Asymptotic implementation of multipartite quantum channels and other quantum instruments using local operations and classical communication

In two recent papers [18, 20], we developed a new approach to the study of quantum measurements implemented by multiple parties utilizing local operations in their respective laboratories and classical communication to their partners to share the results of those local operations, a fundamental paradigm in quantum information processing commonly known as LOCC. We believe our approach provides valuable insights and tools for analyzing what can be accomplished under these circumstances, where quantum systems are not (under practical conditions, often cannot be) transmitted from one party to another. In the first of these papers [18], we identified certain geometric conditions that must hold in order that a measurement can be approximated arbitrarily closely by LOCC, and then in the second paper [20], we refined that approach for use in studying local quantum state discrimination, showing how one can often determine whether success with such a task is impossible, even while knowing only the set of states to be discriminated but nothing about which measurements, if any, might be possible candidates for successfully accomplishing such a task. Here, we generalize our approach to address the question of whether or not a given quantum channel can be closely approximated using LOCC. We then extend the ideas further to apply to any quantum instrument [11].

The paradigm of LOCC has been widely studied [4, 1, 3, 2, 36, 13, 32, 34, 8, 5, 26, 24, 7, 16, 17, 5] since publication of the seminal paper of Peres and Wootters [33]. While much progress has been made, and many important and sometimes surprising results have been obtained, LOCC has also been widely recognized as being quite difficult to characterize. One particularly challenging aspect arises when one is interested in whether or not a task can be accomplished while allowing an arbitrarily small error to be incurred in doing so. In studying such questions, one must allow the parties to share their outcomes back and forth for as many rounds of communication as needed in order to reduce the error as much as possible. With some tasks that can be easily accomplished by acting globally on the entire multipartite system, it is known that when restricted to LOCC, even allowing for an infinite number of rounds is not enough to reduce the error to zero. One such celebrated example involves perfect discrimination of the so-called domino states [3], which demonstrated that separable measurements—ones that involve measurement operators that are simple tensor products on a multipartite system and can be easily implemented globally—cannot always be implemented, or even closely approximated, by LOCC, even with the use of an infinite number of communication rounds. This discovery gave birth to a wide-ranging effort at understanding the surprisingly difficult, and rich, questions concerning the difference between global actions and those restricted to LOCC.

While LOCC has a rich and difficult to characterize structure, one commonly discusses it in terms of a relatively simple picture, that of a rooted tree graph, or LOCC tree. In the case of a finite LOCC tree and starting at the root node, one may follow one of the tree’s many branches to its endpoint, which is known as a leaf node. Each node along any given branch represents an outcome of a (local) measurement performed by one of the parties, and the next node along the branch represents an outcome obtained in a subsequent (local) measurement by a different party. These nodes are connected by edges, and it is easy to envision a path following this branch along these successive edges from the root node all the way to its leaf. We recognized [18] that this path is not just an abstract visualization of the sequence of outcomes, otherwise having no useful meaning, but rather that it corresponds directly to a particular type of path lying within a specific geometric object. Significantly, that object is determined uniquely by the outcomes associated with the leaf nodes appearing at the ends of the collection of all branches in the tree. That is to say the measurement ${\cal M}$ implemented by the LOCC protocol represented by this tree uniquely identifies a geometric object $Z_{\cal M}$, within which lies each and every path associated with the branches in the tree. We have characterized these paths [18] and used that characterization to obtain a powerful condition, reproduced below as Theorem 1, which must hold if it is possible for ${\cal M}$ to be implemented arbitrarily closely by LOCC.

Here, we extend that analysis beyond measurements to encompass the much more general notion of quantum channels and indeed, any quantum instrument [11]. For channels, we find a condition, see Theorem 2 below, which must hold if quantum channel ${\cal E}$ can be closely approximated by LOCC. The condition is a direct generalization of that for measurements in Theorem 1. It also involves characterization of the paths discussed in the preceding paragraph, which in similar fashion must lie within a more general geometric object $Z_{\cal E}$, which itself is uniquely determined by ${\cal E}$. The difference in structures of $Z_{\cal E}$ and $Z_{\cal M}$ reflects the fact that in implementing a quantum channel, no classical information about outcomes can be available, in contrast to measurements, where complete (classical) knowledge of those outcomes must be known. Both channels and measurements are special cases of quantum instruments, which make allowance for that classical information to vary from all to none.

