Operator-valued Measures Research Articles

Separability criteria via some classes of measurements

Mutually unbiased bases (MUBs) and symmetric informationally complete (SIC) positive operator-valued measurements (POVMs) are two related topics in quantum information theory. They are generalized to mutually unbiased measurements (MUMs) and general symmetric informationally complete (GSIC) measurements, respectively, that are both not necessarily rank 1. We study the quantum separability problem by using these measurements and present separability criteria for bipartite systems with arbitrary dimensions and multipartite systems of multi-level subsystems. These criteria are proved to be more effective than previous criteria especially when the dimensions of the subsystems are different. Furthermore, full quantum state tomography is not needed when these criteria are implemented in experiment.

Most incompatible measurements for robust steering tests

We address the problem of characterizing the steerability of quantum states under restrictive measurement scenarios, i.e., the problem of determining whether a quantum state can demonstrate steering when subjected to $N$ measurements of $k$ outcomes. We consider the cases of either general positive operator-valued measures (POVMs) or specific kinds of measurements (e.g., projective or symmetric). We propose general methods to calculate lower and upper bounds for the white-noise robustness of a $d$-dimensional quantum state under different measurement scenarios that are also applicable to the study of the noise robustness of the incompatibility of sets of unknown qudit measurements. We show that some mutually unbiased bases, symmetric informationally complete measurements, and other symmetric choices of measurements are not optimal for steering isotropic states and provide candidates to the most incompatible sets of measurements in each case. Finally, we provide numerical evidence that nonprojective POVMs do not improve over projective ones for this task.

Unambiguous discrimination of nonorthogonal quantum states in cavity QED

We propose an oversimplified scheme to unambiguously discriminate nonorthogonal quantum field states inside high-Q cavities. Our scheme, which is based on positive operator-valued measures (POVM) technique, uses a single three-level atom interacting resonantly with a single mode of a cavity-field and selective atomic state detectors. While the single three-level atom takes the role of the ancilla, the single cavity mode field represents the system we want to obtain information. The efficiency of our proposal is analyzed considering the nowadays achievements in the context of cavity QED. We also analyze the effect of a thermal environment to discrimination of nonorthogonal states.

Two constructions of approximately symmetric informationally complete positive operator-valued measures

Symmetric informationally complete positive operator-valued measures (SIC-POVMs) have many applications in quantum information. However, it is not easy to construct SIC-POVMs and there are only a few known classes of them, and we do not even know whether there exists an infinite class of them, thus constructing approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) has its own meaning. In this paper, we use character sums over finite fields to present two constructions of ASIC-POVMs. We show that there are some classes of infinite families of ASIC-POVMs by using some special functions over finite fields.

An effective iterative method to build the Naimark extension of rank-n POVMs

We revisit the problem of finding the Naimark extension of a probability operator-valued measure (POVM), i.e. its implementation as a projective measurement in a larger Hilbert space. In particular, we suggest an iterative method to build the projective measurement from the sole requirements of orthogonality and positivity. Our method improves existing ones, as it may be employed also to extend POVMs containing elements with rank larger than one. It is also more effective in terms of computational steps.

Equidistribution, ergodicity and irreducibility associated with Gibbs measures

We generalize an equidistribution theorem à la Bader–Muchnik for operator-valued measures constructed from a family of boundary representations associated with Gibbs measures in the context of convex cocompact discrete group of isometries of a simply connected connected Riemannian manifold with pinched negative curvature. We combine a functional analytic tool, namely the property RD of hyperbolic groups, together with a dynamical tool: an equidistribution theorem of Paulin, Pollicott and Schapira inspired by a result of Roblin. In particular, we deduce irreducibility of these new classes of boundary representations.

General construction of reproducing kernels on a quaternionic Hilbert space

A general theory of reproducing kernels and reproducing kernel Hilbert spaces on a right quaternionic Hilbert space is presented. Positive operator-valued measures and their connection to a class of generalized quaternionic coherent states are examined. A Naimark type extension theorem associated with the positive operator-valued measures is proved in a right quaternionic Hilbert space. As illustrative examples, real, complex and quaternionic reproducing kernels and reproducing kernel Hilbert spaces arising from Hermite and Laguerre polynomials are presented. In particular, in the Laguerre case, the Naimark type extension theorem on the associated quaternionic Hilbert space is indicated.

