POVM Research Articles

Separation of measurement uncertainty into quantum and classical parts based on Kirkwood–Dirac quasiprobability and generalized entropy

Abstract Measurement in quantum mechanics is notoriously unpredictable. The uncertainty in quantum measurement can arise from the noncommutativity between the state and the measurement basis which is intrinsically quantum, but it may also be of classical origin due to the agent’s ignorance. It is of fundamental as well as practical importance to cleanly separate the two contributions which can be directly accessed using laboratory operations. Here, we propose two ways of decomposition of the total measurement uncertainty additively into quantum and classical parts. In the two decompositions, the total uncertainty of a measurement described by a positive-operator-valued measure (POVM) over a quantum state is quantified respectively by two generalized nonadditive entropies of the measurement outcomes; the quantum parts are identified, respectively, by the nonreality and the nonclassicality—which captures simultaneously both the nonreality and negativity—of the associated generalized Kirkwood–Dirac quasiprobability relative to the POVM of interest and a projection-valued measure and maximized over all possible choices of the latter; and, the remaining uncertainties are identified as the classical parts. Both decompositions are shown to satisfy a few plausible requirements. The minimum of the total measurement uncertainties in the two decompositions over all POVM measurements are given by the impurity of the quantum state quantified by certain generalized quantum entropies, and are entirely classical. We argue that nonvanishing genuine quantum uncertainty in the two decompositions are sufficient and necessary to prove quantum contextuality via weak measurement with postselection. Finally, we suggest that the genuine quantum uncertainty is a manifestation of a specific measurement disturbance.

Entanglement criteria based on quantum Fisher information

To optimize the entanglement detection, we formulate the metrologically operational entanglement condition in quantum Fisher information by maximizing the QFI on the measurement orbit. Specifically, we consider two classes of typical local observables, i.e. the local orthonormal observables and symmetric informationally complete positive operator-valued measures. Result shows that the symmetric informationally complete positive operator-valued measures are superior to local orthonormal observables in entanglement detection, which in some sense hints the yet unconfirmed generally superiority of symmetric informationally complete positive operator-valued measures in quantum information processing.

Discrete dynamics in the set of quantum measurements

Abstract A quantum measurement, often referred to as positive operator-valued measurement, is a set of positive operators P j = P j † ⩾ 0 summing to identity, ∑ j P j = 𝟙 . This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. We describe transformations in the set of quantum measurements by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity we arrive at blockwise bistochastic matrices (also called quantum magic squares). Analyzing their dynamical properties, we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in the set of quantum measurements.

Scalable quantum detector tomography by high-performance computing

Abstract At large scales, quantum systems may become advantageous over their classical counterparts at performing certain tasks. Developing tools to analyze these systems at the relevant scales, in a manner consistent with quantum mechanics, is therefore critical to benchmarking performance and characterizing their operation. While classical computational approaches cannot perform like-for-like computations of quantum systems beyond a certain scale, classical high-performance computing (HPC) may nevertheless be useful for precisely these characterization and certification tasks. By developing open-source customized algorithms using high-performance computing, we perform quantum tomography on a megascale quantum photonic detector covering a Hilbert space of 106. This requires finding 108 elements of the matrix corresponding to the positive operator valued measure (POVM), the quantum description of the detector, and is achieved in minutes of computation time. Moreover, by exploiting the structure of the problem, we achieve highly efficient parallel scaling, paving the way for quantum objects up to a system size of 1012 elements to be reconstructed using this method. In general, this shows that a consistent quantum mechanical description of quantum phenomena is applicable at everyday scales. More concretely, this enables the reconstruction of large-scale quantum sources, processes and detectors used in computation and sampling tasks, which may be necessary to prove their nonclassical character or quantum computational advantage.

Tight conic approximation of testing regions for quantum statistical models and measurements

Quantum statistical models (i.e., families of normalized density matrices) and quantum measurements (i.e., positive operator-valued measures) can be regarded as linear maps: the former, mapping the space of effects to the space of probability distributions; the latter, mapping the space of states to the space of probability distributions. The images of such linear maps are called the testing regions of the corresponding model or measurement. Testing regions are notoriously impractical to treat analytically in the quantum case. Our first result is to provide an implicit outer approximation of the testing region of any given quantum statistical model or measurement in any finite dimension: namely, a region in probability space that contains the desired image, but is defined implicitly, using a formula that depends only on the given model or measurement. The outer approximation that we construct is minimal among all such outer approximations, and close, in the sense that it becomes the maximal inner approximation up to a constant scaling factor. Finally, we apply our approximation to provide sufficient conditions, that can be tested in a semi-device-independent way, for the ability to transform one quantum statistical model or measurement into another.

