An operational description of the controlled Markov dynamics of quantum-mechanical system is introduced. The feedback control strategies with regard to the dynamical reduction of quantum states in the course of quantum real-time measurements are discribed in terms of quantum filtering of these states. The concept of sufficient coordinates for the description of the a posteriori quantum states from a given class is introduced, and it is proved that they form a classical Markov process with values in either state operators or state vector space. The general problem of optimal control of a quantum-mechanical system is discussed and the corresponding Bellman equation in the space of sufficient coordinates is derived. The results are illustrated in the example of control of the semigroup dynamics of a quantum system that is instantaneously observed at discrete times and evolves between measurement times according to the Schroedinger equation.
We study the changes if any of the expectation value of a general observable in a quantum system, the difficulties associated with the detection of these changes, and the possible methods for correcting the system through unitary control to maintain a constant average expectation value of the observable.
We introduce the probability distributions describing quantum observables in conventional quantum mechanics and clarify their relations to the tomographic probability distributions describing quantum states. We derive the evolution equation for quantum observables (Heisenberg equation) in the probability representation and give examples of the spin-1/2 (qubit) states and the spin observables. We present quantum channels for qubits in the probability representation.
This paper presents some of the basic properties of conditioned observables in finite-dimensional quantum mechanics. We begin by defining the sequential product of quantum effects and use this to define the sequential product of two observables. The sequential product is then employed to construct the conditioned observable relative to another observable. We then show that conditioning preserves mixtures and post-process of observables. We consider conditioning among three observables and a complement of an observable. Corresponding to an observable, we define an observable operator in a natur…
The concept of intrinsic and operational observables in quantum mechanics is introduced. In any realistic description of a quantum measurement that includes a macroscopic detecting device, it is possible to construct from the statistics of the recorded raw data a set of operational quantities that correspond to the intrinsic quantum mechanical observable.
We represent quantum observables as POVMs (normalized positive operator valued measures) and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group $G$. The value space of such observables is a transitive $G$-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference and time observables.
Here we present an expanded analysis of a model for the manipulation and control of observables in a strongly correlated, many-body system, which was first presented in [McCaul et al., eprint: arXiv:1911.05006]. A field-free, non-linear equation of motion for controlling the expectation value of an essentially arbitrary observable is derived, together with rigorous constraints that determine the limits of controllability. We show that these constraints arise from the physically reasonable assumptions that the system will undergo unitary time evolution, and has enough degrees of freedom for the…
Generalised observables (POM observables) are necessary for representing all possible measurements on a quantum system. Useful algebraic operations such as addition and multiplication are defined for these observables, recovering many advantages of the more restrictive Hermitian operator formalism. Examples include new uncertainty relations and metrics, and optical phase applications.
In this paper we investigate the limits of control for mixed-state quantum systems. The constraint of unitary evolution for non-dissipative quantum systems imposes kinematical bounds on the optimization of arbitrary observables. We summarize our previous results on kinematical bounds and show that these bounds are dynamically realizable for completely controllable systems. Moreover, we establish improved bounds for certain partially controllable systems. Finally, the question of dynamical realizability of the bounds for arbitary partially controllable systems is shown to depend on the accessib…
This paper presents an alternative approach to geometric phases from the observable point of view. Precisely, we introduce the notion of observable-geometric phases, which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. The observable-geometric phases are shown to be connected with the quantum geometry of the observable space evolving according to the Heisenberg equation. They are indeed distinct from Berry's phase \cite{Berry1984, Simon1983} as the system evolves adiabatically. It is shown that the observable-geometric phases can be used to …
The existence of maximally incompatible quantum observables in the sense of a minimal joint measurability region is investigated. Employing the universal quantum cloning device it is argued that only infinite dimensional quantum systems can accommodate maximal incompatibility. It is then shown that two of the most common pairs of complementary observables (position and momentum; number and phase) are maximally incompatible.
