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arXiv:quant-ph/0411135 · Physical Review A

Realization of positive-operator-valued measures using measurement-assisted programmable quantum processors

Mário Ziman, Vladimír Bužek

Published Aug 29, 2005• 27 citations
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Abstract

We study possible realizations of generalized quantum measurements on measurement-assisted programmable quantum processors. We focus our attention on the realization of von Neumann measurements and informationally complete positive-operator-valued measures. Nielsen and Chuang [Phys. Rev. Lett. 79, 321 (1997)] have shown that two unitary transformations implementable by the same programmable processor require mutually orthogonal states. We show that two different von Neumann measurements can be encoded into nonorthogonal program states. Nevertheless, given the dimension of a Hilbert space of the program register the number of implementable von Neumann measurements is still limited. As an example of a programmable processor we use the so-called quantum-information distributor.

DOI
10.1103/physreva.72.022343

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We study possible realizations of generalized quantum measurements on measurement-assisted programmable quantum processors. We focus our attention on the realization of von Neumann measurements and informationally complete positive-operator-valued measures. Nielsen and Chuang [Phys. Rev. Lett. 79, 321 (1997)] have shown that two unitary transformations implementable by the same programmable processor require mutually orthogonal states. We show that two different von Neumann measurements can be encoded into nonorthogonal program states. Nevertheless, given the dimension of a Hilbert space of the program register the number of implementable von Neumann measurements is still limited. As an example of a programmable processor we use the so-called quantum-information distributor.
We analyze probabilistic realizations of programmable quantum processors that allow us to realize unitary operations on qubits as well as on qudits. Programmable processors are composed of two inputs — the data register and the program register. In the input state of the program register information about the operation that is supposed to be performed on the data is encoded. At the output of the probabilistic processor a measurement over the program register is performed. An intrinsic property of probabilistic processors is that they sometimes fail, but we know when this happens. We present a complete analysis of two processors: (1) The so-called [Formula: see text] processor that is based on a simple controlled-NOT gate. (2) The so-called [Formula: see text] processor that utilizes the quantum-information-distributor circuit.
Positive operator valued measurements (POVMs) are the most general measurements that can be performed on quantum systems.A POVM is described by set of positive operators instead of orthogonal projections, and it can always be traced back to joint von Neumann measurements on the system and on an ancilla attached to it (Busch et. al, 1991;Peres, 1993).* Note on nomenclature: subscript letters denote the appropriate state spaces for vectors and density operators throughout the document.The same symbol with different subscripts intends to mean that the same quantum information is represented on different systems.
Positive-operator-value measures (POVM's) are the most general class of quantum measurements. We propose a setup in which all possible POVM's of a single-photon polarization state (corresponding to all possible sets of two-dimensional Kraus operators) can be implemented easily using linear optics elements. This method makes it possible to experimentally realize any projective orthogonal, projective nonorthogonal or nonprojective sets of any number of POVM operators. Furthermore, our implementation only requires vacuum ancillas and is deterministic rather than probabilistic. Thus it realizes every POVM with the correct set of output states. We give the settings required to implement two different well-known nonorthogonal projective POVM's.
Abstract In this chapter we first review projective or von Neumann measurements from the point of view of the projection postulate and the Lüders rule, before turning to more general measurements modelled using positive operator-valued measures (POVMs). After examining their mathematical description, we discuss symmetric informationally complete (SIC) POVMs and Zauner’s conjecture, as well as the representation of POVMs as projective measurements on a larger Hilbert space, a result called Naimark dilation. We then a model for non-ideal projective measurements and analyse the properties of such measurements in relation to the properties unbiasedness, faithfulness, and non-invasiveness of ideal projective measurements. Finally, we consider the problem of distinguishing non-orthogonal quantum states and examine how the use of POVMs can be beneficial for this task.
A quantum processor is a device with a data register and a program register. The input to the program register determines the operation, which is a completely positive linear map, that will be performed on the state in the data register. We develop a mathematical description for these devices, and apply it to several different examples of processors. The problem of finding a processor that will be able to implement a given set of mappings is also examined, and it is shown that, while it is possible to design a finite processor to realize the phase-damping channel, it is not possible to do so for the amplitude-damping channel.