Universal programmable unambiguous discrimination of two unknown states with two copies

We investigate the universal programmable unambiguous discrimination of two unknown quantum states with two copies. We first present the optimal positive-operator valued measure (POVM) elements that determine the individual success probabilities. In terms of the positivity of an inconclusive operator, we derive the average success probability, which is higher than the average success probability of discriminating two unknown states with one copy given in the reference [Bergou and Hillery, Phys. Rev. Lett. 94, 160501 (2005)].

Structure of resource theory of block coherence

Emerging from the superposition principle, the resource theory of coherence plays a crucial role in many information-processing tasks. Recently, a generalization to this resource theory was investigated with respect to arbitrary positive operator valued measurement (POVM) based on Naimark's dilation theorem. Here, we introduce the notion of Block Incoherent Operations (BIO), Strictly Block Incoherent Operations (SBIO) and Physically Block Incoherent Operations (PBIO) and provide an analytical expression for Kraus operators of these operations to have a better understanding of the resource theory of block coherence which in turn gives a more clear picture of POVM based resource theory of coherence. A dilation theorem corresponding to SBIO has been introduced to enlighten the proper physical interpretation of this operation. These free operations will be helpful in finding out the conditions of state transformations and could be implemented in various protocols. For a transparent view of this resource theory, we have successfully introduced the concept of state transformation under SBIO.

Assisted cloning of an unknown shared quantum state.

We first propose a novel protocol to realize quantum cloning of an arbitrary unknown shared state with assistance offered by a state preparer. The initial phase of this protocol involves the utilization of quantum teleportation (QT), enabling the transfer of quantum information from an arbitrary number of senders to another arbitrary number of receivers through a maximally entangled GHZ-type state serving as a network channel, without centralizing the information at any specific location. In the second stage of this protocol, the state preparer performs a special single-qubit projective measurement and multiple Z-basis measurements and then communicates a number of classical bits corresponding to measurement results, the perfect copy or orthogonal-complementing copy of an unknown shared state can be produced at senders hands. Then, using a non-maximally entangled GHZ-type state instead of the aforementioned quantum channel, we extend the proposed protocol from three perspectives: projective measurement, positive operator-value measurement (POVM), and a single generalized Bell-state measurement. Our schemes can relay quantum information over a network without requiring fully trusted central or intermediate nodes, and none of participants can fully access the information.

A New Method Based on Locally Optimal Step Length in Accelerated Gradient Descent for Quantum State Tomography.

Quantum state tomography (QST) is one of the key steps in determining the state of the quantum system, which is essential for understanding and controlling it. With statistical data from measurements and Positive Operator-Valued Measures (POVMs), the goal of QST is to find a density operator that best fits the measurement data. Several optimization-based methods have been proposed for QST, and one of the most successful approaches is based on Accelerated Gradient Descent (AGD) with fixed step length. While AGD with fixed step size is easy to implement, it is computationally inefficient when the computational time required to calculate the gradient is high. In this paper, we propose a new optimal method for step-length adaptation, which results in a much faster version of AGD for QST. Numerical results confirm that the proposed method is much more time-efficient than other similar methods due to the optimized step size.

Estimating Molecular Thermal Averages with the Quantum Equation of Motion and Informationally Complete Measurements.

By leveraging the Variational Quantum Eigensolver (VQE), the "quantum equation of motion" (qEOM) method established itself as a promising tool for quantum chemistry on near-term quantum computers and has been used extensively to estimate molecular excited states. Here, we explore a novel application of this method, employing it to compute thermal averages of quantum systems, specifically molecules like ethylene and butadiene. A drawback of qEOM is that it requires measuring the expectation values of a large number of observables on the ground state of the system, and the number of necessary measurements can become a bottleneck of the method. In this work, we focus on measurements through informationally complete positive operator-valued measures (IC-POVMs) to achieve a reduction in the measurement overheads by estimating different observables of interest through the measurement of a single set of POVMs. We show with numerical simulations that the qEOM combined with IC-POVM measurements ensures satisfactory accuracy in the reconstruction of the thermal state with a reasonable number of shots.

