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How to compute the probability distribution of a detection time, i.e., of the time which a detector registers as the arrival time of a quantum particle, is a long-debated problem. In this regard, Bohmian mechanics provides in a straightforward way the distribution of the time at which the particle actually does arrive at a given surface in 3-space in the absence of detectors. However, as we discuss here, since the presence of detectors can change the evolution of the wave function and thus the particle trajectories, it cannot be taken for granted that the arrival time of the Bohmian trajectories in the absence of detectors agrees with the one in the presence of detectors, and even less with the detection time. In particular, we explain why certain distributions that Das and Dürr (Sci. Rep. 9: 2242, 2019) presented as the distribution of the detection time in a case with spin, based on assuming that all three times mentioned coincide, are actually not what Bohmian mechanics predicts. [ABSTRACT FROM AUTHOR]

We introduce a generalization of symmetric measurements to collections of unequinumerous positive, operator-valued measures (POVMs). This provides a uniform description of objects that are more general than symmetric, informationally complete POVMs and mutually unbiased bases, but at the same time less destructive and more noise tolerant. For informationally complete sets, we propose construction methods from orthonormal Hermitian operator bases. The correspondence between operator bases and measurements can be as high as one-to-four, with a one-to-one correspondence following only under additional assumptions. Importantly, it turns out that some of the symmetry properties, lost in the process of generalization, can be recovered without fixing the same number of elements for all POVMs. In particular, for a wide class of unequinumerous symmetric measurements that are conical 2-designs, we derive the index of coincidence, entropic uncertainty relations, and separability criteria for bipartite quantum states. [ABSTRACT FROM AUTHOR]

We generalize the notion of joint measurability to continuous variable systems by extending a recently introduced compression algorithm of quantum measurements to this realm. The extension results in a property that asks for the minimal dimensional quantum system required for representing a given set of quantum measurements. To illustrate the concept, we show that the canonical pair of position and momentum is completely incompressible. We translate the concept of measurement compression to the realm of quantum correlations, where it results in a generalization of continuous variable quantum steering. In contrast to the steering scenario, which detects entanglement, the generalization detects the dimensionality of entanglement. We illustrate the bridge between the concepts by showing that an analogue of the original EPR argument is genuinely infinite-dimensional with respect to our figure of merit, and that a fundamental discrete variable result on preparability of unsteerable state assemblages with separable states does not directly carry over to the continuous variable setting. We further prove a representation result for partially entanglement breaking channels that can be of independent interest. [ABSTRACT FROM AUTHOR]

The accessible information of general signal states is obtained by performing a generalized measurement. In the case that the signal alphabet consists of two states of a qubit system, it is proved that a von Neumann (orthogonal) measurement is sufficient to reach the accessible information. [ABSTRACT FROM AUTHOR]

The purpose of this article is to further investigate the assisted cloning of a d-dimensional arbitrary unknown state by utilizing quantum teleportation (QT) and remote state preparation (RST) techniques. We first propose a scheme for cloning an arbitrary unknown single-qudit state with assistance from a state preparer. In this scheme, the maximally and non-maximally bipartite d-dimensional states are used as quantum channels, respectively. In the first stage of scheme requires usual QT and in the second stage, the state preparer disentangles the left over entangled states by introducing one auxiliary qudit and performing a generalized CNOT, a single-qudit positive operator-valued measurement (POVM) and a single-qudit projective measurement process and conveys some classical messages to different parties so that a copy is produced probabilistically with the unit fidelity. Then by replacing the maximally or non-maximally bipartite d-dimensional entangled channel with a sequence of maximally or non-maximally bipartite d-dimensional entangled states as a quantum channel, we can extend the above scheme to the case of cloning a d-dimensional arbitrary unknown m-qudit state with the unit fidelity and a certain probability in the help of the state preparer. [ABSTRACT FROM AUTHOR]

Existing quantum private query (QPQ) proposals are mostly for single-user scenarios, while recent multi-user proposals assume the existence of a semi-trusted server and only consider the retrieval of the items of common interest. To address these limitations, here we propose a semi-device independent multi-user QPQ scheme exploiting the verification of distributed multipartite GHZ states and the self-testing of a specific class of POVM operators used by the clients. Our scheme allows each user to retrieve different items simultaneously while keeping their choices confidential and also enables the users to retrieve optimal raw key bits during oblivious key generation phase. Additionally, we perform a formal evaluation of the security concerns and are able to determine the maximum likelihood of cheating for both the server and users. [ABSTRACT FROM AUTHOR]

