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arXiv:quant-ph/0209007 · Physical Review Letters

Quantum Filtering and Discrimination between Sets of Boolean Functions

János A. Bergou, Ulrike Herzog, Mark S. Hillery

Published Jun 25, 2003• 75 citations
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Abstract

In quantum state filtering one wants to determine whether an unknown quantum state, which is chosen from a known set of states, [|psi(1)>, em leader,|psi(N)>], is either a specific state, say |psi(1)>, or one of the remaining states, [|psi(2)>, em leader,|psi(N)>]. 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.

DOI
10.1103/physrevlett.90.257901

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In quantum state filtering one wants to determine whether an unknown quantum state, which is chosen from a known set of states, [|psi(1)>, em leader,|psi(N)>], is either a specific state, say |psi(1)>, or one of the remaining states, [|psi(2)>, em leader,|psi(N)>]. 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.
Quantum-state filtering is a variant of the unambiguous state discrimination problem: the states are grouped in sets and we want to determine to which particular set a given input state belongs. The simplest case, when the NNN given states are divided into two subsets and the first set consists of one state only while the second consists of all of the remaining states, is termed quantum-state filtering. We derived previously the optimal strategy for the case of NNN nonorthogonal states, ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$, for distinguishing $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. In a previous paper [Phys. Rev. Lett. 90, 257901 (2003)], we sketched an appplication of the results to probabilistic quantum algorithms. Here we fill the gaps and give the complete derivation of the probabilstic quantum algorithm that can optimally distinguish between two classes of Boolean functions, that of the balanced functions and that of the biased functions. The algorithm is probabilistic, it fails sometimes but when it does it lets us know that it did. Our approach can be considered as a generalization of the Deutsch-Jozsa algorithm that was developed for the discrimination of balanced and constant Boolean functions.
Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the NNN given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.
Abstract Unambiguous discrimination among nonorthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a new variant of that problem. Instead of discriminating among all of the N different states, we now ask for less. We want to unambiguously assign the state to one of two complementary subsets of the set of N given non‐orthogonal quantum states, each occurring with given a priori probabilities. We refer to the special case when one subset contains only one state and the other contains the remaining N‐1 states as unambiguous quantum state filtering. We present an optimal analytical solution for the special case of N=3, and discuss the optimal strategy to unambiguously distinguish |ψ1〉 from the set {|ψ2〉,|ψ3〉}. For unambiguous filtering the subsets need not be linearly independent. We briefly discuss how to construct generalized interferometers (multiports) which provide a fully linear optical implementation of the optimal strategy.
We investigate the unambiguous comparison of quantum states in a scenario that is more general than the one that was originally suggested by Barnett et al. First, we find the optimal solution for the comparison of two states taken from a set of two pure states with arbitrary a priori probabilities. We show that the optimal coherent measurement is always superior to the optimal incoherent measurement. Second, we develop a strategy for the comparison of two states from a set of NNN pure states, and find an optimal solution for some parameter range when N=3N=3N=3. In both cases we use the reduction method for the corresponding problem of mixed-state discrimination, as introduced by Raynal et al., which reduces the problem to the discrimination of two pure states only for N=2N=2N=2. Finally, we provide a necessary and sufficient condition for unambiguous comparison of mixed states to be possible.
We introduce a notion of set discrimination, which is an interesting extension of quantum state discrimination. A state is secretly chosen from a number of quantum states, which are partitioned into some disjoint sets. A set discrimination is required to identify which set the given state belongs to. Several essential problems are addressed in this paper, including the condition of perfect set discrimination, unambiguous set discrimination, and in the latter case, the efficiency of the discrimination. This generalizes some important results on quantum state discrimination in the literature. A combination of state and set discrimination and the efficiency are also studied.