We have found that dealing with $Z_{\cal E}$ is more difficult as compared to $Z_{\cal M}$. As we shall see, the latter is a special case of the former, and it is this greater generality of $Z_{\cal E}$ which makes it harder to analyze. Furthermore, whereas it was quite easy to find examples of previously studied measurements where we could apply, and learn from, our condition for measurements, Theorem 1, examples of quantum channels to which we can fruitfully apply Theorem 2 are much more difficult to find. Perhaps the explanation for this dearth of examples is that quantum channels—in particular, multipartite quantum channels—are just more difficult to study, in general, so comparatively little work has been done in this direction. One may hope that the results presented here will stimulate further efforts in that direction.

A dearth is not the same as a complete absence, however. In fact, there exists a two-qubit example of a quantum instrument for which it is known that it cannot be implemented exactly but can be approximated arbitrarily closely by LOCC [11]. The inspiration for this instrument came from ideas developed in a series of earlier papers on the random distillation of entanglement from a three-qubit system onto two of those qubits [23, 6, 9, 10]. We study the two-qubit example in detail below, first as a quantum channel, then as a general instrument, and finally as a quantum measurement. This analysis leads to insights regarding how the critical property of our paths, that of being piecewise local, see below, disappears in the limit as the number of rounds tends to infinity. It also illustrates other properties of these paths, such as the fact that certain matrices must have a block diagonal form determined by the structure of the given instrument in terms of its constituent completely positive (CP) maps. We also find from this example an interesting result regarding the closure of LOCC measurements.

The relationship between LOCC and its closure, $\overline{\textrm{LOCC}}$, has long been a point of interest to researchers. An early result along these lines was the demonstration, in the context of random distillation of two-party entanglement starting from the three qubit $W$ state, that a convergent sequence of transformations is not achievable by LOCC in the limit [10]. Later, utilizing a two-party generalization of those arguments, it was shown that LOCC is also not closed for two-qubit systems [11]. In each of these cases, the parties must rely upon classical information obtained in these protocols, leaving open the question of whether or not LOCC is closed when considering quantum channels, for which no classical information is retained. It has also been an open question whether LOCC is closed as applied to quantum measurements, where the maximum amount of classical information is needed to uniquely identify each and every measurement outcome. Here, in the context of the two-qubit example just mentioned, we discover a measurement which is not in LOCC but is within its closure. As far as we are aware, this is the first example of a measurement exhibiting such a property.

The remainder of the paper is organized as follows. In the next section, we provide an overview of the tools we will use in the subsequent sections, including a discussion of quantum channels and instruments and their mathematical representation, finite-round LOCC and its extension to infinite rounds, and the geometric objects, $Z_{\cal M}$ and $Z_{\cal E}$, mentioned above. In Section III, we present our main result, Theorem 2. Section IV provides a proof of Theorem 2, which utilizes a result linking convergence of a sequence of quantum channels to convergence of a sequence of the geometric objects, $Z_{\cal E}$. The latter result is proved in Appendix A. In Section V the reader will find an extensive analysis, in the context of Theorem 2, of the two-qubit example channel from Ref. [11]. Section VI provides a generalization of Theorem 2 to the case of arbitrary quantum instruments, encompassing and interpolating between Theorem 1 and Theorem 2. There, the reader will also find a brief discussion in this context of the two-qubit example of Section IV. The important question of whether or not LOCC is a closed set, a question that may depend on the type of circumstances to which it is being applied, is addressed in Section VII where it is shown that LOCC is not closed when applied to quantum measurements. Section VIII presents a generalization of the example of Section IV to any number of qubits, and then finally, we offer our conclusions in Section IX.

In this section, we describe basic ideas underlying the various tools we will use for the analysis presented in this paper. We begin with a discussion of quantum channels and instruments, continue with local operations and classical communication (LOCC), followed by a discussion of infinite-round LOCC, and finally, the geometric objects we will need, known as zonoids.