Various forms of optimality for quantum observables described as normalized positive operator valued measures (POVMs) are studied in this paper. We give characterizations for observables that determine the values of the measured quantity with probabilistic certainty or a state of the system before or after the measurement. We investigate observables which are free from noise caused by classical post-processing, mixing, or pre-processing of quantum nature. Especially, a complete characterization of pre-processing and post-processing clean observables is given, and necessary and sufficient condi…
This paper considers a generalization of the notion of quantum observables in ontological models of quantum mechanics. Within this framework it is possible to construct physical models where quantum noncommutativity can arise dynamically. Unlike quantum systems, the basic entities in this model have definite properties. Relations with no-go theorems and other hidden variable theories are also discussed.
In Quantum Physics there are circumstances where the direct measurement of particular observables encounters diffculties; in some of these cases, however, its value can be evaluated, i.e. it can be inferred by measuring another observable characterized by perfect correlation with the observable of interest. Though an evaluation is often interpreted as a measurement of the evaluated observable, we prove that the two concepts cannot be identified in Quantum Physics, because the identifcation yields contradictions. Then, we establish the conceptual status of evaluations in Quantum Theory and the …
A subclass of dynamical semigroups induced by the interaction of a quantum system with an environment is introduced. Such semigroups lead to the selection of a stable subalgebra of effective observables. The structure of this subalgebra is completely determined.
We begin by defining mutually unbiased (MU) observables on a finite dimensional Hilbert space. We also consider the more general concept of parts of MU observables. The relationships between MU observables, value-complementary observables and two other conditions involving sequential products of observables are discussed. We next present a special motivating case of MU observables called finite position and momentum observables. These are atomic observables related by a finite Fourier transform. Finite position and momentum observables are employed to give examples of parts of MU observables t…
One of the central features of quantum theory is that there are pairs of quantum observables that cannot be measured simultaneously. This incompatibility of quantum observables is a necessary ingredient in several quantum phenomena, such as measurement uncertainty relations, violation of Bell inequalities and steering. Two quantum observables that admit a simultaneous measurement are, in this respect, classical. A finer classification of classicality can be made by formulating four symmetric relations on the set of observables that are stronger than compatibility; they are broadcastability, on…
Information-Theoretical restrictions on the information transfer are applied to Quantum Measurements. For the measurement of quantum object S by information system O this restrictions are calculated in Algebraic QM formalism as the inference map to the (sub)algebra of O observables.
We address the problem of unambiguous discrimination and identification among quantum observables. We set a general framework and investigate in details the case of qubit observables. In particular, we show that perfect discrimination with two shots is possible only for sharp qubit observables (e.g. Stern-Gerlach apparatuses) associated with mutually orthogonal directions. We also show that for sharp qubit observables associated to nonorthogonal directions unambiguous discrimination with an inconclusive result is always possible.
In the Schr{ö}dinger picture, the state of a quantum system evolves in time and the quantum speed limit describes how fast the state of a quantum system evolves from an initial state to a final state. However, in the Heisenberg picture the observable evolves in time instead of the state vector. Therefore, it is natural to ask how fast an observable evolves in time. This can impose a fundamental bound on the evolution time of the expectation value of quantum mechanical observables. We obtain the quantum speed limit time-bound for observable for closed systems, open quantum systems and arbitrary…
In Quantum Physics it is not always possible to directly perform the measurement of an obsevable; in some of these cases, however, its value can be {\sl detected}, i.e. it can be inferred by measuring {\sl another} observable characterized by perfect correlation with the observable of interest. Though a {\sl detection} is often interpreted as a {\sl measurement} of the detected observable, we prove that the two concepts cannot be identified in Quantum Physics. Furthermore, we establish what meaning and role can be ascribed to detections coherently with Quantum Theory.
