Quantum Filtering and Discrimination between Sets of Boolean Functions

Abstract

In quantum state filtering one wants to determine whether an unknown quantum state, which is chosen from a known set of states, {|ψ1 , . . . |ψN }, is either a specific state, say |ψ1 , or one of the remaining states, {|ψ2 , . . . |ψN }. We present the optimal solution to this problem, in terms of generalized measurements, for the case that the filtering is required to be unambiguous. As an application, we propose an efficient, probabilistic quantum algorithm for distinguishing between sets of Boolean functions, which is a generalization of the Deutsch-Jozsa algorithm.PACS numbers: 03. 03.65.Ta, Optimal discrimination among quantum states plays a central role in quantum information theory. Interest in this problem was prompted by the suggestion to use nonorthogonal quantum states for communication in certain secure quantum cryptographic protocols, most notably in the one based on the two-state procedure as developed by Bennett [1]. The reason why until recently the area has shown relatively slow progress within the rapidly evolving field of quantum information is that it poses quite formidable mathematical challenges. Except for a handful of very special cases, no general exact solution has been available involving more than two arbitrary states. In this paper we present an exact solution to an optimum measurement problem involving an arbitrary number of quantum states, with no restriction on the states. The resulting method has the potential for widespread applications in quantum information processing. In particular, it lends itself quite naturally to a quantum generalization of probabilistic classical algorithms. Whenever it is possible to find a one-to-one mapping of classical alternatives onto quantum states, our method can discriminate among these quantum alternatives in a single step with optimum success probability.We illustrate the strength of the method on the example of a probabilistic quantum algorithm to discriminate between sets of Boolean functions. A Boolean function on n bits is one that returns either 0 or 1 as output for every possible value of the input x, where 0 ≤ x ≤ 2 n − 1. The function is uniform (or constant) if it returns the same output on all of its arguments, i.e. either all 0's or all 1's; it is balanced (or even) if it returns 0's on half of its arguments and 1's on the other half; and it is biased if it returns 0's on m 0 of its arguments and 1's on the remaining m 1 = 2 n − m 0 arguments (m 0 = m 1 = 0 or 2 n − 1). Classically, if one is given an unknown function and told that it is either balanced or uniform, one needs 2 (n−1) + 1 measurements to decide which. Deutsch and Jozsa [2] developed a quantum algorithm that can accomplish this task in one step. To discriminate a biased Boolean function from an unknown balanced one, 2 (n−1) + m 1 + 1 measurements are needed classically, where, without loss of generality, we have assumed that m 1 < m 0 . Here we propose a probabilistic quantum algorithm that can unambiguously discriminate a known biased Boolean function from a given se...

Minimum-error discrimination between a pure and a mixed two-qubit state

“…|u d }. The general solution for optimum unambiguous quantum state filtering has been provided in [5]. Assuming equal a-priori-probabilities η = 1/(d + 1) for all states, this solution can be directly applied to our case, yielding the failure probability Q F = 2 ψ /(d + 1).…”

“…In this case the resulting minimum error probability, P E = 1/(d + 1), is achievable by guessing the system always to be in the stateρ 2 , without performing any measurement at all. It is interesting to compare the minimum probability of errors, P E , with the smallest possible failure probability, Q F , that can be obtained in a strategy optimized for unambiguously discriminating between the quantum states given in (5). The solution of the latter problem coincides with the solution to the problem of optimum unambiguous quantum state filtering, where the state of the quantum system is known to be either |ψ , or any state out of the set of pure states {|u 1 , .…”

“…If the first set contains only a single state, the discrimination problem is referred to as quantum state filtering [3 -5]. For optimum unambiguous discrimination, general analytical solutions have been derived to this problem [4,5]. Another recent development consists in studying state discrimination for multipartite systems.…”

Distinguishing mixed quantum states: Minimum-error discrimination versus optimum unambiguous discrimination

“…Analytical solutions for the minimum failure probability, Q F , have been found for distinguishing between two [2,3,4,5] and three [6,7,8] arbitrary pure states, and between any number of pure states that are symmetric and equiprobable [9]. On the other hand, the investigation of unambiguous discrimination involving mixed states, or sets of pure states, respectively, started only recently [10,11,12,13,14,15,16]. So far exact analytical results are known only for simple cases [11,12,13,14].…”

“…On the other hand, the investigation of unambiguous discrimination involving mixed states, or sets of pure states, respectively, started only recently [10,11,12,13,14,15,16]. So far exact analytical results are known only for simple cases [11,12,13,14]. In addition, for unambiguously discriminating between two arbitrary mixed states, general upper and lower bounds have been derived for the minimum failure probability [14].…”