Clean positive operator valued measures

Abstract

In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVM's into POVM's, generally irreversibly, thus loosing some of the information retrieved from the measurement. This poses the problem of which POVM's are "undisturbed", namely they are not irreversibly connected to another POVM. We will call such POVM's clean. In a sense, the clean POVM's would be "perfect", since they would not have any additional "extrinsical" noise. Quite unexpectedly, it turns out that such cleanness property is largely unrelated to the convex structure of POVM's, and there are clean POVM's that are not extremal and vice-versa. In this paper we solve the cleannes classification problem for number n of outcomes n ≤ d (d dimension of the Hilbert space), and we provide a a set of either necessary or sufficient conditions for n > d, along with an iff condition for the case of informationally complete POVM's for n = d 2 .

Dividing Quantum Channels

“…Moreover, when considering quantum channels with a classical output in the sense of the positive operator valued measure (POVM) formalism, then a similar train of thought leads to the notion of clean POVMs which cannot be expressed as a non-trivial concatenation of a quantum channel with a different POVM [19].…”

“…Part 1 of this corollary is a simple consequence of Wigner's theorem and was proven for completely positive maps for instance in [19]. 4 One might wonder whether completely positive maps can have negative determinants.…”

“…As any unital positive map is spectrumwidth decreasing [19] we have for the range of eigenvalues [λ min (P i+1 …”

Equivalence between divisibility and monotonic decrease of information in classical and quantum stochastic processes

“…This fact is reflected in the above definition by the requirement that the guessing probability cannot increase for any ensemble of initial states, i.e., for any finite set X = {x}, any probability distribution on X , and any collection of states ρ x S ∈ D(H S ). This paper builds upon a series of results extending the so-called Blackwell-Sherman-Stein theorem [3,43,44] of classical statistics to quantum statistical decision theory [7,8,10,11,15,40]. In particular, a crucial role in this paper is played by the following result:…”

Optimal Data Processing for Quantum Measurements

“…This more general class of quantum measurements includes also the description of optimal joint measurements of non-commuting observables [2,3], along with the measurements of parameters with no corresponding observable such as the phase of a harmonic oscillator [4], and many other practical measurements such as optimized discrimination of states for quantum communications [5], and, most interesting, the so-called informationally complete measurements [6], i. e. measurements that allow to determine the density matrix of the state or any other desired ensemble average, as for the so-called Quantum Tomography [7]. Moreover POVM's also allow to provide a full description of the measurement apparatus, including noisy channels before detection [8]. The POVM's are not just a theoretical tool, since there is a general quantum calibration procedure in order to determine experimentally the POVM of a measurement device by using a reliable standard [9].…”