Quantum tomography of a photon-number-resolving detector based on continuous time multiplexing: memory effect

We address the characterization of a photon-number-resolving detector based on continuous time multiplexing. We develop and verify a mathematical model of this detector and explore various approaches to its quantum tomography. Additionally, we encounter a memory effect and discuss the feasibility of describing this type of detector using a transfer matrix or positive operator-valued measure (POVM) decomposition.

Informationally overcomplete measurements from generalized equiangular tight frames

Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.

High-Fidelity, Multiqubit Generalized Measurements with Dynamic Circuits

Generalized measurements, also called positive operator-valued measures (POVMs), can offer advantages over projective measurements in various quantum information tasks. Here, we realize a generalized measurement of one and two superconducting qubits with high fidelity and in a single experimental setting. To do so, we propose a hybrid method, the “Naimark-terminated binary tree,” based on a hybridization of Naimark’s dilation and binary tree techniques that leverages emerging hardware capabilities for midcircuit measurements and feed-forward control. Furthermore, we showcase a highly effective use of approximate compiling to enhance POVM fidelity in noisy conditions. We argue that our hybrid method scales better toward larger system sizes than its constituent methods and demonstrate its advantage by performing detector tomography of symmetric, informationally complete POVM (SIC POVM). Detector fidelity is further improved through a composite error-mitigation strategy that incorporates twirling and a newly devised conditional readout error mitigation. Looking forward, we expect improvements in approximate compilation and hardware noise for dynamic circuits to enable generalized measurements of larger multiqubit POVMs on superconducting qubits. Published by the American Physical Society 2024

Performance of entanglement purification including maximally entangled mixed states

Entanglement between distant quantum systems is a critical resource for implementing quantum communication. This property is affected by external agents and can be restored by employing efficient entanglement purification protocols. In this work, we propose an entanglement purification protocol based on two entangling two-qubit operations that replace the usual controlled-NOT (CNOT) gate. These operations arise from a generalised quantum measurement and can be understood as measurement operators in a positive operator-valued measure. Furthermore, two variants of the core protocol are introduced and their performances are studied in terms of the overall success probability of reaching a Bell state and the number of purifiable states. Based on rank-two states, we can obtain analytical expressions for the success probability that we extend and refine using numerical calculations to the case of maximally entangled states. We also consider more general rank-three states to show that our procedure is in general more convenient compared to purification protocols based on Bell diagonal states. Finally, we test the protocols using initial random states. In all cases, we find a better performance and a larger amount of purifiable states using our schemes compared to the CNOT-based protocol.

Group theoretical classification of SIC-POVMs

The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions ⩽ 151 and many larger dimensions as high as 39604 . All known solutions with the exception of the Hoggar solutions are covariant with respect to the Weyl-Heisenberg group and in the case of dimension 3, it has been proven that all SIC-POVMs are Weyl-Heisenberg group covariant. In this work, we explore this by SIC-POVMs in dimensions 4–7 without the assumption of group covariance. We introduce two functions with which SIC-POVM Gram matrices can be generated without the group covariance constraint, and analytically show that the SIC-POVM Gram matrices exist on critical points of the surfaces defined by the two functions on a subspace of Hermitian matrices. In dimensions 4–7, all known SIC-POVM Gram matrices lie in disjoint continuous sets of solution. Thus, we define an equivalent class that relates Gram matrices based on the trivial symmetries of the two functions. In dimensions 4–7, we generated {1.7×106,1.1×105,169,50} Gram matrices, respectively. For each of the Gram matrices, we generate the symmetry group of their respective disjoint sets. In all cases, the symmetry group was isomorphic to a subgroup of the Clifford group containing the Weyl-Heisenberg group matrices and the order-3 unitaries. In dimensions 4–6, All constructed solutions belonged to a single equivalence class of Gram matrices whereas in dimension 7, we find two distinct families of SIC-POVMs. In dimensions 4 and 5, the absence of new classes of SIC-POVM strongly suggests that the functions do not have a non-trivial symmetry. Furthermore, in all the dimensions tested, the only equivalence classes found correspond to the distinct stabilizers of fiducial vectors of each dimension.