Radon–Nikodym theorems for operator-valued measures and continuous generalized frames

In this article we determine that an operator-valued measure (OVM) for Banach spaces is actually a weak∗ measure, and then we show that an OVM can be represented as an operator-valued function if and only if it has σ-finite variation. By the means of direct integrals of Hilbert spaces, we introduce and investigate continuous generalized frames (continuous operator-valued frames, or simply CG frames) for general Hilbert spaces. It is shown that there exists an intrinsic connection between CG frames and positive OVMs. As a byproduct, we show that a Riesz-type CG frame does not exist unless the associated measure space is purely atomic. Also, a dilation theorem for dual pairs of CG frames is given.

Probabilistic Teleportation of an Arbitrary Two-Qubit State via Positive Operator-Valued Measurement with Multi Parties**Supported by the National Natural Science Foundation of China under Grant Nos. 60974037, 61134008, 11074307, and 61273202

We propose a novel scheme to probabilistically transmit an arbitrary unknown two-qubit quantum state via Positive Operator-Valued Measurement with the help of two partially entangled states. In this scheme, the teleportation with two senders and two receives can be realized when the information of non-maximally entangled states is only available for the senders. Furthermore, the concrete implementation processes of this proposal are presented, meanwhile the classical communication cost and the successful probability of our scheme are calculated.

Necessary condition for incompatibility of observables in general probabilistic theories

We quantify the intrinsic noise content of an observable in a general probabilistic theory and derive a noise content inequality for incompatible observables. We apply the derived inequality to standard quantum theory, the quantum theory of processes, and polytope state spaces. The noise content for positive operator-valued measures takes a particularly simple form and equals the sum of minimal eigenvalues of all the effects. We illustrate our findings with a number of examples including the introduced notion of reverse observables.

Direct calibration of click-counting detectors

We introduce and experimentally implement a method for the absolute detector calibration of photon-number-resolving time-bin multiplexing layouts based on the measured click statistics of superconduncting nanowire detectors. In particular, the quantum efficiencies, the dark count rates, and the positive operator-valued measures of these measurement schemes are directly obtained with high accuracy. The method is based on the moments of the click-counting statistics for coherent states with different coherent amplitudes. The strength of our analysis is that we can directly conclude -- on a quantitative basis -- that the detection strategy under study is well described by a linear response function for the light-matter interaction and that it is sensitive to the polarization of the incident light field. Moreover, our method is further extended to a two-mode detection scenario. Finally, we present possible applications for such well characterized detectors, such as sensing of atmospheric loss channels and phase sensitive measurements.

Wigner-Araki-Yanase theorem beyond conservation laws

The ability to measure every quantum observable is ensured by a fundamental result in quantum measurement theory. Nevertheless, additive conservation laws associated with physical symmetries, such as the angular momentum conservation, may lead to restrictions on the measurability of the observables. Such limitations are imposed by the theorem of Wigner, Araki, and Yanase (WAY). In this paper a formulation of the WAY theorem is presented rephrasing the measurability limitations in terms of quantum incompatibility. This broader mathematical basis enables us to both capture and generalize the WAY theorem by allowing us to drop the assumptions of additivity and even conservation of the involved quantities. Moreover, we extend the WAY theorem to the general level of positive operator-valued measures.

Measurement-induced Nonlocality for Gaussian States

We establish an analytic formula of measurement-induced nonlocality (MIN) for two-mode squeezed thermal states and mixed thermal states. Different from the quantum discord case, we show that there is no Gaussian version of MIN by Gaussian positive operator valued measurements.

Spin and localization of relativistic fermions and uncertainty relations

We discuss relations between several relativistic spin observables and derive a Lorentz-invariant characteristic of a reduced spin density matrix.A relativistic position operator that satisfies all the properties of its nonrelativistic analog does not exist. Instead we propose two causality-preserving positive operator-valued measures (POVMs) that are based on projections onto one-particle and antiparticle spaces, and on the normalized energy density. They predict identical expectation values for position. The variances differ by less than a quarter of the squared de Broglie wavelength and coincide in the nonrelativistic limit. Since the resulting statistical moment operators are not canonical conjugates of momentum, the Heisenberg uncertainty relations need not hold. Indeed, the energy density POVM leads to a lower uncertainty. We reformulate the standard equations of the spin dynamics by explicitly considering the charge-independent acceleration, allowing a consistent treatment of backreaction and inclusion of a weak gravitational field.

Informational power of the Hoggar symmetric informationally complete positive operator-valued measure

We compute the informational power for the Hoggar SIC-POVM in dimension 8, i.e. the classical capacity of a quantum-classical channel generated by this measurement. We show that the states constituting a maximally informative ensemble form a twin Hoggar SIC-POVM being the image of the original one under a conjugation.

