In addition, we claim that the choice of index set XNˆ XNˆ XNˆ f g 0† 0 † and bases does not change the results of our approach. The A ⊗ IA¯ ¼ cjK K ¼ V cjVji0 K Ki0 j j ji i reason why the basis choices change nothing has been j¼1 i;i0¼1 j¼1 discussed in footnote 4 of Ref. [48]. Choosing index set S0 XNˆ S C K†K ; instead of , correspondsP to a choice of linearly indepen- ¼ ii0 i i0 ð4Þ † ii0 † 0 0 dent operators, Kj Kj0 ¼ i;i0 ∈ST 0 Ki Ki0 , for ðj;j Þ ∈ S , i;i0¼1 ð Þ jj P where T is a full-rank matrix whose columns are indexed C Nˆ V c V j; j0 i; i0 Q → Q0 and ii0 ¼ j¼1 ji j ji0 . by ð Þ and rows by ð Þ. This changes A A ¼ 0 At this point, it is convenient to choose a set of index QAT, so while the null space of QA differs from that of QA, S⊆ i; i0 Nˆ the dimensions of the two null spaces are identical. Since pairs, fð Þgi;i0¼1, such that the set of operators † † fK Ki0 g i;i0 ∈S is linearly independent. Then with K Kj0 ¼ our results depend only on this dimension (see Theorem 2), P i ð Þ j ii0 † 0 we see that our claim is justified.] In similar fashion, one 0 r 0 K Ki0 for j; j ∉S, ði;i Þ∈S jj i ð Þ can use this approach to obtain all later measurements, X X thereby designing a full LOCC protocol for E whenever A ⊗ I C K†K C rii0 K†K A¯ ¼ ii0 i i0 þ jj0 jj0 i i0 possible. This design approach is described in detail in ði;i0Þ∈S ðj;j0Þ∉S Ref. [48] for the case of implementing a quantum meas- X C~ K†K ; urement rather than a channel. For measurements, the set of ¼ ii0 i i0 ð5Þ V i;i0 ∈S Kraus operators is fixed, and unitary in Eq. (4) is equal to ð Þ Nˆ P the identity; then the free parameters are the coefficients C~ C C rii0 c S C~ where ii0 ¼ ii0 þ ðj;j0Þ∉S jj0 jj0 . This condition must j, rather than the j j coefficients ii0 . Thus, we see that the hold for all sets of Kraus operators fKig representing E. approach described in Ref. [48] can be directly taken over 0 Since the Kraus representation fKjg for E can also be to the case of channels simply by increasing the dimension ˆ ˆ 2 arbitrarily chosen, we see that Eq. (5) with 0 ≤ A ∝ IA of the domain of matrix QA from N [48] to jSj ≤ N , where (or else Alice did not actually measure) is a necessary Nˆ is the number of Kraus operators defining the measure- condition for the existence of a LOCC protocol that ment (in the first case) or representing the channel (in the implements E. second case).
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0.2 By Eq. (2), there is always at least one solution of Eq. (8), which we will denote as c~I corresponding to A ¼ IA in Eq. (5). If this is the only solution, then Alice cannot 0.15 measure first. If this is true for every party, then no LOCC protocol exists for the given channel. † 0.1 Let us augment QA by adding the single row c~I (and, hoping not to cause confusion, we continue to call this QA). Then c~I is no longer in the null space of QA, and we have 0.05 the following theorem. Theorem 2. For a given multipartite quantum channel 0 E, if for every party α, the null space of Qα is empty, then E 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 cannot be implemented by LOCC, even when using an infinite number of rounds. FIG. 1. Impossibility of a first measurement for rotated domino Note that this theorem provides a way to easily prove that states. certain channels are not LOCC, requiring only the solution of a set of linear equations. Below is a list of examples for which least one pair of the dominos is not rotated at all, in which this method provides such a proof. For each example, we give † case there exists a straightforward LOCC protocol for the ratio of the smallest eigenvalue of QαQα ≥ 0 to the ˆ implementing these channels. As seen in Fig. 1 (see largest, minimized over parties α, denoting this ratio as λ. Ref. [51] for further explanation), λˆ is nonzero but smoothly [The computational complexity of finding all eigenvalues approaches zero as the degree of rotation (measured here by n n n O n3 θ of an × matrix in the large- limit is ð Þ by the min) vanishes. Thus, we see a smooth approach to the Householder reduction to tridiagonal form followed by either existence of a first local measurement and, indeed, to a fully QR factorization or direct solution of the resulting character- LOCC channel that can be implemented using three rounds istic equation [50]. We believe this calculation is the limiting of communication in this limit [51]. step in our procedure of checking LOCC impossibility. See (4) Random unitary channels, where the Kraus operators Sec. IVof Ref. [51] for our MATLAB code implementing this are proportional to (randomly chosen) unitaries. The results procedure, where we use MATLAB’s “eig” function to find the in this case depend strongly on the number of random eigenvalues, it being plenty fast enough for the problems we unitaries, Nu. For each number of parties, each set of local ˆ have addressed here.] When λ ≠ 0, the null space of Qα is dimensions, each Nu, and equal a priori probabilities of the empty for all parties α, and then Theorem 2 shows that the random unitaries, we have numerically checked 100 differ- given channel cannot be implemented by LOCC. If λˆ is small, ent sets of randomly chosen unitaries for each of the then Qα is close to singular, implying there is a first following cases: two qubits, three qubits, two qutrits, and a measurement that introduces only a small error into the bipartite qubit-qutrit system. In all cases, we have found protocol. For example, by a small change in the set of Kraus that no party can ever measure first when Nu
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