Positive-Operator Valued Measure (POVM) Quantization

2014

We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on normalized positive operator-valued measure. The latter are built from families of density operators labelled by points of the measure space. We specially focus on various probabilistic aspects of these constructions. Simple or more elaborate examples illustrate the procedure: circle, 2-sphere, plane, half-plane. Links with POVM quantum measurement and quantum statistical inference are sketched.

Physical Review A, 2005

We propose a scheme that can realize a class of positive-operator-valued measures (POVMs) by performing a sequence of projective measurements on the original system, in the sense that for an arbitrary input state the probability distribution of the measurement outcomes is faithfully reproduced. A necessary and sufficient condition for a POVM to be realizable in this way is also derived. In contrast to the canonical approach provided by Neumark's theorem, our method has the advantage of requiring no auxiliary system. Moreover, an arbitrary POVM can be realized by utilizing our protocol on an extended space which is formed by adding only a single extra dimension.

Symmetric informationally complete positive operator valued measures (SIC-POVMs) are studied within the framework of the probability representation of quantum mechanics. A SIC-POVM is shown to be a special case of the probability representation. The problem of SIC-POVM existence is formulated in terms of symbols of operators associated with a star-product quantization scheme. We show that SIC-POVMs (if they do exist) must obey general rules of the star product, and, starting from this fact, we derive new relations on SIC-projectors. The case of qubits is considered in detail, in particular , the relation between the SIC probability representation and other probability representations is established, the connection with mutually unbiased bases is discussed, and comments to the Lie algebraic structure of SIC-POVMs are presented.

2015

2005

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 .

2006

We propose a scheme that can realize a class of positive-operator-valued measures (POVMs) by performing a sequence of projective measurements on the original system, in the sense that for an arbitrary input state the probability distribution of the measurement outcomes is faithfully reproduced. A necessary and sufficient condition for a POVM to be realizable in this way is also derived. In contrast to the canonical approach provided by Neumark's theorem, our method has the advantage of requiring no auxiliary system. Moreover, an arbitrary POVM can be realized by utilizing our protocol on an extended space which is formed by adding only a single extra dimension.

Journal of Physics: Conference Series, 2015

We present quantizations of functions (or distributions) on a measure space viewed as a classical set. They are based on positive operator-valued measures. To illustrate the large range of potentialities of the method, we develop examples where the classical sets are the group SU(2), the (projective) Weyl-Heisenberg group, and the affine group. Applications to quantum cosmology and to the quantization of constraints are outlined.

Physical review, 2023

In Ref. [Phys. Rev. A 100, 062317 (2019)], the authors reported an algorithm to implement, in a circuit-based quantum computer, a general quantum measurement (GQM) of a two-level quantum system, a qubit. Even though their algorithm seems right, its application involves the solution of an intricate non-linear system of equations in order to obtain the angles determining the quantum circuit to be implemented for the simulation. In this article, we identify and discuss a simple way to circumvent this issue and implement GQMs on any d-level quantum system through quantum state preparation algorithms. Using some examples for one qubit, one qutrit and two qubits, we illustrate the easy of application of our protocol. Besides, we show how one can utilize our protocol for simulating quantum instruments, for which we also give an example. All our examples are demonstrated using IBM's quantum processors.

Physical Review A, 2010

In the four-dimensional Hilbert space, there exist 16 Heisenberg-Weyl (HW) covariant symmetric informationally complete positive operator valued measures (SIC POVMs) consisting of 256 fiducial states on a single orbit of the Clifford group. We explore the structure of these SIC POVMs by studying the symmetry transformations within a given SIC POVM and among different SIC POVMs. Furthermore, we find 16 additional SIC POVMs by a regrouping of the 256 fiducial states, and show that they are unitarily equivalent to the original 16 SIC POVMs by establishing an explicit unitary transformation. We then reveal the additional structure of these SIC POVMs when the fourdimensional Hilbert space is taken as the tensor product of two qubit Hilbert spaces. In particular, when either the standard product basis or the Bell basis are chosen as the defining basis of the HW group, in eight of the 16 HW covariant SIC POVMs, all fiducial states have the same concurrence of 2/5. These SIC POVMs are particularly appealing for an experimental implementation, since all fiducial states can be connected to each other with just local unitary transformations. In addition, we introduce a concise representation of the fiducial states with the aid of a suitable tabular arrangement of their parameters.

International Journal of Theoretical Physics, 2017

The histories-based framework of Quantum Measure Theory assigns a generalized probability or measure µ(E) to every (suitably regular) set E of histories. Even though µ(E) cannot in general be interpreted as the expectation value of a selfadjoint operator (or POVM), we describe an arrangement which makes it possible to determine µ(E) experimentally for any desired E. Taking, for simplicity, the system in question to be a particle passing through a series of Stern-Gerlach devices or beam-splitters, we show how to couple a set of ancillas to it, and then to perform on them a suitable unitary transformation followed by a final measurement, such that the probability of a final outcome of "yes" is related to µ(E) by a known factor of proportionality. Finally, we discuss in what sense a positive outcome of the final measurement should count as a minimally disturbing verification that the microscopic event E actually happened.