We propose a single observable to witness the nonzero quantum discord of an unknown quantum state provided that we have four copies of the state. The expectation value of this observable provides a necessary and sufficient condition for nonzero quantum discord in $2\times N$ systems and a necessary condition in higher finite-dimensional bipartite systems. Furthermore, a nontrivial lower bound of quantum discord can be obtained from this expectation value. The proposed observable can be experimentally measured in exactly the same easy manner as that of the entanglement witness, moreover a quant…
Recently a problem concerning the equivalence of joint measurability and coexistence of quantum observables was solved [15]. In this paper we generalize two known joint measurability results from sharp observables to the class of extreme observables and study relationships between coexistence, joint measurability, and post-processing of quantum observables when an extreme observable is involved. We also discuss another notion of compatibility and provide a counterexample separating this from the former notions.
In this short paper we present the main features of a new quantum programming language proposed recently by Peter Selinger which gives a good idea about the difficulties of constructing a scalable quantum computer. We show how some of these difficulties are related to the contextuality of quantum observables and to the abstract and statistical character of quantun theory (QT). We discuss also, in some detail, the statistical interpretation (SI) of QT and the contextuality of observables indicating the importance of these concepts for the whole domain of quantum information.
Quantum speed limit for observables from quantum asymmetry
Research Center for Quantum Physics, National Research and Innovation Agency（量子物理研究中心，国家研究与创新署）；Department of Physics, University of Maryland, Baltimore County（马里兰大学巴尔的摩分校物理系）；Quantum Science Institute, University of Maryland, Baltimore County（马里兰大学巴尔的摩分校量子科学研究所）；National Quantum Laboratory（国家量子实验室）
We introduce the concepts of dual instruments and sub-observables. We show that although a dual instruments measures a unique observable, it determines many sub-observables. We define a unique minimal extension of a sub-observable to an observable and consider sequential products and conditioning of sub-observables. Sub-observable effect algebras are characterized and studied. Moreover, the convexity of these effect algebras is considered. The sequential product of instruments is discussed. These concepts are illustrated with many examples of instruments. In particular, we discuss Lüders, Hole…
We establish a lower bound on the quantum coherence of an arbitrary quantum state in arbitrary dimension, using a noncommutativity estimator of an arbitrary observable of sub-unit norm, where the estimator is the commutator of the observable and its incoherent or classical part. The relation provides a direct method of obtaining an estimate of the quantum coherence of an arbitrary quantum state, without resorting to quantum state tomography or the existing witness operators.
I describe a method of inferring the past of quantum observables given the initial state and the subsequent measurement results using Wigner quasi-probability representations. The method is proved to be compatible with logic for large subclasses of quantum systems, including those that involve incompatible observables and can still exhibit some quantum features, such as the uncertainty relation, measurement backaction, and entanglement.
Some results are reviewed and developments are presented on the study of Time in quantum mechanics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the hermitian (more precisely, maximal hermitian, but non-selfadjoint) operator for Time which appears: (i) for particles, in ordinary non-relativistic quantum mechanics; and (ii) for photons, in first-quantization quantum electrodynamics.
For sharp quantum observables the following facts hold: (i) if we have a collection of sharp observables and each pair of them is jointly measurable, then they are jointly measurable all together; (ii) if two sharp observables are jointly measurable, then their joint observable is unique and it gives the greatest lower bound for the effects corresponding to the observables; (iii) if we have two sharp observables and their every possible two outcome partitionings are jointly measurable, then the observables themselves are jointly measurable. We show that, in general, these properties do not hol…
Malley discussed {[Phys. Rev. A {\bf 69}, 022118 (2004)]} that all quantum observables in a hidden-variable model for quantum events must commute simultaneously. In this comment, we discuss that Malley's theorem is indeed valid for the hidden-variable theoretical assumptions, which were introduced by Kochen and Specker. However, we give an example that the local hidden-variable (LHV) model for quantum events preserves noncommutativity of quantum observables. It turns out that Malley's theorem is not related with the LHV model for quantum events, in general.
