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arXiv:math-ph/0303056 · Journal of Mathematical Physics

Radon–Nikodym derivatives of quantum operations

Maxim Raginsky

Published Oct 24, 2003• 37 citations
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Abstract

Given a completely positive (CP) map T, there is a theorem of the Radon–Nikodym type [W. B. Arveson, Acta Math. 123, 141 (1969); V. P. Belavkin and P. Staszewski, Rep. Math. Phys. 24, 49 (1986)] that completely characterizes all CP maps S such that T−S is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the Radon–Nikodym formalism to study the structure of order intervals of quantum operations, as well as a certain one-to-one correspondence between CP maps and positive operators, already fruitfully exploited in many quantum information-theoretic treatments. We also comment on how the Radon–Nikodym theorem can be used to derive norm estimates for differences of CP maps in general, and of quantum operations in particular.

DOI
10.1063/1.1615697

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Given a completely positive (CP) map T, there is a theorem of the Radon–Nikodym type [W. B. Arveson, Acta Math. 123, 141 (1969); V. P. Belavkin and P. Staszewski, Rep. Math. Phys. 24, 49 (1986)] that completely characterizes all CP maps S such that T−S is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the Radon–Nikodym formalism to study the structure of order intervals of quantum operations, as well as a certain one-to-one correspondence between CP maps and positive operators, already fruitfully exploited in many quantum information-theoretic treatments. We also comment on how the Radon–Nikodym theorem can be used to derive norm estimates for differences of CP maps in general, and of quantum operations in particular.
A well-known theorem of W. Arveson states that a completely positive (CP) map dominated (difference is CP) by a given CP map is described through a positive contraction in the commutant of the homomorphism of Stinespring representation. This contraction plays the role of Radon-Nikodym derivative of measure theory. We look at implications of this to one parameter semigroups of unital completely positive maps, known as quantum Markov semigroups. We focus our attention to quantum Markov semigroups of B(H), the algebra of all bounded operators on a Hilbert space. These semigroups can be dilated to semigroups of unital endomorphisms (E₀-semigroups) of B(K) for some other Hilbert space K. Here we see that dominated CP semigroups dilate to dominated semigroups of the dilation. Moreover at the level of Eo-semigroups the dominated CP semigroups are described through positive contractive local cocycles, which are really families of contractions of 'Radon-Nikodym derivatives' mentioned above. We analyze the structure for quantum semigroups with bounded generators.
A convenient representation for Radon–Nikodym derivatives of completely positive (c.p.) instruments on B(H) with respect to a scalar measure is suggested, similar to the Stinespring–Kraus representation for c.p. maps, but involving possibly nonclosable unbounded operators. The structure of covariant c.p. instruments is studied in detail. In particular, an exhaustive description is given to instruments covariant with respect to shifts or rotations, corresponding to “position” or “angle” measurements.
Quantum supermaps are higher-order maps transforming quantum operations into quantum operations. Here we extend the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps transforming quantum operations with input in a separable von Neumann algebra and output in the algebra of the bounded operators on a given separable Hilbert space. In this setting we prove two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. Finally, we consider the case of quantum superinstruments, namely measures with values in the set of quantum supermaps, and derive a dilation theorem for them that is analogue to Ozawa's theorem for quantum instruments. The three dilation theorems presented here show that all the supermaps defined in this paper can be implemented by connecting devices in quantum circuits.
The Radon-Nikodym formalism is used to study the structure of the set of positive maps from B(H)\mathcal{B}(\mathcal{H})B(H) into itself, where H\mathcal{H}H is a finite dimensional Hilbert space. In particular, this formalism was employed to formulate simple criteria which ensure that certain maps are non decomposable. In that way, a recipe for construction of non decomposable maps was obtained.
Quantum supermaps are higher-order maps transforming quantum operations into quantum operations and satisfying suitable requirements of normality and complete positivity. Here we present the extension of the theory of quantum supermaps, originally formulated in the finite dimensional setting, to the case of higher-order maps transforming quantum operations on generic von Neumann algebras. In this setting, we provide two dilation theorems for quantum supermaps that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. A structure theorem for probability measures with values in the set of quantum supermaps is also illustrated. Finally, some applications are given, and in particular it is shown that all the supermaps defined in this paper can be implemented by connecting quantum devices in quantum circuits.
Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C∗-algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34–54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon–Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon–Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi’s matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained.