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Quantum Detection and Finite Frames

Andrew Kebo

Published Apr 12, 2005• 4 citations
OpenAlex

Abstract

In quantum mechanics, the definition of a Von Neumann measurement can be generalized using positive-operator-valued measures. This modified definition of a quantum measurement allows one to better distinguish between a set of nonorthogo-nal quantum states. In this thesis we examine a quantum detection problem, where we have a physical system whose state is limited to be in only one of a finite number of possibilities. These possible states are not necessarily orthogonal. We want to find the best method of measuring the system in order to distinguish which state the system is in. Mathematically, we want to find a positive-operator-valued measure that minimizes the probability of a detection error. It is shown that all tight-frames with frame constant 1 correspond to positive-operator-valued measures. We reformulate the problem in terms of tight-frames that minimize the detection error. In the finite dimensional case, the problem of finding the tight-frame that minimizes the error can be converted into a Hamiltonian system on the group SO(N). The minimum energy solutions of this Hamiltonian system correspond exactly to the tight-frames that minimize the detection error. In this setting, several numerical methods can be applied to give numerical constructions of the desired tight-frames.

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In quantum mechanics, the definition of a Von Neumann measurement can be generalized using positive-operator-valued measures. This modified definition of a quantum measurement allows one to better distinguish between a set of nonorthogo-nal quantum states. In this thesis we examine a quantum detection problem, where we have a physical system whose state is limited to be in only one of a finite number of possibilities. These possible states are not necessarily orthogonal. We want to find the best method of measuring the system in order to distinguish which state the system is in. Mathematically, we want to find a positive-operator-valued measure that minimizes the probability of a detection error. It is shown that all tight-frames with frame constant 1 correspond to positive-operator-valued measures. We reformulate the problem in terms of tight-frames that minimize the detection error. In the finite dimensional case, the problem of finding the tight-frame that minimizes the error can be converted into a Hamiltonian system on the group SO(N). The minimum energy solutions of this Hamiltonian system correspond exactly to the tight-frames that minimize the detection error. In this setting, several numerical methods can be applied to give numerical constructions of the desired tight-frames.
We consider the problem of constructing measurements optimized to distinguish between a collection of possibly nonorthogonal quantum states. We consider a collection of pure states and seek a positive operator-valued measure (POVM) consisting of rank-one operators with measurement vectors closest in squared norm to the given states. We compare our results to previous measurements suggested by Peres and Wootters (1991) and Hausladen et al. (1996), where we refer to the latter as the square-root measurement (SRM). We obtain a new characterization of the SRM, and prove that it is optimal in a least-squares sense. In addition, we show that for a geometrically uniform state set the SRM minimizes the probability of a detection error. This generalizes a similar result of Ban et al. (see Int. J. Theor. Phys., vol.36, p.1269-88, 1997).
A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.
The tight frames can be regarded as a particular case of POVMs (positive operator-valued measures describing generalized measurements), namely the case when all the operators are rank-one. Each orthonormal basis is a tight frame, and every tight frame, after the embedding into a higher-dimensional space, is the orthogonal projection of an orthonormal basis. There exist several POVM-based definitions of coherence, and they are well-investigated. Our aim is to identify properties specific to the particular case of tight frames, and to look for some applications. All the POVM-based definitions use a Naimark extension. The frame-dependent coherence can be regarded as a particular case of POVM-based coherence, but it can be defined directly, without to use a Naimark extension. Its definition is a direct generalization of the basis-dependent ℓ 1 -norm of coherence, and it offers a more accurate description because we can use a frame containing several orthogonal bases. A quasi frame-independent definition of coherence for qubit systems is presented.
We will give a complete solution to the frame quantum detection problem. We will solve both cases of the problem: the quantum injectivity problem and quantum state estimation problem. We will answer the problem in both the real and complex cases and in both the finite dimensional and infinite dimensional cases. Finite Dimensional Case: (1) We give two complete classifications of the sets of vectors which solve the injectivity problem - for both the real and complex cases. We also give methods for constructing them. (2) We show that the frames which solve the injectivity problem are open and dense in the family of all frames. (3) We show that the Parseval frames which give injectivity are dense in the Parseval frames. (4) We classify all frames for which the state estimation problem is solvable, and when it is not solvable, we give the best approximation to a solution. Infinite Dimensional Case: (1) We give a classification of all frames which solve the injectivity problem and give methods for constructing solutions. (2) We show that the frames solving the injectivity problem are neither open nor dense in all frames. (3) We give necessary and sufficient conditions for the state estimation problem to always have solutions for measurements in ℓ1\ell_1ℓ1​ and show that there is no injective frame for which the state estimation problem is solvable for all measurements in ℓ2\ell_2ℓ2​. (4) When the state estimation problem does not have an exact solution, we give the best approximation to a solution.
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.