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Journal of Modern Optics

Positive-operator and projection valued measurements in quantum key distribution

Howard E. Brandt

Published Nov 10, 2007• 1 citation
Publisher

Abstract

Following a brief review of positive-operator and projection valued measures in quantum mechanics, some applications to quantum key distribution are described. The positive operator valued measure (POVM) can be usefully implemented in a quantum key receiver used in the B92 two-state protocol of quantum key distribution. In this case the measurement is conclusive, at least some of the time. The projection valued (PV) measure is useful in the measurement of a quantum cryptographic entangling probe for tapping quantum key in the BB84 four-state protocol. In the case addressed here, the measurement determines correlations with the measurements made by the legitimate receiver and therewith the maximum information gain by the probe.

DOI
10.1080/09500340701639557

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Following a brief review of positive-operator and projection valued measures in quantum mechanics, some applications to quantum key distribution are described. The positive operator valued measure (POVM) can be usefully implemented in a quantum key receiver used in the B92 two-state protocol of quantum key distribution. In this case the measurement is conclusive, at least some of the time. The projection valued (PV) measure is useful in the measurement of a quantum cryptographic entangling probe for tapping quantum key in the BB84 four-state protocol. In the case addressed here, the measurement determines correlations with the measurements made by the legitimate receiver and therewith the maximum information gain by the probe.
I first review positive-operator and projection valued measures in quantum mechanics, and then address a few applications in quantum key distribution (QKD). The positive operator valued measure (POVM) is useful in the design of a quantum key receiver. The projection valued (PV) measure is useful in the design of QKD eavesdropping devices. In this case the measurement determines correlations with the measurements made by the lefitimate receiver and therewith the maximum information gain by the probe. It is essential to know the latter for privacy amplification.
In the quantum theory of measurement, the positive operator-valued measure (POVM) is an important concept, and its implementation can be useful. A POVM consists of a set of non-negative quantum-mechanical Hermitian operators that add up to the identity. The probability that a quantum system is in a particular state is given by the expectation value of the POVM operator corresponding to that state. Following a brief review of the mathematics and mention of the history of POVMs in quantum theory, a particular implementation of a POVM for use in the measurement of nonorthogonal photon polarization states is reviewed. The implementation consists simply of a Wollaston prism, a mirror, two beam splitters, a polarization rotator and three phototubes arranged in an interferometric configuration, and it is shown analytically that the device faithfully represents the POVM. Based on Neumark's extension theorem, the two-dimensional Hilbert space of the POVM implementation can be embedded in the three-dimensional Hilbert space of an ordinary projective-valued measure. Also, analytical expressions are given for the maximum Renyi information loss from the device to a disturbing probe, and for the error and inconclusive rates induced by the probe. Various aspects of the problem of probe optimization are elaborated.
The positive operator valued measure (POVM), also known as the probability operator valued measure, is useful in quantum information processing. The POVM consists of a set of non-negative quantum-mechanical Hermitian operators that add up to the identity. The probability that a quantum system is in a particular state is given by the expectation value of the POVM operator corresponding to that state. Following a brief review of the mathematics and history of POVMs in quantum theory, and an expository discussion of the quantum mechanics of photonic qubits, a particular implementation of a POVM for use in the measurement of photonic qubits is reviewed.
The positive operator valued measure (POVM), also known as the probability operator valued measure, is useful in quantum information processing. The POVM consists of a set of non-negative quantum-mechanical Hermitian operators that add up to the identity. The probability that a quantum system is in a particular state is given by the expectation value of the POVM operator corresponding to that state. Following a brief review of the mathematics and history of POVMs in quantum theory, and a pedagogical discussion of the quantum mechanics of photonic qubits, a particular implementation of a POVM for use in the measurement of photonic qubits is reviewed.
In the heart of every physical theory lies measurement theory. While classical measurements are passive probes, QMs can be viewed as active physical processes that shape the available information. Depending on the measurement type, different aspects of the information about a system are retrieved. All possible measurements in quantum physics are described by mathematical objects called Positive Operator-Valued Measures (POVMs). The most familiar standard measurements are called Projective-Valued Measures (PVMs) and make up a subset of POVMs. The complementary subset of POVMs is non-projective. However, some non-projective POVMs can be expressed as a probabilistic mixture of PVMs. Such POVMs are called projectively simulable, while non-simulable POVMs are genuinely non-projective. These can be made projectively simulable by adding noise to the measurements. If a genuinely non-projective POVM requires a large amount of noise before it becomes projectively simulable, it is more non-projective than a POVM that requires less noise. Hence, the degree of non-projectiveness is determined by the required noise strength. The act of measuring a quantum system changes the quantum state. While quantum measurements only take into account the readable classical output value, quantum instruments account for both the classical outcome and the post-measurement state. Hence, instruments are associated with measurements, and in an analogous way, projective and projectively simulable instruments can be defined. For many purposes, non-projective POVMs are more useful than PVMs, and the more non-projective a POVM is, the greater the advantage of using them over PVMs. Therefore, the aim of the thesis is to investigate the genuinely non-projective POVMs. More precisely, the degree of non-projectiveness in qubit POVMs is studied. This is done by convex optimization, specifically by solving semidefinite programs. Furthermore, the thesis addresses the open problem of finding the most non-projective qubit quantum instrument by a witness-based see-saw algorithm.
It is demonstrated that the eavesdropping optimization obtained recently by Slutsky et al. [Phys. Rev. A 57, 2383 (1998)] for Bennett's two-state protocol of key distribution in quantum cryptography holds, not only for the case in which an ordinary von Neumann projective measure is implemented by the legitimate receiver, but also for the case in which a positive operator-valued measure is implemented, as in Brandt et al. [Phys. Rev. A 56, 4456 (1997)]. In both cases, identical expressions hold for both the Renyi information gained by the eavesdropper and for the error rate induced by the eavesdropper in terms of the parameters characterizing the key distribution system and the eavesdropper's probe.