In this section we present our main result, a necessary condition that ${\cal E}\in\overline{\textrm{LOCC}}$, Theorem 2, the proof of which is given in the subsequent section. We begin by recalling the main theorem of Ref. [18]. There, we proved that if measurement ${\cal M}=\{E_{j}\}$ can be approximated by LOCC arbitrarily closely, then for each $j$, there exists a continuous, monotonic path, $\Pi_{j}(s)$, of positive semidefinite product operators from $I_{\cal H}$ to each outcome $E_{j}$ of that measurement and lying entirely within the zonotope, $Z_{{\cal M}}$, generated by those POVM elements, $\{E_{j}\}$. Here, monotonicity means that the trace of $\Pi_{j}(s)$ is non-increasing along the path. The proof of that result relied on the observation that every finite-round LOCC protocol implements a measurement for which there exists a continuous, monotonic, piecewise local¹¹1The notion of a piecewise local path is borrowed from the piecewise constant curves we all learned about when studying calculus. Instead of being constant, however, the pieces here correspond to local measurement outcomes and correspond to the edges between a parent node and one of its children in an LOCC tree. Along this edge, only one party’s part of the full measurement operator changes, justifying use of the word local. See Ref. [18] for a more detailed explanation of this concept. path of positive semidefinite product operators to each of the outcomes of that measurement. Using this observation, it was then argued that from any sequence of finite-round LOCC protocols implementing measurements ${\cal M}^{\nu}$ for which $\lim_{\nu\to\infty}{\cal M}^{\nu}={\cal M}$, there exists a sequence of piecewise local paths whose limit is a (not necessarily piecewise local) path lying within $Z_{{\cal M}}$ and ending at one of the outcomes of ${\cal M}$, and that this holds for each such outcome. The resulting theorem reads as follows.

In this section, we are concerned with quantum channels rather than measurements. Since any LOCC protocol implementing channel ${\cal E}$ may just as well be viewed as implementing the generalized measurement consisting of a specific set of Kraus operators which represents ${\cal E}$, Theorem 1 provides a solid foundation for the following theorem, which is our main result, providing a necessary condition that a quantum channel may be simulated arbitrarily closely by LOCC.

The condition of Theorem 2 that ${\cal E}$ can be implemented by LOCC with vanishingly small error means there exists a sequence of finite-round protocols ${\cal P}^{\nu}$ implementing quantum channel ${\cal E}^{\nu}$ such that $\lim_{\nu\to\infty}{\cal E}^{\nu}={\cal E}$. Since as discussed in Section II.2, each of these finite-round protocols may equally well be viewed as implementing a measurement ${\cal M}^{\nu}$, which implies the existence of the concomitant paths of product operators in Theorem 1 [18], this (at least) suggests the existence of the set of limiting paths alluded to in Theorem 2. Thus, one should not be surprised to find that the results for approximating measurements by LOCC [18] can be generalized, as we do here, to apply in similar fashion to approximating quantum channels by LOCC. One notable difference was discussed in Section II.4, that while the approximation of measurements involves paths lying entirely within a zonotope, $Z_{{\cal M}}$, for the approximation of channels one must utilize the zonoid, $Z_{{\cal E}}$.

Recall from Section II.4 that in order to implement a quantum channel, say ${\cal E}^{\nu}$, by LOCC, one may implement any measurement ${\cal M}^{\nu}$ compatible with ${\cal E}^{\nu}$. This means that simply knowing one can implement a given channel leaves one without knowledge of which compatible measurement(s) can be implemented. Thus, we also do not know the set of Kraus operators implemented, whereas that set is what determines the endpoints of the paths introduced in Theorem 1 [18]. Therefore, we are left with the question, how do we know what the endpoints of all the paths of Theorem 2 are? The answer is, we do not. Instead, as is indicated in Item 2 of Theorem 2, we only know that each of these endpoints corresponds to a rank-$1$ matrix $\hat{C}_{j}$ and the collection of these matrices constitutes a rank-$1$ decomposition of the $\kappa\times\kappa$ identity matrix. This is the extent to which we are able to characterize those endpoints without knowing in advance what specific set of Kraus operators is to be implemented. Nonetheless, this is sufficient to ensure that the resulting set of Kraus operators are a representation of ${\cal E}$.

We now turn to the proof of Theorem 2.

