Complete Positive Operator-valued Measures Research Articles

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.

Dynamical quantum state tomography with time-dependent channels

Abstract In this paper, we establish a dynamical quantum state tomography framework. Under this framework, it is feasible to obtain complete knowledge of any unknown state of a d-level system via only one operator of a special type of positive operator-valued measure (POVM) in dimension d. We define a new channel, referred to as the time-dependent average channel. Utilizing this channel, we show that we can acquire a collection of projective operators that is informationally complete (IC) and thus obtain the corresponding informationally complete POVMs (IC-POVMs). Zauner conjectured that for any dimension d there exists a fiducial vector, such that all remaining d 2 − 1 elements of the desired symmetric informationally complete POVM (SIC-POVM) can be obtained by acting on said vector with unitary matrices representing elements of the Weyl-Heisenberg group. We show that under certain condition, it is possible to obtain infinite families of projective operators that are IC, and obtain infinite families of corresponding IC-POVMs; otherwise, Zauner’s conjecture is incorrect. We also show how to simulate a SIC-POVM on any unknown quantum state by using the time-dependent average channel.

Operator theory and quantum information theory—POVM and entangled state

This survey paper introduces several topics in quantum information theory, especially positive operator valued measures (POVMs) and entangled states. POVMs are needed to mathematically express measurements in the quantum world, and there are two special POVMs, named symmetric informationally complete POVMs and mutually unbiased bases. These two POVMs have very beautiful symmetry. We give the definitions and examples of these POVMs and discuss the existence or non-existence problem of these POVMs. Entanglement is some kind of relationship between two or more quanta. When several quanta have entanglement, the state of the quanta is called an entangled state. In this paper, quantum teleportation is introduced as an application using entangled states. Moreover, we investigate the problem of determining whether a given quantum state is an entangled state.

Quantum detection problem for fusion frames

A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.

Characterizing the uncertainty relation via a class of measurements

The connection between uncertainty and entanglement is a prevalent topic in quantum information processing. Based on a broad class of informationally complete symmetric measurements, which can be viewed as a common generalization of symmetric, informationally complete positive operator-valued measures and mutually unbiased bases, a conical 2-design is calculated. This design plays a crucial role in quantum measurement theory. Subsequently, the relation between the uncertainty and the entanglement for a set of measurements is portrayed using conditional collision entropy. Furthermore, a tighter lower bound of the uncertainty relation is discussed according to the characterization of the entropic bound. Finally, the relation is applied to entanglement witnesses. It is demonstrated that the present results are unified and comprehensive.

Enhancing some separability criteria in many-body quantum systems

Quantum measurements based on mutually unbiased bases, mutually unbiased measurements and general symmetric informationally complete positive operator valued measurements, which are commonly used in quantum information and quantum computation. In this paper, by a broad class of informationally complete symmetric measurements and correlation matrices, some separability criteria are proposed for different quantum systems that are more robust and comprehensive than the corresponding criteria previously introduced L M Lai, T Li, S M Fei, and Z X Wang, (2018 Quantum Inf. Process., 17, 314); J W Shang, A Asadian, and H J Zhu, (2018 Phys. Rev. A 98, 022309). Furthermore, we extend the separability criteria of tripartite quantum systems to many-body systems. The proposed entanglement criteria may further enrich the relationship between quantum entanglement and measurement.

Detecting genuine tripartite entanglement via general symmetric informationally complete measurements.

We study multipartite entanglement and genuine tripartite entanglement based on general symmetric informationally complete positive operator valued measurements (GSIC-POVMs). By representing the bipartite density matrices in terms of GSIC-POVMs, we obtain the lower bound of the sum of squares of the corresponding probability. We then construct a special matrix with the correlation probability of GSIC-POVMs to derive useful and operational criteria to detect genuine tripartite entanglement. We also generalize the results to obtain a sufficient criterion to detect entanglement for multipartite quantum states in arbitrary dimensions. Detailed examples show that the new method can detect more entangled and genuine entangled states than previous criteria.

Generalized polarization measurement and its connection with information energy

We generalize the polarization measurement to the projective measurements of mutually unbiased bases (MUBs). Based on this result, we reveal a direct connection between polarization measurement and information energy function. We also express the polarization measurement by means of symmetric informationally complete positive operator-valued measures (SIC-POVMs). We present an argument suggesting that polarization measurement by means of SIC-POVMs requires fewer resources than the projective measurements of MUBs.

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.

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.

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.

Experimental Single-Setting Quantum State Tomography

Quantum computers solve ever more complex tasks using steadily growing system sizes. Characterizing these quantum systems is vital, yet becoming increasingly challenging. The gold-standard is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge. Measurement and classical computing costs, however, increase exponentially in the system size - a bottleneck given the scale of existing and near-term quantum devices. Here, we demonstrate a scalable and practical QST approach that uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVM). We implement these nonorthogonal measurements on an ion trap device by utilizing more energy levels in each ion - without ancilla qubits. More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, providing full tomographic information in every shot. Combining this SIC tomography with the recently developed randomized measurement toolbox ("classical shadows") proves to be a powerful combination. SIC tomography alleviates the need for choosing measurement settings at random ("derandomization"), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods. The latter enables in-depth entanglement studies, which we experimentally showcase on a 5-qubit absolutely maximally entangled (AME) state. Moreover, the fact that the full tomography information is available in every shot enables online QST in real time. We demonstrate this on an 8-qubit entangled state, as well as for fast state identification. All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.