It is shown that in presence of certain external fields a well-defined self-adjoint time operator exists, satisfying the standard canonical commutation relations with the Hamiltonian. Examples include uniform electric and gravitational fields with nonrelativistic and relativistic Hamiltonians. The physical intepretation of these operators is proposed in terms of time of arrival in the momentum space. [ABSTRACT FROM AUTHOR]

We present a principle to determine whether a matrix is unitary. Through this principle, we can exclude some zero entries distribution in unitary matrices. As a result, we can construct higher order unitary matrices with a given zero entries distribution. We apply our results to construct CP-equivalent gates. [ABSTRACT FROM AUTHOR]

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 region σg > 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 region σ̃g>0 in momentum space. The usual generalised uncertainty principle and extended uncertainty principle are recovered by setting σg≃ℏG/c3, the Planck length, and σ̃g≃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.

Given an ontological model of a quantum system, a “genuine measurement,” as opposed to a quantum measurement, means an experiment that determines the value of a beable, i.e., of a variable that, according to the model, has an actual value in nature before the experiment. We prove a theorem showing that in every ontological model, it is impossible to measure all beables. Put differently, there is no experiment that would reliably determine the ontic state. This result shows that the positivistic idea that a physical theory should only involve observable quantities is too optimistic. [ABSTRACT FROM AUTHOR]

By utilizing the non-maximally entangled four-qubit cluster states as the quantum channel, we first propose a hierarchical quantum information splitting scheme of arbitrary three-qubit states among three agents with a certain probability. Then we generalize the scheme to arbitrary multi-qubit states. Hierarchy is reflected on the different abilities of agents to restore the target state. The high-grade agent only needs the help of one low-grade agent, while the low-grade agent requires all the other agents' assistance. The designated receiver performs positive operator-valued measurement (POVM) which is elaborately constructed with the aid of Hadamard matrix. It is worth mentioning that a general expression of recovery operation is derived to disclose the relationship with measurement outcomes. Moreover, the scheme is extended to multiple agents by means of the symmetry of cluster states. [ABSTRACT FROM AUTHOR]

Large-scale entangled states are important physical resources, which can be generated from small-scale ones via quantum state fusion. Measurement in fusion operation will cause particle and efficiency loss, so a fusion scheme without qubit loss is expected to be the most efficient one, but counter-intuitively it is not the case. So in this paper we reveal the essence of the fusion mechanism, and answer this counter-intuitive question. The results show that any fusion mechanism is basically a generalized measurement (POVM), and the POVM essence shows why the qubit-loss-free fusion scheme for standard W state is not the most efficient one. A more efficient fusion mechanism targeting at large-scale W state is proposed, where two unequally-weighted W states are fed into the fusion mechanism, and it beats all the current fusion schemes. The universality of the POVM essence of the fusion mechanism makes it applicable to any multipartite entangled state. • The essence of quantum state fusion mechanism was revealed: generalized measurement. • The mechanism is applicable to any kind of multipartite entangled states. • Reveal why the qubit-loss scheme is more efficient than the qubit-loss-free one. • More efficient fusion schemes for generating large-scale W/W-like states. • Beat all the current fusion schemes by a huge margin in the resource cost. [ABSTRACT FROM AUTHOR]

Heisenberg's uncertainty relations have played an essential role in quantum physics since its very beginning. The uncertainty relations in the modern quantum formalism have become a fundamental limitation on the joint measurements of general quantum mechanical observables, going much beyond the original discussion of the trade-off between knowing a particle's position and momentum. Recently, the uncertainty relations have generated a considerable amount of lively debate as a result of the new inequalities proposed as extensions of the original uncertainty relations. We report an experimental test of one of the new Heisenberg's uncertainty relations using a single 40Ca+ ion trapped in a harmonic potential. By performing unitary operations under carrier transitions, we verify the uncertainty relation proposed by Busch, Lahti, and Werner (BLW) based on a general error-trade-off relation for joint measurements on two compatible observables. The positive operator-valued measure, required by the compatible observables, is constructed by single-qubit operations, and the lower bound of the uncertainty, as observed, is satisfied in a state-independent manner. Our results provide the first evidence confirming the BLW-formulated uncertainty at a single-spin level and will stimulate broad interests in various fields associated with quantum mechanics.

Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, number theory and probability. Among our results are some equivalent conditions for unbiased MICs, a characterization of rank-1 MICs through the Hadamard product, several ways in which immediate properties of MICs capture the abandonment of classical phase space intuitions, and a numerical study of MIC Gram matrix spectra. We also present, to our knowledge, the first example of an unbiased rank-1 MIC which is not group covariant. This work provides further context to the discovery that the symmetric informationally complete quantum measurements (SICs) are in many ways optimal among MICs. In a deep sense, the ideal measurements of quantum physics are not orthogonal bases. [ABSTRACT FROM AUTHOR]

We define a p-norm in the context of quantum random variables, measurable operator-valued functions with respect to a positive operator-valued measure. This norm leads to a operator-valued L p space that is shown to be complete. Various other norm candidates are considered as well as generalizations of Hölder's inequality to this new context. [ABSTRACT FROM AUTHOR]

We consider error correction, based on the theory of non-commutative graphs, for a model of a qubit interacting with quantum oscillator. The dynamics of the composite system is governed by the Schrödinger equation which generates positive operator-valued measure (POVM) for the system dynamics. We construct a quantum channel generating the non-commutative graph as a linear envelope of the POVM. The idea is based on applying a generalized version of a quantum channel using the apparatus of von Neumann algebras. The results are analyzes for a non-commutative graph generated by a qubit interacting with quantum oscillator. For this model the quantum anticlique which determines the error correcting subspace has an explicit expression. [ABSTRACT FROM AUTHOR]

Coherence theory has been studied for many years. It plays an important role in quantum information and quantum computation. Recently, Bischof established POVM-based coherence measures and POVM-incoherent operations that coincide for the case of von Neumann measurements with their counterparts in standard coherence theory. Following this theoretical framework, we study whether the geometric coherence based on fidelity meet the constraint for quantifying coherence. [ABSTRACT FROM AUTHOR]

By utilizing the non-maximally entangled four-qubit cluster states as the quantum channel, we first propose a hierarchical quantum information splitting scheme of arbitrary three-qubit states among three agents with a certain probability. Then we generalize the scheme to arbitrary multi-qubit states. Hierarchy is reflected on the different abilities of agents to restore the target state. The high-grade agent only needs the help of one low-grade agent, while the low-grade agent requires all the other agents’ assistance. The designated receiver performs positive operator-valued measurement (POVM) which is elaborately constructed with the aid of Hadamard matrix. It is worth mentioning that a general expression of recovery operation is derived to disclose the relationship with measurement outcomes. Moreover, the scheme is extended to multiple agents by means of the symmetry of cluster states.

The article establishes a framework for dynamic generation of informationally complete POVMs in quantum state tomography. Assuming that the evolution of a quantum system is given by a dynamical map in the Kraus representation, one can switch to the Heisenberg picture and define the measurements in the time domain. Consequently, starting with an incomplete set of positive operators, one can obtain sufficient information for quantum state reconstruction by multiple measurements. The framework has been demonstrated on qubits and qutrits. For some types of dynamical maps, it suffices to initially have one measurement operator. The results demonstrate that quantum state tomography is feasible even with limited measurement potential. [ABSTRACT FROM AUTHOR]

The Mach–Zehnder interferometer is a particularly simple device for demonstrating the wave-particle duality. This duality is particularly exhibited in the quantum delayed choice experiment where one realizes a superposition of wave- and particle-like behaviors. Considering this experiment as a generalized measurement described by a positive operator–valued measure (POVM), it realizes a joint measurement of path and interference obeying some duality relations. [ABSTRACT FROM AUTHOR]

This study aimed to extend the use of video-based Instructions (VBI) by using point-of-view video modeling (POVM), schema based-word problem solving method and peer tutoring to teach addition-subtraction word problem solving to students with ADHD. When core mathematic instruction is not enough to address skill deficits of students with ADHD, more intensive interventions are required to support student learning and comprehensive interventions with strong instructional components may use to help improve achievement of students with mathematics difficulties. The video presentation of the instruction via POVM allows for development of procedural knowledge as participants can view instruction of problem-solving steps. A single-case multiple probe across participants, experimental design was used with three participants. As shown on the single subject graphic all three participants not only demonstrated mastery of the skill, but also maintained the skill several weeks after the intervention ended and they were able to teach the skill to their tutees. This means participants completed the three different types of word addition and subtraction problems independently. All participants were successful and shared that they enjoyed learning about word problem solving using the video instruction, supporting the social validity of this intervention. [ABSTRACT FROM AUTHOR]

