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Journal of Mathematical Physics

Radon–Nikodym derivatives of quantum instruments

Alexander Semenovich Holevo

Published Mar 1, 1998• 39 citations
Publisher

Abstract

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.

DOI
10.1063/1.532385

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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.
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.
We study completely positive (CP) [Formula: see text]-sesquilinear-form-valued maps on a unital [Formula: see text]-algebra [Formula: see text], where the sesquilinear forms operate on a module over a [Formula: see text]-algebra [Formula: see text]. We also study the cases when either one or both of the algebras are von Neumann algebras. Moreover, we assume that the CP maps are covariant with respect to actions of a symmetry group. This allows us to view these maps as generalizations of covariant quantum instruments. We determine minimal covariant dilations (KSGNS constructions) for covariant CP maps to find necessary and sufficient conditions for a CP map to be extreme in convex subsets of normalized covariant CP maps. As a special case, we study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. Finally, we discuss the case of instruments that are covariant with respect to a square-integrable representation.
Understanding Quantum Instruments Through the Analysis of C∗C^*C∗-Convexity and Their Marginals
Quantum instruments are mathematical devices introduced to describe the conditional state change during a quantum process. They are completely positive map valued measures on measurable spaces. We may also view them as non-commutative analogues of joint probability measures. We analyze the C∗C^*C∗-convexity structure of spaces of quantum instruments. A complete description of the C∗C^*C∗-extreme instruments in finite dimensions has been established. Further, the implications of C∗C^*C∗-extremity between quantum instruments and their marginals has been explored.
We develop a theory of integration with respect to quantum instruments through two complementary approaches: a vector measure formulation based on Bartle's integration theory and a tensor product construction. As a principal application, we establish a CP map-valued Lyapunov theorem by characterizing the convexity of the range of non-atomic completely positive instruments via the associated integration map. This extends the work of Plosker and Ramsey~\cite{Plosker_Ramsey} from POVMs to completely positive instruments and gives a general characterization of the phenomenon exhibited in their setting. We also establish a correspondence between completely positive instruments and completely positive maps, prove a Krein--Milman type theorem for the C∗C^*C∗-convex set of unital completely positive instruments, and show that the tensor product construction connects the theory of CP instruments with the integration theory for POVMs due to Farenick \emph{et al}(\cite{douglus_plosker_ramsey_povmintegration_1}).