Complete positive operator-valued measure description of multichannel quantum electro-optic sampling with monochromatic field modes

We propose a multichannel version of quantum electro-optic sampling involving monochromatic field modes. It allows for multiple simultaneous measurements of arbitrarily many $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ field quadratures for a single quantum-state copy, while independently tuning the interaction strengths at each channel. In contrast to standard electro-optic sampling, the sampled midinfrared (MIR) mode undergoes a nonlinear interaction with multiple near-infrared pump beams. We present a complete positive operator-valued measure description for quantum states in the MIR mode. The probability distribution of the electro-optic signal outcomes is shown to be related to an $s$-parametrized phase-space quasiprobability distribution of the indirectly measured MIR state, with the parameter $s$ depending solely on the quantities characterizing the nonlinear interaction. Furthermore, we show that the quasiprobability distributions for the sampled and postmeasurement states are related to each other through a renormalization and a change in the parametrization. This result is then used to demonstrate that two consecutive measurements of both $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ quadratures can outperform eight-port homodyne detection.

Quantum Tomography of Two-Qutrit Werner States

In this article, we introduce a framework for two-qutrit Werner states tomography with Gaussian noise. The measurement scheme is based on the symmetric, informationally complete positive operator-valued measure. To make the framework realistic, we impose the Gaussian noise on the measured states numbers. Through numerical simulation, we successfully reconstructed the two-qutrit Werner states in various experimental scenarios and analyzed the optimal scenario from four aspects: fidelity, purity, entanglement, and coherence.

Ancilla-free implementation of generalized measurements for qubits embedded in a qudit space

Informationally complete (IC) positive operator-valued measures (POVMs) are generalized quantum measurements that offer advantages over the standard computational basis readout of qubits. For instance, IC-POVMs enable efficient extraction of operator expectation values, a crucial step in many quantum algorithms. POVM measurements are typically implemented by coupling one additional ancilla qubit to each logical qubit, thus imposing high demands on the device size and connectivity. Here, we show how to implement a general class of IC-POVMs without ancilla qubits. We exploit the higher-dimensional Hilbert space of a qudit in which qubits are often encoded. POVMs can then be realized by coupling each qubit to two of the available qudit states, followed by a projective measurement. We develop the required control pulse sequences and numerically establish their feasibility for superconducting transmon qubits through pulse-level simulations. Finally, we present an experimental demonstration of a qudit-space POVM measurement on IBM Quantum hardware. This paves the way to making POVM measurements broadly available to quantum computing applications.

Efficiency of photonic state tomography affected by fiber attenuation

In this article, we investigate the efficiency of photonic state tomography in the presence of fiber attenuation. The theoretical formalism of the photon loss is provided by implementing methods from the theory of open quantum systems. The quantum state is reconstructed from photon counts obtained for symmetric informationally complete positive operator-valued measures. The number of photons that reach the detectors is numerically modeled by the binomial distribution, which describes the loss of light caused by the medium. This approach allows us to study the quality of state tomography versus the length of the fiber. In particular, we focus on entangled qubits and qutrits, which are sent through fibers of different lengths. The amount of entanglement detected by the measurement scheme is quantified and presented on graphs. The results demonstrate how the quality of photonic tomography depends on the distance between the source and the receiver.

Learning to Measure: Adaptive Informationally Complete Generalized Measurements for Quantum Algorithms

Many prominent quantum computing algorithms with applications in fields such as chemistry and materials science require a large number of measurements, which represents an important roadblock for future real-world use cases. We introduce a novel approach to tackle this problem through an adaptive measurement scheme. We present an algorithm that optimizes informationally complete positive operator-valued measurements (POVMs) on the fly in order to minimize the statistical fluctuations in the estimation of relevant cost functions. We show its advantage by improving the efficiency of the variational quantum eigensolver in calculating ground-state energies of molecular Hamiltonians with extensive numerical simulations. Our results indicate that the proposed method is competitive with state-of-the-art measurement-reduction approaches in terms of efficiency. In addition, the informational completeness of the approach offers a crucial advantage, as the measurement data can be reused to infer other quantities of interest. We demonstrate the feasibility of this prospect by reusing ground-state energy-estimation data to perform high-fidelity reduced state tomography.

Shadow tomography based on informationally complete positive operator-valued measure

Recently introduced shadow tomography protocols use ``classical shadows'' of quantum states to predict many target functions of an unknown quantum state. Unlike full quantum state tomography, shadow tomography does not insist on accurate recovery of the density matrix for high rank mixed states. Yet, such a protocol makes multiple accurate predictions with high confidence, based on a moderate number of quantum measurements. One particular influential algorithm, proposed by Huang et al. [Huang, Kueng, and Preskill, Nat. Phys. 16, 1050 (2020)], requires additional circuits for performing certain random unitary transformations. In this paper, we avoid these transformations but employ an arbitrary informationally complete positive operator-valued measure and show that such a procedure can compute $k$-bit correlation functions for quantum states reliably. We also show that, for this application, we do not need the median of means procedure of Huang et al. Finally, we discuss the contrast between the computation of correlation functions and fidelity of reconstruction of low rank density matrices.

Towards a classification of incomplete Gabor POVMs in d?>

Every (full) finite Gabor system generated by a unit-norm vector is a finite unit-norm tight frame (FUNTF) and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the corresponding rank 1 matrices form a basis for the space of matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process, we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank 1 matrices generated by a finite Gabor frame. We also use these results to construct some sets of unit vectors in with a relatively smaller number of distinct inner products.