The evolution equations of quantum observables are derived from the classical Hamiltonian equations of motion with the only additional assumption that the phase space is non-commutative. The demonstration of the quantum evolution laws is quite general; it does not rely on any assumption on the operator nature of x and p and is independent of the quantum mechanical formalism.
For quantum systems with two dimensional configuration space we construct a physical radial momentum observable. Rescaling the radius we find the dilatonic degrees of freedom form a Weyl algebra. With this we construct the radial Wigner quasi-probability distribution function.
We consider a general symplectic transformation (also known as linear canonical transformation) of quantum-mechanical observables in a quantized version of a finite-dimensional system with configuration space isomorphic to $ \mathbb{R}^{q} $. Using the formalism of rigged Hilbert spaces, we define eigenstates for all the observables. Then we work out the explicit form of the corresponding transformation of these eigenstates. A few examples are included at the end of the paper.
We consider the problem of estimating the ensemble average of an observable on an ensemble of equally prepared identical quantum systems. We show that, among all kinds of measurements performed jointly on the copies, the optimal unbiased estimation is achieved by the usual procedure that consists in performing independent measurements of the observable on each system and averaging the measurement outcomes.
We describe an algorithm for using a quantum computer to calculate mean values of observables and the partition function of a quantum system. Our algorithm includes two sub-algorithms. The first sub-algorithm is for calculating, with polynomial efficiency, certain diagonal matrix elements of an observable. This sub-algorithm is performed on a quantum computer, using quantum phase estimation and tomography. The second sub-algorithm is for sampling a probability distribution. This sub-algorithm is not polynomially efficient. It can be performed either on a classical or a quantum computer, but a …
It is pointed out that there exists an unambiguous definition of locality that enables one to distinguish local and nonlocal quantities. Observables of both types coexist in quantum optics but one must be very careful when attempting to measure them. A nonlocal observable which formally depends on the spatial position $\bi r$ cannot be {\em locally} measured without disturbing the measurements of this observable at all other positions.
Whereas complementarity manifests itself via two incompatible observables, quantum contextuality can only be revealed via the joint measurements among at least three observables. By incorporating unsharp measurements and joint measurements into a realistic model, we reestablish an inequality due to Liang, Spekkens, and Wiseman rigorously based on the assumption of noncontextuality alone. Its violation therefore unambiguously pinpoints the quantum contextuality of a two-level system. The maximal violation is attained by three triplewise jointly measurable observables that are pairwise jointly m…
Recently authors of a paper (arXiv:2105.04407) claim that quantum energy teleportation is unobservable due to a time-energy uncertainty relation. In this short note, I will point out that their argument is wrong. They misuse the uncertainty relation.
We study the limitations of deterministic programmability of quantum circuits, e.g., quantum computer. More precisely, we analyse the programming of quantum observables and channels via quantum multimeters. We show that the programming vectors for any two different sharp observables are necessarily orthogonal, whenever post-processing is not allowed. This result then directly implies that also any two different unitary channels require orthogonal programming vectors. This approach generalizes the well-known orthogonality result first proven by Nielsen and Chuang. In addition, we give size-boun…
Quantum information-theoretic approach has been identified as a way to understand the foundations of quantum mechanics as early as 1950 due to Shannon. However there hasn't been enough advancement or rigorous development of the subject. In the following paper we try to find relationship between a general quantum mechanical observable and it von Neumann entropy. We find that the expectation values and the uncertainties of the observables have bounds which depend on the entropy. The results also show that von Neumann entropy is not just the uncertainty of the state but also encompasses the infor…
The broad success of theoretical and experimental quantum optimal control is intimately connected to the topology of the underlying control landscape. For several common quantum control goals, including the maximization of an observable expectation value, the landscape has been shown to lack local optima if three assumptions are satisfied: (i) the quantum system is controllable, (ii) the Jacobian of the map from the control field to the evolution operator is full-rank, and (iii) the control field is not constrained. In the case of the observable objective, this favorable analysis shows that th…
A complete theory of overmeasurement by measuring refinements of observables is presented. It encompasses a wider set of functions of observ- ables (coarsenings) . Thus the theory has a broad potential application.It is applied to a thorough investigation of simultaneous measurements. In partic- ular, the set of all simultaneous measurements for a given pair of compatible observables is determined.
We study the problem of representation of statistical data (of any origin) by a complex probability amplitude. This paper is devoted to representation of data collected from measurements of two trichotomous observables. The complexity of the problem eventually increased comparing to with the case of dichotomous observables. We see that only special statistical data (satisfying s number of nonlinear constraints) have the quantum--like representation.
The advent of cloud quantum computing has led to the rapid development of quantum algorithms. In particular, it is necessary to study variational quantum-classical hybrid algorithms, which are executable on noisy intermediate-scale quantum (NISQ) computers. Evaluations of observables appear frequently in the variational quantum-classical hybrid algorithms for NISQ computers. By speeding up the evaluation of observables, it is possible to realize a faster algorithm and save resources of quantum computers. Grouping of observables with separable measurements has been conventionally used, and the …
A normalized positive operator measure $X\mapsto E(X)$ has the norm-1-property if $\no{E(X)}=1$ whenever $E(X)\ne O$. This property reflects the fact that the measurement outcome probabilities for the values of such observables can be made arbitrary close to one with suitable state preparations. Some general implications of the norm-1-property are investigated. As case studies, localization observables, phase observables, and phase space observables are considered.
Observables and instruments have played significant roles in recent studies on the foundations of quantum mechanics. Sequential products of effects and conditioned observables have also been introduced. After an introduction in Section~1, we review these concepts in Section~2. Moreover, it is shown how these ideas can be unified within the framework of measurement models. In Section~3, we illustrate these concepts and their relationships for the simple example of a qubit Hilbert space. Conditioned observables and their distributions are studied in Section~4. Section~5 considers joint probabili…
Suppose we are given two graphs on $n$ vertices. We define an observable in the Hilbert space $\Co[(S_n \wr S_2)^m]$ which returns the answer ``yes'' with certainty if the graphs are isomorphic and ``no'' with probability at least $1-n!/2^m$ if the graphs are not isomorphic. We do not know if this observable is efficiently implementable.
A reference frame F is described by the element g of the Poincare' group P which connects F with a given fixed frame F_0. If F is a quantum frame, defined by a physical object following the laws of quantum physics, the parameters of g have to be considered as quantum observables. However, these observables are not compatible and some of them, namely the coordinates of the origin of F, cannot be represented by self-adjoint operators. Both these difficulties can be overcome by considering a positive-operator-valued measure (POVM) on P, covariant with respect to the left translations of the group…
We raise the problem of constructing quantum observables that have classical counterparts without quantization. Specifically we seek to define and motivate a solution to the quantum-classical correspondence problem independent from quantization and discuss the general insufficiency of prescriptive quantization, particularly the Weyl quantization. We demonstrate our points by constructing time of arrival operators without quantization and from these recover their classical counterparts.
It is argued that the partition of a quantum system into subsystems is dictated by the set of operationally accessible interactions and measurements. The emergence of a multi-partite tensor product structure of the state-space and the associated notion of quantum entanglement are then relative and observable-induced. We develop a general algebraic framework aimed to formalize this concept. We discuss several cases relevant to quantum information processing and decoherence control.
An observable effects a schematization of the Quantum event structure by correlating Boolean algebras picked by measurements with the Borel algebra of the real line. In a well-defined sense Boolean observables play the role of coordinatizing objects in the Quantum world, by picking Boolean figures and subsequently opening Boolean windows for the perception of the latter, interpreted as local measurement charts. A mathematical scheme for the implementation of this thesis is being proposed based on Category theoretical methods. The scheme leads to a manifold representation of Quantum structure i…
In this work simple and effective quantization procedure of classical dynamical systems is proposed and illustrated by a number of examples. The procedure is based entirely on differential equations which describe time evolution of systems.
