POVM Research Articles

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.

Quantum coherence of steered states.

Lying at the heart of quantum mechanics, coherence has recently been studied as a key resource in quantum information theory. Quantum steering, a fundamental notion originally considered by Schödinger, has also recently received much attention. When Alice and Bob share a correlated quantum system, Alice can perform a local measurement to ‘steer’ Bob’s reduced state. We introduce the maximal steered coherence as a measure describing the extent to which steering can remotely create coherence; more precisely, we find the maximal coherence of Bob’s steered state in the eigenbasis of his original reduced state, where maximization is performed over all positive-operator valued measurements for Alice. We prove that maximal steered coherence vanishes for quantum-classical states whilst reaching a maximum for pure entangled states with full Schmidt rank. Although invariant under local unitary operations, maximal steered coherence may be increased when Bob performs a channel. For a two-qubit state we find that Bob’s channel can increase maximal steered coherence if and only if it is neither unital nor semi-classical, which coincides with the condition for increasing discord. Our results show that the power of steering for coherence generation, though related to discord, is distinct from existing measures of quantum correlation.

Quantum state tomography with noninstantaneous measurements, imperfections, and decoherence

Tomography of a quantum state is usually based on positive operator-valued measure (POVM) and on their experimental statistics. Among the available reconstructions, the maximum-likelihood (MaxLike) technique is an efficient one. We propose an extension of this technique when the measurement process cannot be simply described by an instantaneous POVM. Instead, the tomography relies on a set of quantum trajectories and their measurement records. This model includes the fact that, in practice, each measurement could be corrupted by imperfections and decoherence, and could also be associated with the record of continuous-time signals over a finite amount of time. The goal is then to retrieve the quantum state that was present at the start of this measurement process. The proposed extension relies on an explicit expression of the likelihood function via the effective matrices appearing in quantum smoothing and solutions of the adjoint quantum filter. It allows to retrieve the initial quantum state as in standard MaxLike tomography, but where the traditional POVM operators are replaced by more general ones that depend on the measurement record of each trajectory. It also provides, aside the MaxLike estimate of the quantum state, confidence intervals for any observable. Such confidence intervals are derived, as the MaxLike estimate, from an asymptotic expansion of multi-dimensional Laplace integrals appearing in Bayesian Mean estimation. A validation is performed on two sets of experimental data: photon(s) trapped in a microwave cavity subject to quantum non-demolition measurements relying on Rydberg atoms; heterodyne fluorescence measurements of a superconducting qubit.

Adaptive optimal measurement for the squeezed vacuum and coherent state

The output phase of the Sagnac interferometer has been measured with quantum balanced homodyne technique when coherent light and squeezed vacuum light are fed into the Sagnac interferometer simultaneously [Chen Kun et al., Acta Phys. Sin. 65 054203(2016)]. Nevertheless, there exist two deficiencies: 1) the phase sensitivity is related to the phase itself; 2) there are strict requirements for the phases of local oscillator light, coherent light and squeezed vacuum light. For overcoming these deficiencies, an adaptive optimal measurement scheme is suggested for the phase estimation. Firstly, we calculate that the quantum Fisher information (QFI) of the squeezed vacuum and coherent state is sinh2r+||2e2r by treating them as a quantum pure state, for they satisfy a condition of the quantum pure state, namely ()=()2. The QFI is related to quantum Cramer-Rao lower bound which can be used to evaluate the performance of the estimator. Secondly, we make an analysis of positive operator-valued measure (POVM) and design a set of the optimal measurement operators for reaching the quantum Cramer-Rao lower bound, whereas the optimal measurement operators depend on the true value of the phase which is what we want to estimate. In order to solve the problem and estimate the parameter effectively, an adaptive method is suggested. We set an initial value of the phase parameter to obtain a set of measurement operators which are not optimal at the first step. And then the initial measurement operators are used for POVM and to obtain a conditional probability function, from which we can obtain a new value of the phase with maximum likelihood estimator. Therefore, the measurement operators and conditional probability function will be updated with the new value. As the measurement operators and conditional probability function are updated step by step, we can estimate the value adaptively. In fact, the results of the maximum likelihood estimator will converge at the true value of the phase parameter gradually, which is then proved with the theoretical analysis. All in all, an adaptive measurement method of estimating the phase parameter of the squeezed vacuum and coherent state in Sagnac interferometer is suggested, and is proved theoretically to be that the scheme will converge at the true value of the phase with a probability of 1 and can reach the quantum Cramer-Rao lower bound.