In this work we propose a scheme in the context of superconducting cavities to unambiguously discriminate non-orthogonal quantum field states when one of them is one of the four Bell states. The present work, which generalizes an earlier proposal [1] dealing with non-entangled states, also makes use of a single high Q cavity, a sample of three-level atoms, Ramsey zones and selective detectors, thus consisting of an oversimplification from the experimental view. • No explicit POVM technique is used to unambiguously discriminate nonorthogonal Bell states into a bimodal high-Q cavity. • Our proposal circumvents the impossibility to directly project the cavity states onto the computational Fock states. • Our protocol without explicit POVM achieves a rate of success probability greater than the direct projective techniques. [ABSTRACT FROM AUTHOR]

In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions, which are obtained via minimal informationally complete quantum measurements. The suggested method for probability representation of quantum dynamics preserves the tensor product structure, which makes it favourable for the analysis of multi-qubit systems. A key advantage of the suggested approach is that minimal informationally complete positive operator-valued measures (MIC-POVMs) are easier to construct in comparison with their symmetric versions (SIC-POVMs). We establish a correspondence between the standard quantum-mechanical formalism and the MIC-POVM-based probability formalism. Within the latter approach, we derive equations for the unitary von-Neumann evolution and the Markovian dissipative evolution, which is governed by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generator. We apply the MIC-POVM-based probability representation to the digital quantum computing model. In particular, for the case of spin-1/2 evolution, we demonstrate identifying a transition of a dissipative quantum dynamics to a completely classical-like stochastic dynamics. One of the most important findings is that the MIC-POVM-based probability representation gives more strict requirements for revealing the non-classical character of dissipative quantum dynamics in comparison with the SIC-POVM-based approach. Our results give a physical interpretation of quantum computations and pave a way for exploring the resources of noisy intermediate-scale quantum devices.

We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on various probabilistic aspects of these constructions. Simple ormore elaborate examples illustrate the procedure: circle, two-sphere, plane and half-plane. Links with Positive-Operator Valued Measure (POVM) quantum measurement and quantum statistical inference are sketched.

We present new results concerning simulation of general quantum measurements (POVMs) by projective measurements (PMs) for the task of Unambiguous State Discrimination (USD). We formulate a problem of finding optimal strategy of simulation for given quantum measurement. The problem can be solved for qubit and qutrits measurements by Semi-Definite Programming (SDP) methods.

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

We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the -dimensional subspace of the -dimensional Hilbert space of qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch-Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones. [ABSTRACT FROM AUTHOR]

In this work, we study the wave particle duality and the Aharonov-Bohm effect in a Mach-Zehnder interferometer in a magnetic field which is in a superposition of two opposite directions. In addition, a 'classical' experimental scheme is proposed which mimics such an interferometer. [ABSTRACT FROM AUTHOR]

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

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

We study the weak-convergence properties of random variables generated by unsharp quantum measurements. More precisely, for a sequence of random variables generated by repeated unsharp quantum measurements, we study the limit distribution of relative frequency. We provide a representation theorem for all separable states, showing that the distribution can be well approximated by a mixture of normal distributions. Furthermore, we investigate the convergence rates and show that the relative frequency can stabilize to some constant at best at the rate of order $1/\sqrt{N}$ for all separable inputs. On the other hand, we provide an example of a strictly unsharp quantum measurement where the better rates are achieved by using entangled inputs. This means that in certain cases the noise generated by the measurement process can be suppressed by using entanglement. We deliver our result in the form of quantum information task where the player achieves the goal with certainty in the limiting case by using entangled inputs or fails with certainty by using separable inputs.

We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension 2 (for qubits). In this case, we prove that the entropy is minimal, and hence, the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation, and the structure of invariant polynomials for unitary-antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy, and the quantum dynamical entropy are described. [ABSTRACT FROM AUTHOR]