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IEEE International Symposium on Information Theory (ISIT)

An algebraic approach to quantum information theory with applications in quantum cryptography

Pim Tuyls

Published Jun 25, 2003• 1 citation
Publisher

Abstract

We present an algebraic approach to quantum information theory and its application to quantum cryptography. Therefore, we model the measurement process in terms of an algebraic object: a partition of unity. The POVM associated to such a partition describes the measurement results and the completely positive map associated to it the disturbance of the state. This approach leads to a compact description of quantum cryptography in which general measurements, noisy channels as well as measurement apparatus of the receiver that are out of sync with those of the sender can be described.

DOI
10.1109/isit.2002.1023374

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The projection postulate and spectral representation of observables remain central in quantum theory but describe only an idealized measurement limit. Realistic measurements may be inefficient, unsharp, weak, indirect, adaptive, destructive, or informationally incomplete. Positive-operator-valued measures (POVMs) provide the probability rule for processes, whereas quantum instruments specify state transformation. This review traces the hierarchy from von Neumann projective measurement to generalized measurement, emphasizing distinctions among effects, instruments, channels, and implementations. Naimark dilation realizes arbitrary POVMs through unitary coupling to an ancillary meter and projective readout. Stinespring dilation represents completely positive trace-preserving dynamics by unitary evolution on an enlarged system followed by partial trace. A qubit–ancilla interaction yields a partial-measurement instrument, its POVM effects, and its information–disturbance trade-off. It compares projective, generalized, and ancilla-mediated schemes; examines compatibility, instrument incompatibility, and disturbance; and surveys applications to state discrimination, tomography, control, and error correction. Recent developments in dynamic-circuit measurements and noisy-POVM compatibility are integrated with questions concerning measurement back-action and outcome definiteness. The review stresses that a POVM does not uniquely determine state update, dilation theorems establish realizability rather than unique mechanisms, and operational measurement theory does not settle definite outcomes, arguing quantum instruments provide a complete description of measurement.