Unsharp Measurements and Joint Measurability

We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice's pair of non-commuting observables in the presence of Bob's entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice-Bob state, implementing incompatible POVMs at Bob's end is necessary to beat the uncertainty bound and hence, predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to {\em validate} a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix -- which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables -- could be realized if and only if compatible POVMs are employed.

Unsharp Measurements and Joint Measurability

We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice’s pair of non-commuting observables in the presence of Bob’s entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice–Bob state, implementing incompatible POVMs at Bob’s end is necessary to beat the uncertainty bound and hence predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to validate a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix – which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables – could be realized if and only if compatible POVMs are employed.

Remotely and Conclusively Mapping One Finite Set of Qudit States onto Another Assisted by Qubit Entanglements

Alice and Bob are two remote parties. We propose a probabilistic method which allows Alice to map remotely and conclusively Bob’s set of nonorthogonal symmetric d-level quantum states onto another. The procedure we use is a remote positive operator valued measurement (POVM) in Bob’s (2d-1)-level direct sum space. We construct a quantum network for implementing this (2d-1)-level remote nonunitary POVM with (d-1) two-level remote unitary rotations. The fact that the two-level remote rotation, which is hired to rotate remotely a basis vector, can been implementing rapidly using only one ebit (a two-level Einstein-Podolsky-Rosen (EPR) pair) and one cbit (classical communication) is notable. This scheme is simpler but with less resource, which will make it more feasible and suitable for large-scale quantum network.

Geometric characterization of true quantum decoherence

Surprisingly often decoherence is due to classical fluctuations of ambient fields and may thus be described in terms of random unitary (RU) dynamics. However, there are decoherence channels where such a representation cannot exist. Based on a simple and intuitive geometric measure for the distance of an extremal channel to the convex set of RU channels we are able to characterize the set of true quantum phase-damping channels. Remarkably, using the Caley-Menger determinant, our measure may be assessed directly from the matrix representation of the channel. We find that the channel of maximum quantumness is closely related to a symmetric, informationally-complete positive operator-valued measure (SIC-POVM) on the environment. Our findings are in line with numerical results based on the entanglement of assistance.

Controlled Dense Coding Using the Maximal Slice States

In this paper we investigate the controlled dense coding with the maximal slice states. Three schemes are presented. Our schemes employ the maximal slice states as quantum channel, which consists of the tripartite entangled state from the first party(Alice), the second party(Bob), the third party(Cliff). The supervisor(Cliff) can supervises and controls the channel between Alice and Bob via measurement. Through carrying out local von Neumann measurement, controlled-NOT operation and positive operator-valued measure(POVM), and introducing an auxiliary particle, we can obtain the success probability of dense coding. It is shown that the success probability of information transmitted from Alice to Bob is usually less than one. The average amount of information for each scheme is calculated in detail. These results offer deeper insight into quantum dense coding via quantum channels of partially entangled states.

Conclusive qudit states transformation implemented by remote parters

Alice, Cindy, and Bob are three remote parties. We present a scheme which allows Alice and Cindy to transform probabilistically and conclusively Bobs unknown d -level quantum (qudit) state into another. In this scheme, the remote positive operator valued measurements (POVM) lies at the heart. Based on the knowledge of probability amplitudes and phase factors of Bobs original state and target state, Alice and Cindy could construct jointly the required optimal POVM, which acts remotely on Bobs (2 d -1)-dimensional direct sum space. We construct a quantum network for realizing the remote POVM with 2( d -1) remote two-level unitary rotations, and thus provide a feasible physical means to realize the remote POVM. The fact that the remote POVM could be realized using only 2( d -1) two-level Bell states (ebits) and 3( d -1) two-level classical communications (cbits) is notable. This scheme can be generalized to implement conclusively the ( N +2)-party remote qudit state transformation via ( d -1) ( N +1)-body two-level GHZ states, in which the phase factors of Bobs original state and target state are shared jointly by N agents. The success probability of implementing this remote state transformation is also investigated. This scheme is simpler but conclusive, and more efficient but with less resource, which will make it more feasible with the current experimental technology and more suitable for large-scale quantum network.

Experimental Realization of Quantum Tomography of Photonic Qudits via Symmetric Informationally Complete Positive Operator-Valued Measures

Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension. In this work, we implement state tomography using symmetric informationally complete positive operator-valued measures for both pure and mixed photonic qudit states in Hilbert spaces of orbital angular momentum, including spaces whose dimension is not power of a prime. Fidelities of reconstruction within the range of 0.81-0.96 are obtained for both pure and mixed states. These results are relevant to high-dimensional quantum information and computation experiments, especially to those where a complete set of mutually unbiased bases is unknown.

Minimal sufficient positive-operator valued measure on a separable Hilbert space

We introduce a concept of a minimal sufficient positive-operator valued measure (POVM), which is the least redundant POVM among the POVMs that have the equivalent information about the measured quantum system. Assuming the system Hilbert space to be separable, we show that for a given POVM, a sufficient statistic called a Lehmann-Scheffé-Bahadur statistic induces a minimal sufficient POVM. We also show that every POVM has an equivalent minimal sufficient POVM and that such a minimal sufficient POVM is unique up to relabeling neglecting null sets. We apply these results to discrete POVMs and information conservation conditions proposed by the author.

Non-Markovian quantum measurements in the non-invasive limit

In many real quantum systems the measurement is much less invasive than described by the projection postulate. For example, it is performed by weakly coupling a detector for a finite time to the system. The finite coupling time might lead to different measurement outcomes depending on the details of the interaction. We first investigate a generalized measurements scheme by means of positive operator-valued measures in the weak coupling limit and discuss the consequences appearing due to the non-Markovian interaction. Second we discuss the resulting quasiprobabilities, which in the case of a Markovian measurement scheme obey the so-called weak positivity, i.e. second-order correlation functions are positive semi-definite. In particular, we show a possible violation of weak positivity by second-order correlation functions for a frequency dependent detector measuring a spin-1/2 system. When both are themselves in an equilibrium state but at different respective temperatures, we find that the detection of the quantum behavior vanishes if the system temperature is larger than that of the detector.

Construction of the least informative observable conserved by a given quantum instrument

For a quantum measurement process described by a quantum instrument I and a system observable corresponding to a positive-operator valued measure (POVM) E, I is said to conserve the information of E if the joint successive measurement of I followed by E is equivalent to a single measurement of E. We show that for any quantum instrument I we can construct a POVM conserved by I. Intuitively, the construction gives the infinite joint successive measurement of I. We also show that the constructed POVM is the least informative observable among POVMs conserved by I, i.e., the constructed POVM can be realized by a classical post-processing of any POVM conserved by I. As typical examples of quantum instruments, we explicitly evaluate POVMs of infinite successive measurements for photon counting and quantum counter instruments.

Characterizing the dynamics of quantum discord under phase damping with POVM measurements**Project supported by the National Natural Science Foundation of China (Grant Nos. 11305074, 11135002, and 11275083).

In the analysis of quantum discord, the minimization of average entropy traditionally involved over orthogonal projective measurements may be attained at more optimal decompositions by using the positive-operator-valued measure (POVM) measurements. Taking advantage of the quantum steering ellipsoid in combination with three-element POVM optimization, we show that, for a family of two-qubit X states locally interacting with Markovian non-dissipative environments, the decay rates of quantum discord show smooth dynamical evolutions without any sudden change. This is in contrast to two-element orthogonal projective measurements, in which case the sudden change of the decay rates of quantum and classical decoherences may be a common phenomenon. Notwithstanding this, we find that a subset of X states (including the Bell diagonal states) involving POVM optimization can still preserve the sudden change character as usual.

Noise robustness of the incompatibility of quantum measurements

The existence of incompatible measurements is a fundamental phenomenon having no explanation in classical physics. Intuitively, one considers given measurements to be incompatible within a framework of a physical theory, if their simultaneous implementation on a single physical device is prohibited by the theory itself. In the mathematical language of quantum theory, measurements are described by POVMs (positive operator valued measures), and given POVMs are by definition incompatible if they cannot be obtained via coarse-graining from a single common POVM; this notion generalizes noncommutativity of projective measurements. In quantum theory, incompatibility can be regarded as a resource necessary for manifesting phenomena such as Clauser-Horne-Shimony-Holt (CHSH) Bell inequality violations or Einstein-Podolsky-Rosen (EPR) steering which do not have classical explanation. We define operational ways of quantifying this resource via the amount of added classical noise needed to render the measurements compatible, i.e., useless as a resource. In analogy to entanglement measures, we generalize this idea by introducing the concept of incompatibility measure, which is monotone in local operations. In this paper, we restrict our consideration to binary measurements, which are already sufficient to explicitly demonstrate nontrivial features of the theory. In particular, we construct a family of incompatibility monotones operationally quantifying violations of certain scaled versions of the CHSH Bell inequality, prove that they can be computed via a semidefinite program, and show how the noise-based quantities arise as special cases. We also determine maximal violations of the new inequalities, demonstrating how Tsirelson's bound appears as a special case. The resource aspect is further motivated by simple quantum protocols where our incompatibility monotones appear as relevant figures of merit.

