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arXiv:quant-ph/0308111 · Physical Review A

Probabilistic quantum multimeters

Jaromı́r Fiurášek, Miloslav Dušek

Published Mar 8, 2004• 26 citations
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Abstract

We propose quantum devices that can realize probabilistically different projective measurements on a qubit. The desired measurement basis is selected by the quantum state of a program register. First we analyze the phase-covariant multimeters for a large class of program states and then the universal multimeters for a special choice of program. In both cases we start with deterministic but error-prone devices and then proceed to devices that never make a mistake but from time to time give an inconclusive result. These multimeters are optimized (for a given type of program) with respect to the minimum probability of an inconclusive result. This concept is further generalized to multimeters that minimize the error rate for a given probability of an inconclusive result (or vice versa). Finally, we propose a generalization for qudits.

DOI
10.1103/physreva.69.032302

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We propose quantum devices that can realize probabilistically different projective measurements on a qubit. The desired measurement basis is selected by the quantum state of a program register. First we analyze the phase-covariant multimeters for a large class of program states and then the universal multimeters for a special choice of program. In both cases we start with deterministic but error-prone devices and then proceed to devices that never make a mistake but from time to time give an inconclusive result. These multimeters are optimized (for a given type of program) with respect to the minimum probability of an inconclusive result. This concept is further generalized to multimeters that minimize the error rate for a given probability of an inconclusive result (or vice versa). Finally, we propose a generalization for qudits.
We propose a quantum device that can approximate any projective measurement on a qubit-a quantum "multimeter." The desired measurement basis is selected by the quantum state of a "program register." Two different kinds of programs are considered and in both cases the device is optimized with respect to maximal average fidelity (assuming uniform distribution of measurement bases). Quantum multimeters exhibiting the covariance property are introduced and an optimal covariant multimeter with a single-qubit program register is found. Possible experimental realization of the simplest proposed device is presented.
We extend the concept of probabilistic unambiguous discrimination of quantum states to quantum state estimation. We consider a scenario where the measurement device can output either an estimate of the unknown input state or an inconclusive result. We present a general method how to evaluate the maximum fidelity achievable by the probabilistic estimation strategy. We illustrate our method on two explicit examples: estimation of a qudit from a pair of conjugate qudits and phase covariant estimation of a qubit from N copies. We show that by allowing for inconclusive results it is possible to reach estimation fidelity higher than that achievable by the best deterministic estimation strategy.
We describe a class of programmable devices that can discriminate between two quantum states. We consider two cases. In the first, both states are unknown. One copy of each of the unknown states is provided as an input, or program, for the two program registers, and the data state, which is guaranteed to be prepared in one of the program states, is fed into the data register of the device. This device will then tell us, in an optimal way, which of the templates stored in the program registers the data state matches. In the second case, we know one of the states while the other is unknown. One copy of the unknown state is fed into the single program register, and the data state which is guaranteed to be prepared in either the program state or the known state, is fed into the data register. The device will then tell us, again optimally, whether the data state matches the template or is the known state. We determine two types of optimal devices. The first performs discrimination with minimum error, and the second performs optimum unambiguous discrimination. In all cases we first treat the simpler problem of only one copy of the data state and then generalize the treatment to nnn copies. In comparison to other works we find that providing n>1n>1n>1 copies of the data state yields higher success probabilities than providing n>1n>1n>1 copies of the program states.
We consider a group covariant quantum state set with a group in which each element corresponds to a unitary or antiunitary operator. This type of quantum state set can express a broad class of quantum state sets, including geometrically uniform state sets and self-symmetric state sets. We derive that for any quantum measurement for a group-covariant state set (both with and without a certain fraction of inconclusive results), a group covariant quantum measurement exists with respect to the same group with the same performance, that is, the same probability of correct detection and the same rate of inconclusive results. We then show that a group covariant optimal measurement exists for any group-covariant state set. In some cases, we derive an optimal measurement for a group-covariant state set.
Abstract We analyze how to improve performance of probabilistic programmable quantum processors. We show how the probability of success of the probabilistic processor can be enhanced by using the processor in loops. In addition, we show that an arbitrary SU(2) transformations of qubits can be encoded in program state of a universal programmable probabilistic quantum processor. The probability of success of this processor can be enhanced by a systematic correction of errors via conditional loops. Finally, we show that all our results can be generalized also for qudits.