We begin the proof of Theorem 2 with the following lemma showing that convergence of a sequence of quantum channels implies convergence of the corresponding sequence of zonoids.

The next step is to recall that each LOCC protocol ${\cal P}^{\nu}$, which implements channel ${\cal E}^{\nu}$, may also be thought of as implementing a POVM, ${\cal M}^{\nu}$, consisting of elements $E_{j}^{\nu}=K_{j}^{\nu{\dagger}}K_{j}^{\nu}$, where with $W^{\nu}$ an isometry, $K_{j}^{\nu}=\sum_{m}W_{jm}^{\nu}\hat{K}_{m}^{\nu}$ is a Kraus operator in a collection representing channel ${\cal E}^{\nu}$. Therefore, by Theorem 1, we know that there exists a continuous, monotonic path of product operators from $I_{\cal H}$ to a point on the half-open line segment $(0,E_{j}^{\nu}]$, for each outcome, $j$, and that these paths lie entirely within the zonotope $Z_{{\cal M}^{\nu}}$ generated by the collection of these operators, ${E_{j}^{\nu}}$. In fact, by Theorem $2$ of Ref. [18], these paths are piecewise local. As discussed at the end of the preceding section, we do not know ${\cal M}^{\nu}$. Therefore, we consider all possible ${\cal M}^{\nu}$ compatible with channel ${\cal E}^{\nu}$, which means that instead of considering our continuous paths as lying within zonotope $Z_{{\cal M}^{\nu}}$, we instead view these paths as lying within zonoid $Z_{{\cal E}^{\nu}}$. According to Lemma 2, $Z_{{\cal M}^{\nu}}\subseteq Z_{{\cal E}^{\nu}}$ for every such ${\cal M}^{\nu}$, so this is always possible. We could, of course, restrict to measurements that are implementable by LOCC, so that such paths of product operators actually exist, but it is sufficient to follow the easier route and include all compatible measurements, as we do here.

Thus, we have a sequence of zonoids, $\{Z_{{\cal E}^{\nu}}\}$, which converges to zonoid $Z_{\cal E}$, and within each $Z_{{\cal E}^{\nu}}$, we have a collection of continuous paths of product operators starting at $I_{\cal H}$. This situation is very similar to the one encountered in Ref. [18], including that the paths are all Lipschitz continuous with Lipschitz constant equal to unity [18]. By the Arzéla-Ascoli theorem [22], this tells us that there exists a subsequence of these paths which converges uniformly on the range, $s\in[d,0]$, to a continuous limiting path. However, to this point, we have many paths for each $\nu$ and have not actually identified a sequence of paths to begin with. We now turn to the task of identifying such a sequence. In fact we will identify, and characterize, many such sequences.

Whereas in our earlier work, we had a sequence of known measurements, ${\cal M}^{\nu}$, whose limit was another known measurement, ${\cal M}$, here we do not know what the ${\cal M}^{\nu}$ or ${\cal M}$ are. Instead, we only know that each ${\cal M}^{\nu}$ is compatible with channel ${\cal E}^{\nu}$, the sequence of which has the limit ${\cal E}$. In the previous case, knowing ${\cal M}^{\nu}$ meant that we also knew the final outcomes, $E_{j}^{\nu}$, of the corresponding LOCC protocol, and also that there was an ordering of these outcomes, provided by labels $\nu$ for fixed $j$, such that for each $j$, the sequence $\{E_{j}^{\nu}\}_{\nu}$ satisfied $\lim_{\nu\to\infty}E_{j}^{\nu}=E_{j}$, with $E_{j}$ an outcome of the limiting measurement ${\cal M}$. This allowed us to identify sequences of paths—i.e., those ending at $E_{j}^{\nu}$ with the aforementioned fixed label, $j$—from which, according to Arzéla-Ascoli, it followed that there was a limiting path from $I_{\cal H}$ to a point along $(0,E_{j}]$. In our present case, with $\lim_{\nu\to\infty}{\cal E}^{\nu}={\cal E}$, we do not know $E_{j}^{\nu}$ or the limiting outcomes $E_{j}$. Thus, we need a way to partition the various paths we’ve identified as lying within the $Z_{{\cal E}^{\nu}}$ so that each subset provides a sequence of paths whose limit we can characterize. In particular, we wish to characterize the endpoints of the limiting paths. We will discuss how to partition the paths forthwith. The following lemma concerning the Kraus ranks of the ${\cal E}^{\nu}$ will be useful, and it may be of interest in other contexts, as well.

