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arXiv:2012.13845

Discrimination of symmetric states in operational probabilistic theory

Kenji Nakahira

Published Dec 27, 2020• 0 citations
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Abstract

A state discrimination problem in an operational probabilistic theory (OPT) is investigated in diagrammatic terms. It is well-known that, in the case of quantum theory, if a state set has a certain symmetry, then there exists a minimum-error measurement having the same type of symmetry. However, to our knowledge, it is not yet clear whether this property also holds in a more general OPT. We show that it also holds in OPTs, i.e., for a symmetric state set, there exists a minimum-error measurement that has the same type of symmetry. It is also shown that this result can be utilized to optimize over a restricted class of measurements, such as sequential or separable measurements.

DOI
10.48550/arxiv.2012.13845

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A state discrimination problem in an operational probabilistic theory (OPT) is investigated in diagrammatic terms. It is well-known that, in the case of quantum theory, if a state set has a certain symmetry, then there exists a minimum-error measurement having the same type of symmetry. However, to our knowledge, it is not yet clear whether this property also holds in a more general OPT. We show that it also holds in OPTs, i.e., for a symmetric state set, there exists a minimum-error measurement that has the same type of symmetry. It is also shown that this result can be utilized to optimize over a restricted class of measurements, such as sequential or separable measurements.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
We consider optimal state discrimination in a general convex operational framework, so-called generalized probabilistic theories (GPTs), and present a general method of optimal discrimination by applying the complementarity problem from convex optimization. The method exploits the convex geometry of states but not other detailed conditions or relations of states and effects. We also show that properties in optimal quantum state discrimination are shared in GPTs in general: (i) no measurement sometimes gives optimal discrimination, and (ii) optimal measurement is not unique.
State discrimination is a useful test problem with which to clarify the power and limitations of different classes of measurement. We consider the problem of discriminating between given states of a bipartite quantum system via sequential measurement of the subsystems, with classical feed-forward of measurement results. Our aim is to understand when sequential measurements, which are relatively easy to implement experimentally, perform as well, or almost as well, as optimal joint measurements, which are in general more technologically challenging. We construct conditions that the optimal sequential measurement must satisfy, analogous to the well-known Helstrom conditions for minimum error discrimination in the unrestricted case. We give several examples and compare the optimal probability of correctly identifying the state via global versus sequential measurement strategies.
Abstract Minimum error discrimination (MED) and Unambiguous discrimination (UD) are two common strategies for quantum state discrimination that can be modified by imposing a finite error margin on the error probability. Error margins 0 and 1 correspond to two common strategies. In this paper, for an arbitrary error margin m , the discrimination problem of equiprobable quantum symmetric states is analytically solved for four distinct cases. A generating set of irreducible and reducible representations of a subgroup of a unitary group are considered, separately, as unitary operators that produce one set of the symmetric states. In the irreducible case, for N ⩾ d mixed and pure qudit states, one critical m which divides the parameter space into two domains is obtained. The number of critical values m in the reducible case is two, for both N mixed and pure qubit states. The reason for this difference between numbers of critical values m is explained. The optimal set of measurements and corresponding maximum success probability in fully analytical form are determined for all values of the error margin. The relationship between the amount of error that is imposed on error probability and geometrical situation of states with changes in rank of element corresponding to inconclusive result is determined. The behaviors of elements of measurement are explained geometrically in order to decrease the error probability in each domain. Furthermore, the problem of the discrimination with error margin among elements of two different sets of symmetric quantum states is studied. The number of critical values m is equivalent to one set in both reducible and irreducible cases. In addition, optimal measurements in each domain are obtained.
This is a book about operational probabilistic theories. The standard approach in such theories is from a time forward perspective. In this book we mostly take a time symmetric perspective. This presents a branding problem. Is this a niche book merely about time symmetry? No. This is a comprehensive book about operational probabilistic theories, but mostly from a time symmetric perspective. In fact, this book consists of (1) a simple book about simple operations having simple causal structure (where all the inputs are before all the outputs), and (2) a complex book about complex operations that can have complicated causal structure (a complex operation is equipped with a causal diagram). For the simple case we are able to show that the time symmetric perspective is equivalent to the time forward perspective. In each book we set up (A) operational probabilistic theories (OPTs) in terms of operations, (B) Operational Quantum Theory (OQT) in terms of operator tensors which correspond to operations, and (C) the theory of Hilbert objects which can be doubled up to give operator tensors. Operations are required to be physical which guarantees that circuits built out of operations have probabilities between 0 and 1 and that certain causality conditions are met. We prove that when we wire together operations the resulting networks are also physical. We model Sorkin's impossible measurements with complex operations and show that physicality prevents anomalous signalling. We develop diagrammatic notation for Hilbert objects. This includes mirrors for doubling up and mirror theorems. We use this framework to prove time symmetric causal dilation theorems for various causal diagrams.
Abstract It is a key issue to characterize the model of standard quantum theory out of general models by an operational condition. The framework of general probabilistic theories (GPTs) is a new information theoretical approach to single out standard quantum theory. It is known that traditional properties, for example, Bell-CHSH inequality, are not sufficient to single out standard quantum theory among possible models in GPTs. As a more precise property, we focus on the bound of the performance for an information task called state discrimination in general models. We give an equivalent condition for outperforming the minimum discrimination error probability under the standard quantum theory given by the trace norm. Besides, by applying the equivalent condition, we characterize standard quantum theory out of general models in GPTs by the bound of the performance for state discrimination.