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arXiv:1910.04646 · Journal of Physics A Mathematical and Theoretical

Volume of the set of LOCC-convertible quantum states

Fabio Deelan Cunden, Paolo Facchi A, Giuseppe Florio, Giovanni Gramegna

Published Mar 2, 2020• 5 citations
arXivHTMLPDF

Abstract

Abstract The class of quantum operations known as local operations and classical communication (LOCC) induces a partial ordering on quantum states. We present the results of systematic numerical computations related to the volume (with respect to the unitarily invariant measure) of the set of LOCC-convertible bipartite pure states, where the ordering is characterised by an algebraic relation known as majorization. The numerical results, which exploit a tridiagonal model of random matrices, provide quantitative evidence that the proportion of LOCC-convertible pairs vanishes in the limit of large dimensions, and therefore support a previous conjecture by Nielsen. In particular, we show that the problem is equivalent to the persistence of a non-Markovian stochastic process and the proportion of LOCC-convertible pairs decays algebraically with a nontrivial persistence exponent. We extend this analysis by investigating the distribution of the maximal success probability of LOCC-conversions. We show a dichotomy in behaviour between balanced and unbalanced bipartitions. In the latter case the asymptotics is somehow surprising: in the limit of large dimensions, for the overwhelming majority of pairs of states a perfect LOCC-conversion is not possible; nevertheless, for most states there exist local strategies that succeed in achieving the conversion with a probability arbitrarily close to one. We present strong evidence of a universal scaling limit for the maximal probability of successful LOCC-conversions and we suggest a connection with the typical fluctuations of the smallest eigenvalue of Wishart random matrices.

DOI
10.1088/1751-8121/ab7b21

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Abstract The class of quantum operations known as local operations and classical communication (LOCC) induces a partial ordering on quantum states. We present the results of systematic numerical computations related to the volume (with respect to the unitarily invariant measure) of the set of LOCC-convertible bipartite pure states, where the ordering is characterised by an algebraic relation known as majorization. The numerical results, which exploit a tridiagonal model of random matrices, provide quantitative evidence that the proportion of LOCC-convertible pairs vanishes in the limit of large dimensions, and therefore support a previous conjecture by Nielsen. In particular, we show that the problem is equivalent to the persistence of a non-Markovian stochastic process and the proportion of LOCC-convertible pairs decays algebraically with a nontrivial persistence exponent. We extend this analysis by investigating the distribution of the maximal success probability of LOCC-conversions. We show a dichotomy in behaviour between balanced and unbalanced bipartitions. In the latter case the asymptotics is somehow surprising: in the limit of large dimensions, for the overwhelming majority of pairs of states a perfect LOCC-conversion is not possible; nevertheless, for most states there exist local strategies that succeed in achieving the conversion with a probability arbitrarily close to one. We present strong evidence of a universal scaling limit for the maximal probability of successful LOCC-conversions and we suggest a connection with the typical fluctuations of the smallest eigenvalue of Wishart random matrices.
Abstract We advance on the conversion of bipartite quantum states via local operations and classical communication (LOCC) for infinite-dimensional systems. We introduce δ-LOCC convertibility based on the observation that any pure state can be approximated by a state with finite-support Schmidt coefficients. We show that δ-LOCC convertibility of bipartite states is fully characterized by a majorization relation between the sequences of squared Schmidt coefficients, providing a novel extension of Nielsen’s theorem for infinite-dimensional systems. Hence, our definition is equivalent to the one of ε-LOCC convertibility (Owari et al 2008 Quantum Inf. Comput. 8 0030), but deals with states having finitely supported sequences of Schmidt coefficients. Additionally, we discuss the notions of optimal common resource and optimal common product in this scenario. The optimal common product always exists, whereas the optimal common resource depends on the existence of a common resource. This highlights a distinction between the resource-theoretic aspects of finite versus infinite-dimensional systems. Our results rely on the order-theoretic properties of majorization for infinite sequences, applicable beyond the LOCC convertibility problem.
We study the exact, non-deterministic conversion of multipartite pure quantum states into one-another via local operations and classical communication (LOCC) and asymptotic entanglement transformation under such channels. In particular, we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in the success probability, known as the converse error exponent. We give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of LOCC conversion. A full understanding of exact asymptotic extraction rates between pure states in the converse regime thus depends on a full understanding of this spectrum. We present a characterization of spectral points and use it to describe the spectrum in the bipartite case. This leads to a full description of the spectrum and thus an explicit formula for the asymptotic extraction rate between pure bipartite states, given a converse error exponent. This extends the result on entanglement concentration in [1], where the target state is fixed as the Bell state. In the limit of vanishing converse error exponent, the rate formula provides an upper bound on the exact asymptotic extraction rate between two states, when the probability of success goes to 1. In the bipartite case, we prove that this bound holds with equality.
Abstract We develop the theory of local operations and classical communication (LOCC) for bipartite quantum systems represented by commuting von Neumann algebras. Our central result is the extension of Nielsen’s Theorem, stating that the LOCC ordering of bipartite pure states is equivalent to the majorization of their restrictions to arbitrary factors. As a consequence, we find that in bipartite system modeled by commuting factors in Haag duality, (a) all states have infinite single-shot entanglement if and only if the local factors are not of type I, (b) type III factors are characterized by LOCC transitions of arbitrary precision between any two pure states, and c) the latter holds even without classical communication for type III1\hbox {III}_{1}III1​ III 1 factors. In the case of semifinite factors, the usual construction of pure state entanglement monotones carries over. Together with recent work on embezzlement of entanglement, this gives a one-to-one correspondence between the classification of factors into types and subtypes and operational entanglement properties. In the appendix, we provide a self-contained treatment of majorization on semifinite von Neumann algebras and σ\sigma σ σ -finite measure spaces.
Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.
Recently, entanglement concentration was explicitly shown to be irreversible. However, it is still not clear what kind of states can be reversibly converted in the asymptotic setting by local operations and classical communication (LOCC) when neither the initial nor the target state is maximally entangled. We derive the necessary and sufficient condition for the reversibility of LOCC conversions between two bipartite pure entangled states in the asymptotic setting. In addition, we show that conversion can be achieved perfectly with only local unitary operation under such condition except for special cases. Interestingly, our result implies that an error-free reversible conversion is asymptotically possible even between states whose copies can never be locally unitarily equivalent with any finite numbers of copies, although such a conversion is impossible in the finite setting. In fact, we show such an example. Moreover, we establish how to overcome the irreversibility of LOCC conversion in two ways. As for the first method, we evaluate how many copies of the initial state are to be lost to overcome the irreversibility of LOCC conversion. The second method is to add a supplementary state appropriately, which also works for local unitary conversion unlike the first method. Especially, for the qubit system, any nonmaximally pure entangled state can be a universal resource for the asymptotic reversibility when copies of the state are sufficiently many. More interestingly, our analysis implies that far-from-maximally entangled states can be better than nearly maximally entangled states as this type of resource. This fact brings new insight to the resource theory of state conversion.
Some progress is reported on conditions for convertibility among bipartite 2x2 entangled states: An inconvertibility condition related to the rank of an entangled state is given that it is impossible to convert to an entangled state with lower rank under separable operations; a particular set of local operations and classical communication (LOCC) is used to analyze convertibility of three subclasses of states - Werner states, Bell diagonal states and maximally entangled mixed states (MEMS). It is conjectured that MEMS may lie on the bottom of entangled state ordering for given entanglement of formation. A plausible way is suggested of systematically calculating convertibility in a general subclass of bipartite states whose density matrices are defined to be diagonal in a common basis. The set of LOCC adopted in this work is argued to be generalizable to provide sufficient conditions for convertibility among a large range of general 2x2 entangled states.