Families of Schmidt-number witnesses for high dimensional quantum states

Higher dimensional entangled states demonstrate significant advantages in quantum information processing tasks. The Schmidt number is a quantity of the entanglement dimension of a bipartite state. Here we build families of k-positive maps from the symmetric information complete positive operator-valued measurements and mutually unbiased bases, and we also present the Schmidt number witnesses, correspondingly. At last, based on the witnesses obtained from mutually unbiased bases, we show the distance between a bipartite state and the set of states with a Schmidt number less than k.

Relative Entropy of Coherence Quantifies Performance in Bayesian Metrology

The ability of quantum states to be in superposition is one of the key features that sets them apart from the classical world. This “coherence” is rigorously quantified by resource theories, which aim to understand how such properties may be exploited in quantum technologies. There has been much research on what the resource theory of coherence can reveal about quantum metrology, almost all of which has been from the viewpoint of Fisher information. We prove, however, that the relative entropy of coherence, and its recent generalization to positive operator-valued measures (POVMs), naturally quantify the performance of Bayesian metrology. In particular, we show how a coherence measure can be applied to an ensemble of states. We then prove that during parameter estimation, the ensemble relative entropy of coherence (C) is equal to the difference between the optimal Holevo information (X), and the mutual information attained by a measurement (I). We call this relation the CXI equality. The ensemble coherence lets us visualize how much information is locked away in superposition and hence is inaccessible with a given measurement scheme and quantifies the advantage that would be gained by using a joint measurement on multiple states. Our results hold regardless of how the parameter is encoded in the state, encompassing unitary, dissipative, and discrete settings. We consider both projective measurements and general POVMs. This work suggests new directions for research in coherence, provides a novel operation interpretation for the relative entropy of coherence and its POVM generalization, and introduces a new tool to study the role of quantum features in metrology. Published by the American Physical Society 2024

Compatibility of Generalized Noisy Qubit Measurements.

It is a crucial feature of quantum mechanics that not all measurements are compatible with each other. However, if measurements suffer from noise they may lose their incompatibility. Here, we consider the effect of white noise and determine the critical visibility such that all qubit measurements, i.e., all positive operator-valued measures (POVMs), become compatible, i.e., jointly measurable. In addition, we apply our methods to quantum steering and Bell nonlocality. We obtain a tight local hidden state model for two-qubit Werner states of visibility 1/2. This determines the exact steering bound for two-qubit Werner states and also provides a local hidden variable model that improves on previously known models. Interestingly, this proves that POVMs are not more powerful than projective measurements to demonstrate quantum steering for these states.

Exact Steering Bound for Two-Qubit Werner States.

Whether positive operator-valued measures (POVMs) provide advantages in demonstrating Bell nonlocality has remained unknown, even in the simple scenario of Einstein-Podolsky-Rosen steering with noisy singlet state, known as Werner states. Here we resolve this long-standing open problem by constructing a local hidden state model for Werner states with any visibility r≤1/2 under general POVMs, thereby closing the so-called Werner gap. This construction is based on an exact measurement compatibility model for the set of all noisy POVMs and also provides a local hidden variable model for a larger range of Werner states than previously known.

Dual-frame optimization for informationally complete quantum measurements

Randomized measurement protocols such as classical shadows represent powerful resources for quantum technologies, with applications ranging from quantum state characterization and process tomography to machine learning and error mitigation. Recently, the notion of measurement dual frames, in which classical shadows are generalized to dual operators of positive operator-valued measure (POVM) effects, resurfaced in the literature. This brought attention to additional degrees of freedom in the postprocessing stage of randomized measurements that are often neglected by established techniques. In this work, we leverage dual frames to construct improved observable estimators from informationally complete measurement samples. We introduce novel classes of parametrized frame superoperators and optimization-free dual frames based on empirical frequencies, which offer advantages over their canonical counterparts while retaining computational efficiency. Remarkably, this comes at almost no quantum or classical cost, thus rendering dual frame optimization a valuable addition to the randomized measurement toolbox. Published by the American Physical Society 2024