We study the power of measurements implementable with local quantum operations and classical communication (LOCC) measurements in the setting of quantum channel discrimination. More precisely, we consider discrimination procedures that attempt to identify an unknown channel, chosen uniformly from two known alternatives, that take the following form: (i) the input to the unknown channel is prepared in a possibly entangled state with an ancillary system, (ii) the unknown channel is applied to the input system, and (iii) an LOCC measurement is performed on the output and ancillary systems, resulting in a guess for which of the two channels was given. The restriction of the measurement in such a procedure to be an LOCC measurement is of interest because it isolates the entanglement in the initial input-ancillary systems as a resource in the setting of channel discrimination. We prove that there exist channel discrimination problems for which restricted procedures of this sort can be at either of the two extremes: they may be optimal within the set of all discrimination procedures (and simultaneously outperform all strategies that make no use of entanglement), or they may be no better than unentangled strategies (and simultaneously suboptimal within the set of all discrimination procedures).
We study the power of measurements implementable with local quantum operations and classical communication (LOCC) measurements in the setting of quantum channel discrimination. More precisely, we consider discrimination procedures that attempt to identify an unknown channel, chosen uniformly from two known alternatives, that take the following form: (i) the input to the unknown channel is prepared in a possibly entangled state with an ancillary system, (ii) the unknown channel is applied to the input system, and (iii) an LOCC measurement is performed on the output and ancillary systems, resulting in a guess for which of the two channels was given. The restriction of the measurement in such a procedure to be an LOCC measurement is of interest because it isolates the entanglement in the initial input-ancillary systems as a resource in the setting of channel discrimination. We prove that there exist channel discrimination problems for which restricted procedures of this sort can be at either of the two extremes: they may be optimal within the set of all discrimination procedures (and simultaneously outperform all strategies that make no use of entanglement), or they may be no better than unentangled strategies (and simultaneously suboptimal within the set of all discrimination procedures).
Entanglement is known to be a powerful resource that improves performance in various quantum information and computational tasks. A standard example of such a phenomenon is the possibility of perfectly discriminating all four Pauli operations in a single shot via the superdense coding protocol. While entanglement is often a powerful resource for quantum channel discrimination, this is not necessarily the case. In this work, we identify scenarios in which the maximally entangled state is a bad choice of input state and, more generally, show that excessive entanglement can reduce channel discriminability dramatically. To do so, we present an explicit pair of unitary channels which are perfectly discriminable without entanglement, but for which any strategy with maximally entangled input states is ε-close to a blind uniform guessing strategy. To develop a systematic approach, we introduce the concepts of Maximal Entanglement Worst Case (MEWC) and Maximal Entanglement Best Case (MEBC) pairs of channels, and present conditions for a pair of channels to be MEWC or MEBC. With these conditions, we show that the optimal input states for discriminating MEWC pairs of channels are necessarily separable, and provide non-trivial examples of measurement channels for which entanglement necessarily reduces the maximum probability of discrimination.
We consider the problem of local operations and classical communication (LOCC) discrimination between two bipartite pure states of fermionic systems. We show that, contrary to the case of quantum systems, for fermionic systems it is generally not possible to achieve the ideal state discrimination performances through LOCC measurements. On the other hand, we show that an ancillary system made of two fermionic modes in a maximally entangled state is a sufficient additional resource to attain the ideal performances via LOCC measurements. The stability of the ideal results is studied when the probability of preparation of the two states is perturbed, and a tight bound on the discrimination error is derived.
The paper studies unambiguous discrimination of fermionic states through local operations and classical communication (LOCC). In the task of unambiguous discrimination, no error is tolerated but an inconclusive result is allowed. We show that contrary to the quantum case, it is not always possible to distinguish two fermionic states through LOCC unambiguously with the same success probability as if global measurements were allowed. Furthermore, we prove that we can overcome such a limit through an ancillary system made of two fermionic modes, independently of the dimension of the system, prepared in a maximally entangled state: in this case, LOCC protocols achieve the optimal success probability.
We consider an infinite class of unambiguous quantum state discrimination problems on multipartite systems, described by Hilbert space H, of any number of parties. Restricting consideration to measurements that act only on H, we find the optimal global measurement for each element of this class, achieving the maximum possible success probability of 1/2 in all cases. This measurement turns out to be both separable and unique, and by our recently discovered necessary condition for local quantum operations and classical communication (LOCC) it is easily shown to be impossible by any finite-round LOCC protocol. We also show that, quite generally, if the input state is restricted to lie in H, then any LOCC measurement on an enlarged Hilbert space is effectively identical to an LOCC measurement on H. Therefore, our necessary condition for LOCC demonstrates directly that a higher success probability is attainable for each of these problems using general separable measurements as compared to that which is possible with any finite-round LOCC protocol.
