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Journal of Physics Conference Series

On the irreversibility of measurements of correlations

Masato Koashi

Published Jan 7, 2009• 5 citations
Publisher

Abstract

Measurements on a bipartite system AB are classified into ones that are freely implementable with only classical communication between A and B (LOCC measurements), and the others that require consumption of entanglement if they are to be implemented with classical communication. When we notice that measurements on a bipartite system can also be used to create entanglement, we have another natural classification: measurements that are capable of creating entanglement, and the others that have no ability of producing entanglement (separable measurements). Interestingly, there exists a separable measurement that is not an LOCC measurement, namely, a measurement that requires entanglement to implement, but is not capable of creating entanglement at all. Such an example was found for a pair of three-level systems, in conjunction with a discrimination task of an intricate set of eight or nine states. Here we show an example of such a measurement on a pair of two-level systems (qubits), which arises in a discrimination task of just two states that look quite simple and have little intricacy. Such an example suggests that this kind of irreversibility is not only shown by a limited set of cleverly constructed measurements, but is also exhibited by a much larger class of measurements than we had expected.

DOI
10.1088/1742-6596/143/1/012007

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Measurements on a bipartite system AB are classified into ones that are freely implementable with only classical communication between A and B (LOCC measurements), and the others that require consumption of entanglement if they are to be implemented with classical communication. When we notice that measurements on a bipartite system can also be used to create entanglement, we have another natural classification: measurements that are capable of creating entanglement, and the others that have no ability of producing entanglement (separable measurements). Interestingly, there exists a separable measurement that is not an LOCC measurement, namely, a measurement that requires entanglement to implement, but is not capable of creating entanglement at all. Such an example was found for a pair of three-level systems, in conjunction with a discrimination task of an intricate set of eight or nine states. Here we show an example of such a measurement on a pair of two-level systems (qubits), which arises in a discrimination task of just two states that look quite simple and have little intricacy. Such an example suggests that this kind of irreversibility is not only shown by a limited set of cleverly constructed measurements, but is also exhibited by a much larger class of measurements than we had expected.
Any separable measurement with a finite number of outcomes on a finite dimensional multiparticle quantum system can, with non-zero probability, be implemented with local operations and classical communication (LOCC). I use this fact to obtain a necessary and sufficient condition for a finite set of states of a finite dimensional multiparticle quantum system to be amenable to LOCC unambiguous state discrimination. This condition is valid for states which may be be mixed, entangled or both. It is also found to have an intriguing connection with the theory of entanglement witnesses.
Quantum nonlocality is usually associated with entangled states by their violations of Bell-type inequalities. However, even unentangled systems, whose parts may have been prepared separately, can show nonlocal properties. In particular, a set of product states is said to exhibit "quantum nonlocality without entanglement" if the states are locally indistinguishable; i.e., it is not possible to optimally distinguish the states by any sequence of local operations and classical communication. Here, we present a stronger manifestation of this kind of nonlocality in multiparty systems through the notion of local irreducibility. A set of multiparty orthogonal quantum states is defined to be locally irreducible if it is not possible to locally eliminate one or more states from the set while preserving orthogonality of the postmeasurement states. Such a set, by definition, is locally indistinguishable, but we show that the converse does not always hold. We provide the first examples of orthogonal product bases on C^{d}⊗C^{d}⊗C^{d} for d=3, 4 that are locally irreducible in all bipartitions, where the construction for d=3 achieves the minimum dimension necessary for such product states to exist. The existence of such product bases implies that local implementation of a multiparty separable measurement may require entangled resources across all bipartitions.
We explore the question of entanglement as a universal resource for implementing quantum measurements by local operations and classical communication (LOCC). We show that for most multipartite systems (consisting of three or more subsystems), there is no entangled state from the same space that can enable all measurements by LOCC. This is in direct contrast to the bipartite case, where a maximally entangled state is an universal resource. Our results are obtained by considering the problem of local state discrimination using entanglement as a resource; and a key ingredient in our proofs is showing an equivalence between the problem of local state discrimination and that of unambiguous local state transformation. We also present partial characterization of universal resource states in multipartite systems.
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.
We now have all the basic tools we need to describe systems of one and two qubits. In this final chapter of Part I we look at some of the consequences of the peculiar properties of qubits when they are used to encode information. Many of these results can be traced to the properties of quantum measurement. Measuring a single qubit A key result in quantum information theory is that it is impossible to accurately characterize a single qubit. In other words, there is no experiment, or sequence of experiments, which allows us to find out the state of a single quantum bit. The reason for this problem is twofold. Firstly, we have to make some sort of decision about the basis we will use for the measurement. For example, when measuring a single qubit the most popular choice is to make a measurement in the computational basis. This is equivalent to asking the qubit whether it is in state |0⧽ or state |1⧽. If the qubit is indeed in one of the basis states the measurement process is simple, and we will get the obvious answer of 0 if it is in |0⧽ and 1 if it is in |1⧽. If, however, the qubit is in a superposition, such as α|0⧽+β|1⧽, then the situation is more difficult. Characterizing the state now means determining the values of the two complex numbers α and β, but the measurement can only return the answer 0 or 1, and for a superposition state one of these two answers will be returned at random, with probabilities |α| 2 and |β| 2 , respectively.