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Chinese Journal of Quantum Electronics

Construction methods of tight frame bsaed on quantum measurement

Lin Ni

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The construction methods of tight frame in quantum field are investigated according to the finite-dimensional tight frame in classic field.As one-to-one correspondence between tight frames and rank-one generalized quantum measurement is proved,two construction methods of tight frame in quantum field based on generalized quantum measurement are designed.The results show that two optimal quantum tight frame can separately solve two basic strategies of state discrimination which are minimum error discrimination and optimal unambiguous discrimination.The minimum error discrimination of a given set of vectors using generalized quantum measurement method with the least-squares criteria is precisely quantum tight frame based on least-squares measurement,which can be divided into two situations,constrained least-squares frame(CLSF) that has specified scale and unconstrained least-squares frame(ULSF) whose scale is chosen. to minimize the least-squares error.Quantum tight frame based on optimal unambiguous discrimination is also given using quantum generalized measurement method.Finally,an example is given to compare these optimal construction methods of tight frame and results show that ULSF is the best.

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The construction methods of tight frame in quantum field are investigated according to the finite-dimensional tight frame in classic field.As one-to-one correspondence between tight frames and rank-one generalized quantum measurement is proved,two construction methods of tight frame in quantum field based on generalized quantum measurement are designed.The results show that two optimal quantum tight frame can separately solve two basic strategies of state discrimination which are minimum error discrimination and optimal unambiguous discrimination.The minimum error discrimination of a given set of vectors using generalized quantum measurement method with the least-squares criteria is precisely quantum tight frame based on least-squares measurement,which can be divided into two situations,constrained least-squares frame(CLSF) that has specified scale and unconstrained least-squares frame(ULSF) whose scale is chosen. to minimize the least-squares error.Quantum tight frame based on optimal unambiguous discrimination is also given using quantum generalized measurement method.Finally,an example is given to compare these optimal construction methods of tight frame and results show that ULSF is the best.
Tight frames and rank-one quantum measurements are shown to be intimately related. In fact, the family of normalized tight frames for the space in which a quantum-mechanical system lies is precisely the family of rank-one generalized quantum measurements on that space. Using this relationship, frame-theoretical analogs of various quantum-mechanical concepts and results are developed. The analog of a least-squares quantum measurement is a tight frame that is closest in a least-squares sense to a given set of vectors. The least-squares tight frame is found for both the case in which the scaling of the frame is specified (constrained least-squares frame (CLSF)) and the case in which the scaling is chosen to minimize the least-squares error (unconstrained least-squares frame (ULSF)). The well-known canonical frame is shown to be proportional to the ULSF and to coincide with the CLSF with a certain scaling.
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 study an optimal quantum measurement for generalized quantum state discrimination problems, which include the problem of finding an optimal measurement maximizing the average success probability with and without a fixed rate of inconclusive results and the problem of finding an optimal measurement in the Neyman-Pearson strategy. We show that for any generalized quantum state discrimination problem, there is a corresponding minimum error discrimination problem sharing the same optimal measurement. We propose an approach in which the optimal measurement is obtained by solving the corresponding minimum error discrimination problem, and clarify the relationship between optimal solutions to the original and the corresponding problems, with which one can obtain an optimal solution to the original problem in some cases. Moreover, as an example of application of our approach, we present an algorithm for numerically obtaining optimal solutions to generalized quantum state discrimination problems.
To construct a tight frame for a given vector,joint quantum measurement of teleportation was investigated,and a construction method of tight frame was proposed.The generalized construction method of tight frame is based on joint quantum measurement.Firstly,arrange the given vector into a coefficient matrix,then obtain its left unitary,right unitary and singular value matrix by singular value decomposition. Secondly,find a base including a column whose coefficient matrix is just the non-zero part of the singular value matrix.At the end,arrage the rest colomns of the base into non-zero part of the singular value matrix, then get the rest tight frame columns by singular value reconstruction.
We derive the optimal measurement for quantum-state discrimination without a priori probabilities, i.e., in a minimax strategy instead of the usually considered Bayesian one. We consider both minimal-error and unambiguous discrimination problems, and provide the relation between the optimal measurements according to the two schemes. We show that there are instances in which the minimum risk cannot be achieved by an orthogonal measurement, and this is a common feature of the minimax estimation strategy.
We consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint.