POVM Research Articles

Experimentally accessible nonseparability criteria for multipartite-entanglement-structure detection

The description of the complex separability structure of quantum states in terms of partially ordered sets has been recently put forward. In this work, we address the question of how to efficiently determine these structures for unknown states. We propose an experimentally accessible and scalable iterative methodology that identifies, on solid statistical grounds, sufficient conditions for nonseparability with respect to certain partitions. In addition, we propose an algorithm to determine the minimal partitions (those that do not admit further splitting) consistent with the experimental observations. We test our methodology experimentally on a 20-qubit IBM quantum computer by inferring the structure of the 4-qubit Smolin and an 8-qubit W state. In the first case, our results reveal that, while the fidelity of the state is low, it nevertheless exhibits the partitioning structure expected from the theory. In the case of the W state, we obtain very disparate results in different runs on the device, which range from nonseparable states to very fragmented minimal partitions with little entanglement in the system. Furthermore, our work demonstrates the applicability of informationally complete positive operator-valued measurements for practical purposes on current noisy intermediate-scale quantum devices.

Multi-class classification based on quantum state discrimination

We present a general framework for the problem of multi-class classification using classification functions that can be interpreted as fuzzy sets. We specialize these functions in the domain of Quantum-inspired classifiers, which are based on quantum state discrimination techniques. In particular, we use unsharp observables (Positive Operator-Valued Measures) that are determined by the training set of a given dataset to construct these classification functions. We show that such classifiers can be tested on near-term quantum computers once these classification functions are “distilled” (on a classical platform) from the quantum encoding of a training dataset. We compare these experimental results with their theoretical counterparts and we pose some questions for future research.

Classical Cost of Transmitting a Qubit.

We consider general prepare-and-measure scenarios in which Alice can transmit qubit states to Bob, who can perform general measurements in the form of positive operator-valued measures (POVMs). We show that the statistics obtained in any such quantum protocol can be simulated by the purely classical means of shared randomness and two bits of communication. Furthermore, we prove that two bits of communication is the minimal cost of a perfect classical simulation. In addition, we apply our methods to Bell scenarios, which extends the well-known Toner and Bacon protocol. In particular, two bits of communication are enough to simulate all quantum correlations associated to arbitrary local POVMs applied to any entangled two-qubit state.

Exploring postselection-induced quantum phenomena with time-bidirectional state formalism

Here we present the time-bidirectional state formalism (TBSF) unifying in a general manner the standard quantum mechanical formalism with no postselection and the time-symmetrized two-state (density) vector formalism, which deals with postselected states. In the proposed approach, a quantum particle's state, called a time-bidirectional state, is equivalent to a joined state of two particles propagating in opposite time directions. For a general time-bidirectional state, we derive outcome probabilities of generalized measurements, as well as mean and weak values of Hermitian observables. We also show how the obtained expressions reduce to known ones in the special cases of no postselection and generalized two-state (density) vectors. Then we develop tomography protocols based on mutually unbiased bases and a symmetric informationally complete positive operator-valued measure, allowing experimental reconstruction of an unknown single qubit time-bidirectional state. Finally, we employ the developed techniques for tracking of a qubit's time-reversal journey in a quantum teleportation protocol realized with a cloud-accessible noisy superconducting quantum processor. The obtained results justify an existence of a postselection-induced qubit's proper time-arrow, which is different from the time-arrow of a classical observer, and demonstrate capabilities of the TBSF for exploring quantum phenomena brought forth by a postselection in the presence of noise.

Certification of the maximally entangled state using nonprojective measurements

In recent times, device-independent certification of quantum states has been one of the intensively studied areas in quantum information. However, all such schemes utilize projective measurements which are practically difficult to generate. In this paper, we consider the one-sided device-independent scenario and propose a self-testing scheme for the two-qubit maximally entangled state using nonprojective measurements, in particular, three three-outcome extremal positive operator-valued measures. We also analyze the robustness of our scheme against white noise.

Quantum-Inspired Multi-Parameter Adaptive Bayesian Estimation for Sensing and Imaging

It is well known in Bayesian estimation theory that the conditional estimator attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest. In quantum, e.g., optical and atomic, imaging and sensing tasks the user has access to the quantum state that encodes the parameter. The choice of a measurement operator, i.e. a positive-operator valued measure (POVM), leads to a measurement outcome on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM that attains the MMSE= over all possible physically allowable measurements and the resulting MMSE [1]. This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements without considering any prior probability. For multi-parameter estimation, in quantum Fisher estimation theory the inverse of the QFI matrix provides an operator lower bound on the covariance of an unbiased estimator, and this bound is understood in the positive semidefinite sense. However, there has been little work on quantifying the quantum limits and measurement designs, for multi-parameter quantum estimation in a Bayesian setting. In this work, we build upon Personick's result to construct a Bayesian adaptive (greedy) measurement scheme for multiparameter estimation. We illustrate our proposed measurement scheme with the application of localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs superior to quantum-noise-limited focal-plane direct imaging.

On the Extension of a Family of Projections to a Positive Operator-Valued Measure

On the Extension of a Family of Projections to a Positive Operator-Valued Measure

Simulation of positive operator-valued measures and quantum instruments via quantum state-preparation algorithms

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.

Generalised uncertainty relations from finite-accuracy measurements

In this short note we show how the Generalised Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP), two of the most common generalised uncertainty relations proposed in the quantum gravity literature, can be derived within the context of canonical quantum theory, without the need for modified commutation relations. A generalised uncertainty principle-type relation naturally emerges when the standard position operator is replaced by an appropriate Positive Operator Valued Measure (POVM), representing a finite-accuracy measurement that localises the quantum wave packet to within a spatial regionσg> 0. This length scale is the standard deviation of the envelope function,g, that defines the positive operator valued measure elements. Similarly, an extended uncertainty principle-type relation emerges when the standard momentum operator is replaced by a positive operator valued measure that localises the wave packet to within a regionσ̃g>0in momentum space. The usual generalised uncertainty principle and extended uncertainty principle are recovered by settingσg≃ℏG/c3, the Planck length, andσ̃g≃ℏΛ/3, where Λ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schrödinger and Heisenberg equations, remain unchanged. This demonstrates that generalised uncertainty principle and extended uncertainty principle phenomenology can be obtained without modified commutators, which are known to lead to various pathologies, including violation of the equivalence principle, violation of Lorentz invariance in the relativistic limit, the reference frame-dependence of the “minimum” length, and the so-called soccer ball problem for multi-particle states.

Bounds on positive operator-valued measure based coherence of superposition

Quantum coherence is a fundamental feature of quantum physics and plays a significant role in quantum information processing. By generalizing the resource theory of coherence from von Neumann measurements to positive operator-valued measures (POVMs), POVM-based coherence measures have been proposed with respect to the relative entropy of coherence, the l 1 norm of coherence, the robustness of coherence and the Tsallis relative entropy of coherence. We derive analytically the lower and upper bounds on these POVM-based coherence of an arbitrary given superposed pure state in terms of the POVM-based coherence of the states in superposition. Our results can be used to estimate range of quantum coherence of superposed states. Detailed examples are presented to verify our analytical bounds.

Quantum measurements constrained by the third law of thermodynamics

In the quantum regime, the third law of thermodynamics implies the unattainability of pure states. As shown recently, such unattainability implies that a unitary interaction between the measured system and a measuring apparatus can never implement an ideal projective measurement. In this paper, we introduce an operational formulation of the third law for the most general class of physical transformations, the violation of which is both necessary and sufficient for the preparation of pure states. Subsequently, we investigate how such a law constrains measurements of general observables, or positive operator valued measures. We identify several desirable properties of measurements which are simultaneously enjoyed by ideal projective measurements -- and are hence all ruled out by the third law in such a case -- and determine if the third law allows for these properties to obtain for general measurements of general observables and, if so, under what conditions. It is shown that while the third law rules out some of these properties for all observables, others may be enjoyed by observables that are sufficiently "unsharp".

Device-independent certification of degeneracy-breaking measurements

In a device-independent Bell test, the devices are considered to be black boxes and the dimension of the system remains unspecified. The dichotomic observables involved in such a Bell test can be degenerate and one may invoke a suitable measurement scheme to lift the degeneracy. However, the standard Bell test cannot account for whether or up to what extent the degeneracy is lifted, as the effect of lifting the degeneracy can only be reflected in the postmeasurement states, which the standard Bell tests do not certify. In this work, we demonstrate the device-independent certification of degeneracy-breaking measurement based on the sequential Bell test by multiple observers who perform degeneracy-breaking unsharp measurements characterized by positive-operator-valued measures (POVMs)---the noisy variants of projectors. The optimal quantum violation of Clauser-Horne-Shimony-Holt inequality by multiple sequential observers eventually enables us to certify up to what extent the degeneracy has been lifted. In particular, our protocol certifies the upper bound on the number of POVMs used for performing such measurements along with the entangled state and measurement observables. We use an elegant sum-of-squares approach that powers such certification of degeneracy-breaking measurements.

Quantum State Tomography in Nonequilibrium Environments

We generalize an approach to studying the quantum state tomography (QST) of open systems in terms of the dynamical map in Kraus representation within the framework of dynamic generation of informationally complete positive operator-valued measures. As applications, we use the generalized approach to theoretically study the QST of qubit systems in the presence of nonequilibrium environments which exhibit nonstationary and non-Markovian random telegraph noise statistical properties. We derive the time-dependent measurement operators for the quantum state reconstruction of the single qubit and two-qubit systems in terms of the polarization operator basis. It is shown that the behavior of the time-dependent measurement operators is closely associated with the dynamical map of the qubit systems.