COVARIANT INTEGRAL QUANTIZATIONS AND THEIR APPLICATIONS TO QUANTUM COSMOLOGY

We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on group representation and probabilistic aspects of these constructions. Simple phase space examples illustrate the procedure: plane (Weyl-Heisenberg symmetry), half-plane (affine symmetry). Interesting applications to quantum cosmology (“smooth bouncing”) for Friedmann-Robertson-Walker metric are presented and those for Bianchi I and IX models are mentioned.

Quantum Fisher information of the GHZ state due to classical phase noise lasers under non-Markovian environment

The dynamics of N-qubit GHZ state quantum Fisher information (QFI) under phase noise lasers (PNLs) driving is investigated in terms of non-Markovian master equation. We first investigate the non-Markovian dynamics of the QFI of N-qubit GHZ state and show that when the ratio of the PNL rate and the system–environment coupling strength is very small, the oscillations of the QFIs decay slower which corresponds to the non-Markovian region; yet when it becomes large, the QFIs monotonously decay which corresponds to the Markovian region. When the atom number N increases, QFIs in both regions decay faster. We further find that the QFI flow disappears suddenly followed by a sudden birth depending on the ratio of the PNL rate and the system–environment coupling strength and the atom number N, which unveil a fundamental connection between the non-Markovian behaviors and the parameters of system–environment couplings. We discuss two optimal positive operator-valued measures (POVMs) for two different strategies of our model and find the condition of the optimal measurement. At last, we consider the QFI of two atoms with qubit–qubit interaction under random telegraph noises (RTNs).

Gleason-Type Theorem for Projective Measurements, Including Qubits: The Born Rule Beyond Quantum Physics

Born’s quantum probability rule is traditionally included among the quantum postulates as being given by the squared amplitude projection of a measured state over a prepared state, or else as a trace formula for density operators. Both Gleason’s theorem and Busch’s theorem derive the quantum probability rule starting from very general assumptions about probability measures. Remarkably, Gleason’s theorem holds only under the physically unsound restriction that the dimension of the underlying Hilbert space \(\mathcal {H}\) must be larger than two. Busch’s theorem lifted this restriction, thereby including qubits in its domain of validity. However, while Gleason assumed that observables are given by complete sets of orthogonal projectors, Busch made the mathematically stronger assumption that observables are given by positive operator-valued measures. The theorem we present here applies, similarly to the quantum postulate, without restricting the dimension of \(\mathcal {H}\) and for observables given by complete sets of orthogonal projectors. We also show that the Born rule applies beyond the quantum domain, thereby exhibiting the common root shared by some quantum and classical phenomena.

The classical-quantum channel with random state parameters known to the sender

We study an analog of the well-known Gel’fand Pinsker channel which uses quantum states for the transmission of the data. We consider the case where both the sender’s inputs to the channel and the channel states are to be taken from a finite set (the cq-channel with state information at the sender). We distinguish between causal and non-causal channel state information input by the sender. The receiver remains ignorant, throughout. We give a single-letter description of the capacity in the first case. In the second case we present two different regularized expressions for the capacity. It is an astonishing and unexpected result of our work that a simple change from causal to non-causal channel state information by the encoder causes the complexity of a numerical computation of the capacity formula to change from trivial to seemingly difficult. Still, even the non-single letter formula allows one to draw nontrivial conclusions, for example regarding the continuity of the capacity with respect to changes in the system parameters. The direct parts of both coding theorems are based on a special class of positive operator valued measurements (POVMs) which are derived from orthogonal projections onto certain representations of the symmetric groups. This approach supports a reasoning that is inspired by the classical method of types. In combination with the non-commutative union bound these POVMs yield an elegant method of proof for the direct part of the coding theorem in the first case.

Universal optimal estimation of the polarization of light with arbitrary photon statistics

A universal and optimal method for the polarimetry of light with arbitrary photon statistics is presented. The method is based on the continuous maximum-likelihood positive operator-valued measure (ML-POVM) for pure polarization states over the surface of the Bloch sphere. The success probability and the mean fidelity are used as the figures of merit to show its performance. The POVM is found to attain the collective bound of polarization estimation with respect to the mean fidelity. As demonstrations, explicit results for the N photon Fock state, the phase-randomized coherent state (Poisson distribution), and the thermal light are obtained. It is found that the estimation performances for the Fock state and the Poisson distribution are almost identical, while that for the thermal light is much worse. This suggests that thermal light leaks less information to an eavesdropper and hence could potentially provide more security in polarization-encoded quantum communication protocols than a single-mode laser beam as customarily considered. Finally, comparisons against an optimal adaptive measurement with classical communications are made to show the better and more stable performance of the continuous ML-POVM.