In this section, we consider the example [11] of a two-qubit state transformation which can be arbitrarily closely approximated, but cannot be implemented exactly by LOCC. In Section VIII, we consider a generalization of this example to any number of qubits. Consider a quantum instrument [11], which consists of the following three CP maps, the sum of which form the quantum channel ${\cal E}$ we consider here, but are not individually trace preserving,

where $T_{1}=I/\sqrt{3}$ and $T_{2}=\left([0]+2[1]\right)/\sqrt{6}$. This suggests describing channel ${\cal E}$ by the five product Kraus operators,

It turns out that there is a smaller set of (product) Kraus operators that represent the same channel. For the sake of simplicity, we will use this smaller set, which is

It is easily checked that $K_{j}^{\prime}=\sum_{j}W_{jl}\hat{K}_{l}$, with $W$ the isometry,

The following LOCC protocol [23, 11], ${\cal P}^{\nu}$, implements a quantum channel, ${\cal E}^{\nu}$, which approximates the channel ${\cal E}$ of Eq. (V). Given single qubit (Hermitian) operators, $A_{0}=\sqrt{1-\epsilon}[0]_{A}+[1]_{A}$ and $A_{1}=\sqrt{\epsilon}[0]_{A}$, Alice begins by measuring $\{A_{0},A_{1}\}$ on her system $A$, with corresponding POVM elements ${\cal A}_{j}=A_{j}^{\dagger}A_{j}=A_{j}^{2},~{}j=0,1$. She shares her outcome with Bob, who follows with the same measurement—having POVM elements ${\cal B}_{0}=(1-\epsilon)[0]_{B}+[1]_{B}$ and ${\cal B}_{1}=\epsilon[1]_{B}$—on his system $B$, sharing his outcome in turn with Alice. They halt at this point unless they both obtained outcome $0$, in which case they repeat their local measurements as just described. Then, after a maximum of $\nu$ iterations of this procedure, they stop. Noting that $A_{0}^{n}=(1-\epsilon)^{n/2}[0]_{A}+[1]_{A}$ and $A_{1}A_{0}^{n}=\epsilon^{1/2}(1-\epsilon)^{n/2}[0]_{A}$, and similarly for Bob’s measurement operators, then the first few rounds of this LOCC protocol—represented as a tree graph with nodes labeled by the corresponding POVM elements implemented to that point in the protocol—is depicted in Figure 1 (note that the notation is condensed slightly in this figure by defining $\eta=1-\epsilon$). In the figure, there is a single main branch, stretching down along the upper right side, which continues (note ellipses at bottom right) for $2\nu$ segments, while all other branches extend either one or two segments off of that main branch and then terminate. As discussed in [11], the limiting channel is ${\cal E}$, as desired, as long as we choose $\epsilon=\nu^{-c}$ with $0<c<1$, which we will do in what follows.

Consider that main branch in each protocol and the sequence of these main branches formed as we increment $\nu$. Each of these main branches describes a path, $\Pi^{\nu}(s)$, of product operators in operator space, as described in Sections III and IV. The sequence of such paths of product operators has a limiting path, which we will now identify. Defining $s=\textrm{Tr}\left({\cal A}_{0}^{n}\otimes{\cal B}_{0}^{n}\right)=[(1-\epsilon)^{n}+1]^{2}$, we have $\sqrt{s}-1=(1-\epsilon)^{n}$, at least at those particular nodes along this main branch. This suggests the quantity $\sqrt{s}-1$ may play a role. Noting that we will be considering the limit as $\epsilon\to 0$, and that in this limit, we will also have the length of the branch, $\nu\to\infty$, care must be taken in considering how the quantity $(1-\epsilon)^{n}$ behaves in these limits (since $n$ ranges up to $\nu$). Nonetheless, the quantity $\sqrt{s}-1$ is perfectly well-behaved in these limits, and in fact, we can prove the following Lemma.