Revealing quantum contextuality using a single measurement device

In this paper we analyze the notion of measurement noncontextuality (MNC) and identify contextual scenarios which involve sequential measurements of only a single measurement device. We show that any noncontextual ontological model fails to explain the statistics of outcomes of a single carefully constructed positive-operator-valued measure (POVM) executed sequentially on a quantum system. The context of measurement arises from the different configurations in which the device can be used. We develop an inequality from the noncontextual (NC) ontic model, and construct a quantum situation involving measurements from the Klyachko-Can-Binicio\ifmmode \check{g}\else \v{g}\fi{}lu-Shumovsky (KCBS) inequality. We show that the resultant statistics arising from this device violate our NC inequality. This device can be generalized by incorporating measurements from arbitrary $n$-cycle contextuality inequalities of which $n=5$ corresponds to the KCBS inequality. We show that the NC and quantum bounds for various scenarios can be derived more easily using only the functional relationships between the outcomes for larger values $n$. This makes it one of the simpler contextual inequalities to analyze.

On extension of the family of projections to positive operator-valued measure

The problem of constructing a measure on a discrete set X taking values in a positive cone of bounded operators in a Hilbert space is considered. It is assumed that a projectionvalued function defined on a subset of X0 of the original set X is initially given. The aim of the study is to find such a scalar measure μ on the set X and the continuation of a projector-valued function from X0 to X, which results in an operator-valued measure having a projector-valued density relative to μ. In general, the problem is solved for |X| = 4 and |X0| = 2. As an example, we consider a function on X0 that takes values in a set of projections on coherent states. For this case, the question of the information completeness of the measurement determined by the constructed measure is investigated. In other words, is it possible to reconstruct a quantum state (a positive unit trace operator) from the values of the matrix trace from the product of a measure with a quantum state. It is shown that for the constructed measure it is possible to restore the quantum state only if it is a projection. A restriction on the probability distribution is also found, at which it can be obtained as a result of measuring a certain quantum state.

Homodyne measurement with a Schrödinger cat state as a local oscillator

Homodyne measurements are a widely used quantum measurement. Using a coherent state of large amplitude as the local oscillator, it can be shown that the quantum homodyne measurement limits to a field quadrature measurement. In this work, we give an example of a general idea: injecting nonclassical states as a local oscillator can lead to nonclassical measurements. Specifically, we consider injecting a superposition of coherent states, a Schr\"odinger cat state, as a local oscillator. We derive the Kraus operators and the positive operator-valued measure in this situation.

Quantum discord and entropic measures of two relativistic fermions

In the present work, we study the interplay between relativistic effects and quantumness in the system of two relativistic fermions. In particular, we explore entropic measures of quantum correlations and quantum discord before and after application of a boost and subsequent Wigner rotation. We also study the positive operator-valued measurements (POVMs) invasiveness before and after the boosts. While the relativistic principle is universal and requires Lorentz invariance of quantum correlations in the entire system, we have found specific partitions where quantum correlations stored in particular subsystems are not invariant. We calculate quantum discords corresponding of the states before and after applying a boost, and observe that the state gains extra discord after the boost. When analyzing the invasiveness of the POVMs, we have found that the POVM applied to the initial entangled state reduces the discord to zero. However, discord of the boosted state survives after the same POVM. Thus we conclude that the quantum discord generated by Lorentz boost is robust concerning the protective POVM, while the measurement exerts an invasive effect on the discord of the initial state. Finally, we discuss potential implementation of the ideas of this work using top quarks as a benchmark scenario.

On a quantum martingale convergence theorem

It is well known in quantum information theory that a positive operator-valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalized POVM, namely, a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator-valued function which is measurable with respect to a quantum probability. In this work, we study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of the Lebesgue-dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits nonclassical behavior; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.

Reconstructing quantum states via unambiguous state discrimination

Abstract In this paper, we introduce an analytical framework for the reconstruction of quantum states. The reconstruction of an unknown quantum state requires the information of a complete set of observables, obtained through experimental measurements of Hermitian operators usually defined as positive-operator-valued measures (POVMs). The scheme involves a single-qubit unambiguous state discrimination POVM, which can be generalized to perform n-qubit measurements. We also use maximum likelihood estimation as a method in the reconstruction of the density matrix from experimental data and show that the expected value of the cleaner is independent of the parameter of the density operator.

Unbounded randomness from uncharacterized sources

Randomness is a central feature of quantum mechanics and an invaluable resource for both classical and quantum technologies. Commonly, in Device-Independent and Semi-Device-Independent scenarios, randomness is certified using projective measurements, and its amount is bounded by the quantum system’s dimension. Here, we propose a Source-Device-Independent protocol, based on Positive Operator Valued Measurement (POVM), which can arbitrarily increase the number of certified bits for any fixed dimension. Additionally, the proposed protocol doesn’t require an initial seed and active basis switching, simplifying its experimental implementation and increasing the generation rates. A tight lower-bound on the quantum conditional min-entropy is derived using only the POVM structure and the experimental expectation values, taking into account the quantum side-information. For symmetric POVM on the Bloch sphere, we derive closed-form analytical bounds. Finally, we experimentally demonstrate our method with a compact and simple photonic setup that employs polarization-encoded qubits and POVM up to 6 outcomes.