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arXiv:quant-ph/0105016 · Physical Review A

Unambiguous discrimination between linearly dependent states with multiple copies

Anthony Chefles

Published Nov 9, 2001• 66 citations
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Abstract

A set of quantum states can be unambiguously discriminated if and only if they are linearly independent. However, for a linearly dependent set, if C copies of the state are available, then the resulting C particle states may form a linearly independent set, and be amenable to unambiguous discrimination. We obtain one necessary and one sufficient condition for the possibility of unambiguous discrimination among N states given that C copies are available and that the single copies span a D-dimensional space. These conditions are found to be identical for qubits. We then examine in detail the linearly dependent trine ensemble. The set of $C>1$ copies of each state is a set of linearly independent lifted trine states. The maximum unambiguous discrimination probability is evaluated for all $C>1$ with equal a priori probabilities.

DOI
10.1103/physreva.64.062305

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A set of quantum states can be unambiguously discriminated if and only if they are linearly independent. However, for a linearly dependent set, if C copies of the state are available, then the resulting C particle states may form a linearly independent set, and be amenable to unambiguous discrimination. We obtain one necessary and one sufficient condition for the possibility of unambiguous discrimination among N states given that C copies are available and that the single copies span a D-dimensional space. These conditions are found to be identical for qubits. We then examine in detail the linearly dependent trine ensemble. The set of C>1C>1C>1 copies of each state is a set of linearly independent lifted trine states. The maximum unambiguous discrimination probability is evaluated for all C>1C>1C>1 with equal a priori probabilities.
We investigate the unambiguous discrimination of three linearly independent symmetric states. We derive the minimum inconclusive probability by using an ancillary system and introducing a unitary transformation which acts on the input states and produces the output measured on the ancilla. We generalize to the case of discriminating the symmetric states with multiple copies. Our results show that the minimum inconclusive probabilities can be expressed as a form, not only dependent on the modular and the phase factors but also dependent on the number of the copies.
In this paper, we consider the quantum unambiguous discrimination problem of a set of linearly independent pure states. With a constructive procedure, we establish a necessary and sufficient condition for feasible POVM measurements for unambiguous quantum state discrimination. We develop methods to calculate the optimal discrimination efficiencies for minimizing the inconclusive probability and the optimal discrimination efficiencies for the minimax criterion. Finally, we prove a necessary and sufficient condition to have a feasible POVM measurement which achieves the minimum inconclusive probability criterion and the minimax criterion simultaneously and show that geometrically uniform states with equal a priori probabilities meet this condition.
Given nnn linearly independent pure states and their prior probabilities, we study the optimum unambiguous state discrimination problem. We derive the conditions for the optimum measurement strategy to achieve the maximum average success probability and establish two sets of equations that must be satisfied by the optimum solution in different situations. We also provide the detailed steps to find the optimum measurement strategy. The method and results we obtain are given a geometrical illustration with a numerical example. Furthermore, using these equations, we derive a formula which shows a clear analytical relation between the optimum solution and the nnn states to be discriminated. We also solve a generalized equal-probability measurement problem analytically. Finally, as another application of our result, the unambiguous discrimination problem of three pure states is studied in detail and analytical solutions are obtained for some interesting cases.
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We consider the problem of determining the state of an unknown quantum sequence without error. The elements of the given sequence are drawn with equal probability from a known set of linearly independent pure quantum states with the property that their mutual inner products are all real and equal. This problem can be posed as an instance of unambiguous state discrimination where the states correspond to that of all possible sequences having the same length as the given one. We calculate the optimum probability by solving the optimality conditions of a semidefinite program. The optimum value is achievable by measuring individual members of the sequence, and no collective measurement is necessary.