In this thesis we study the subset of quantum operations that can be implemented using only local quantum operations and classical communication (LOCC). This restricted paradigm serves as a tool to study not only quantum correlations and other nonlocal quantum effects, but also resource transformations such as channel capacities. \n \nThe mathematical structure of LOCC is complex and difficult to characterize. In the first part of this thesis we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. This perspective allows us to measure the distance between two LOCC instruments and hence discuss the closure of LOCC in a rigorous way. While the set of LOCC is not topologically closed, we show that the operations that can be implemented using some fixed number rounds of communication constitute a compact subset of all quantum operations. We also exhibit a subset of LOCC measurements that is closed. Additionally we establish the existence of an open ball around the completely depolarizing map consisting entirely of LOCC implementable maps. \n \nIn the second part of this thesis we focus on the task of discriminating states from some known set S by LOCC. Building on the work in the paper "Quantum nonlocality without entanglement", we provide a framework for lower bounding the error probability of any LOCC protocol aiming at discriminating the states from S. We apply our framework to an orthonormal product basis known as the domino states. This gives an alternative and simplified bound quantifying how well these states can be discriminated using LOCC. We generalize this result for similar bases in larger dimensions, as well as the "rotated" domino states, resolving a long-standing open question. These results give new examples of quantitative gaps between the classes of separable and LOCC operations. \n \nIn the last part of this thesis, we ask what differentiates separable from LOCC operations. Both of these classes play a key role in the study of entanglement. Separable operations are known to be strictly more powerful than LOCC ones, but no simple explanation of this phenomenon is known. We show that, in the case of bipartite von Neumann measurements, the ability to interpolate is an operational principle that separates LOCC and separable operations.
Single-shot quantum channel discrimination is the fundamental task of determining, given only a single use, which of two known quantum channels is acting on a system. In this thesis we investigate the well-known phenomenon that entanglement to an auxiliary system can provide an advantage in this task. In particular, we consider the questions: (1) How much entanglement is in general necessary to achieve an optimal discrimination strategy? (2) What is the maximal advantage provided by entanglement? \n \nGiven a linear map Ψ : L(Cn) → L(Cm), its multiplicity maps are defined as the family of linear maps Ψ ⊗ 1L(Ck) : L(Cn ⊗ Ck) → L(Cm ⊗ Ck), where 1L(Ck) is the identity on L(Ck). Due to the Holevo-Helstrom theorem, the optimal performance using an auxiliary system of dimension k is quantified in terms of the norm of Ψ⊗1L(Ck), where Ψ is a linear map that depends on the parameters of the discrimination problem. Hence, the advantage provided by entanglement is represented in the growth of the norm of Ψ ⊗ 1L(Ck ) with k, a classic phenomenon in the theory of operator algebras. \n \nWe formalize question (1) by investigating, relative to the input and output dimensions of the channels to be discriminated, how large of an auxiliary system is necessary to achieve an optimal strategy. Mathematically, this is connected to when the norm of Ψ ⊗ 1L(Ck) stops growing with k. It is well- known that an auxiliary system of dimension equal to the input is always sufficient to achieve an optimal strategy, and that this is sometimes necessary when the output dimension is at least as large as the input. We prove that, even when the output dimension is arbitrarily small compared to the input, it is still sometimes necessary to use an auxiliary system as large as the input to achieve an optimal strategy. \n \nFor question (2), we investigate, with respect to a fixed input dimension, how large the gap between the optimal performances with and without entanglement can be. Mathematically, this is quantified by the rate of growth of the norm of Ψ ⊗ 1L(Ck ) in k. It is known that matrix transposition has the fastest possible growth, and we prove that it is essentially the unique linear map with this property. We use this to prove that a discrimination problem defined in terms of the Werner-Holevo channels is essentially the unique game satisfying a norm relation that states that the game can be won with certainty using entanglement, but is hard to win without entanglement. \n \nAlong the way, we prove characterizations of the structure of maximal entanglement, as measured according to the entanglement negativity, as well as relative to a large class of entanglement measures. We also give various characterizations of complete trace-norm isometries, and reversible quantum channels.