Examples of conditional SIC-POVMs

The state of a quantum system is a density matrix with several parameters. The concern herein is how to recover the parameters. Several possibilities exist for the optimal recovery method, and we consider some special cases. We assume that a few parameters are known and that the others are to be recovered. The optimal positive-operator-valued measure (POVM) for recovering unknown parameters with an additional condition is called a conditional symmetric informationally complete POVM (SIC-POVM). In this paper, we study the existence or nonexistence of conditional SIC-POVMs. We provide a necessary condition for existence and some examples.

On the Possibility to Combine the Order Effect with Sequential Reproducibility for Quantum Measurements

In this paper we study the problem of a possibility to use quantum observables to describe a possible combination of the order effect with sequential reproducibility for quantum measurements. By the order effect we mean a dependence of probability distributions (of measurement results) on the order of measurements. We consider two types of the sequential reproducibility: adjacent reproducibility ($A-A$) and separated reproducibility($A-B-A$). The first one is reproducibility with probability 1 of a result of measurement of some observable $A$ measured twice, one $A$ measurement after the other. The second one, $A-B-A$, is reproducibility with probability 1 of a result of $A$ measurement when another quantum observable $B$ is measured between two $A$'s. Heuristically, it is clear that the second type of reproducibility is complementary to the order effect. We show that, surprisingly, for an important class of quantum observables given by positive operator valued measures (POVMs), this may not be the case. The order effect can coexist with a separated reproducibility as well as adjacent reproducibility for both observables $A$ and $B.$ However, the additional constraint in the form of separated reproducibility of the $B-A-B$ type makes this coexistence impossible. Mathematically, this paper is about the operator algebra for effects composing POVMs. The problem under consideration was motivated by attempts to apply the quantum formalism outside of physics, especially, in cognitive psychology and psychophysics. However, it is also important for foundations of quantum physics as a part of the problem about the structure of sequential quantum measurements.

Mutually unbiased bases as minimal Clifford covariant 2-designs

Mutually unbiased bases (MUB) are interesting for various reasons. The most attractive example of (a complete set of) MUB is the one constructed by Ivanovi\'c as well as Wootters and Fields, which is referred to as the canonical MUB. Nevertheless, little is known about anything that is unique to this MUB. We show that the canonical MUB in any prime power dimension is uniquely determined by an extremal orbit of the (restricted) Clifford group except in dimension 3, in which case the orbit defines a special symmetric informationally complete measurement (SIC), known as the Hesse SIC. Here the extremal orbit is the one with the smallest number of pure states. Quite surprisingly, this characterization does not rely on any concept that is related to bases or unbiasedness. As a corollary, the canonical MUB is the unique minimal 2-design covariant with respect to the Clifford group except in dimension 3. In addition, these MUB provide an infinite family of highly symmetric frames and positive-operator-valued measures (POVMs), which are of independent interest.

Single-photon observables and preparation uncertainty relations

We propose a procedure to define all single-photon observables in a consistent and unified picture based on operational approach to quantum mechanics. We identify the suppression of zero-helicity states as a projection from an extended Hilbert space onto the physical single-photon Hilbert space. We show that all single-photon observables are in general described by positive-operator valued measures (POVMs), obtained by applying this projection to opportune projection-valued measures (PVMs) defined on the extended Hilbert space. The POVMs associated to momentum and helicity reduce to PVMs, unlike those associated to position and spin. This fact reflects the intrinsic unsharpness of these observables. We apply this formalism to study the preparation uncertainty relations for position and momentum and to compute the probability distribution of spin, for a broad class of Gaussian states. Results show quantitatively the enhancement of the statistical character of the theory.