The concept of intrinsic and operational observables in quantum mechanics is introduced. It is argued that, in any realistic description of a quantum measurement that includes a 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. Using the concept of the propensity and the associated operational positive operator valued measure (POVM) a general description of the operational algebra of quantum mechanical observables is derived for a wide class of realistic detection schemes. This general approach is illustrated by the example of an operational Malus measurement of the spin phases and by an analysis of the operational homodyne detection of the phase of an optical field with a squeezed vacuum in the unused ports.
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.
Quantum mechanics predicts the joint probability distribution of the outcomes of simultaneous measurements of commuting observables, but, in the state of the art, has lacked the operational definition of simultaneous measurements. The question is answered as to when the consecutive applications of measuring apparatuses give a simultaneous measurement of their observables. For this purpose, all the possible state reductions caused by measurements of an observable is also characterized by their operations.
A C*-algebra formulation of Quantum Mechanics is derived from purely operational axioms in which the primary role is played by the "transformations" that the system undergoes in the course of an "experiment". The notion of the {\em adjoint} of a transformation is based on the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus.
This paper outlines a mathematical framework of quantum probability in which the time asymmetry in describing measuring processes is avoided. The main objects of the framework are hyperfinite operations, which are constructed by using nonstandard analysis and the operational approach by Davies and Lewis. Then the notions of Bayesian conditional probability are defined, and Bayes-type theorems in terms of the probability are showed.
In this work a generalization of the consistent histories approach to quantum mechanics is presented. We first critically review the consistent histories approach to nonrelativistic quantum mechanics in a mathematically rigorous way and give some general comments about it. We investigate to what extent the consistent histories scheme is compatible with the results of the operational formulation of quantum mechanics. According to the operational approach nonrelativistic quantum mechanics is most generally formulated in terms of effects, states and operations. We formulate a generalized consiste…
The role of operational quantum mechanics, quantum axiomatics and quantum structures in general is presented as a contribution to a compendium on quantum physics, its history and philosophy.
Understanding the core content of quantum mechanics requires us to disentangle the hidden logical relationships between the postulates of this theory. Here we show that the mathematical structure of quantum measurements, the formula for assigning outcome probabilities (Born's rule) and the post-measurement state-update rule, can be deduced from the other quantum postulates, often referred to as "unitary quantum mechanics", and the assumption that ensembles on finite-dimensional Hilbert spaces are characterised by finitely many parameters. This is achieved by taking an operational approach to p…
The debate on the nature of quantum probabilities in relation to Quantum Non Locality has elevated Quantum Mechanics to the level of an "Operational Epistemic Theory". In such context the quantum superposition principle has an extraneous non epistemic nature. This leads us to seek purely operational foundations for Quantum Mechanics, from which to derive the current mathematical axiomatization based on Hilbert spaces. In the present work I present a set of axioms of purely operational nature, based on a general definition of "the experiment", the operational/epistemic archetype of information …
Entanglement without Quantum Mechanics: Operational Constraints on the Quantum Signature
University of Vienna（维也纳大学）；Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；University of Geneva（日内瓦大学）；Constructor University（Constructor 大学）
A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in this context are operational theories, and categorical quantum mechanics. The aim of the present paper is to establish strong connections between these two formalisms. We show how models of categorical quantum mechanics have representations as operati…
The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of "physical experiment" and from five simple Postulates concerning "experimental accessibility and simplicity". For the infinite dimensional case, on the other hand, a C*-algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand-Naimark-Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in version 1. The main ingredient of the axiomatiza…
The standard presentation of the principles of quantum mechanics is critically reviewed both from the experimental/operational point and with respect to the request of mathematical consistency and logical economy. A simpler and more physically motivated formulation is discussed. The existence of non commuting observables, which characterizes quantum mechanics with respect to classical mechanics, is related to operationally testable complementarity relations, rather than to uncertainty relations. The drawbacks of Dirac argument for canonical quantization are avoided by a more geometrical approa…
The Born rule is derived from operational assumptions, independent of the normalization of the state. Unlike Gleason's theorem, the argument applies even if probabilities are defined for only a single resolution of the identity, so it applies to all the major foundational approaches to quantum mechanics. There are important points of contact with Deutsch's program for deducing the probabilistic interpretation of quantum mechanics from decision thoery, as recently completed by Wallace. Decision theory can be used to supplement the present derivation, in application to the Everett interpretation…
Classical and quantum mechanics across representations: an operational reading of the Wigner Weyl correspondence
University of Vienna, Faculty of Physics（维也纳大学物理学院）；Vienna Center for Quantum Science and Technology（维也纳量子科学与技术中心）；Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；Constructor University（Constructor大学）；University of Geneva, Group of Applied Physics（日内瓦大学应用物理组）
The mathematical formulation of Quantum Mechanics is derived from purely operational axioms based on a general definition of "experiment" as a set of transformations. The main ingredient of the mathematical construction is the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the "adjoint" of a transformation.
In this article, I use an operational formulation of the Choi-Jamiołkowski isomorphism to explore an approach to quantum mechanics in which the state is not the fundamental object. I first situate this project in the context of generalized probabilistic theories and argue that this framework may be understood as a means of drawing conclusions about the intratheoretic causal structure of quantum mechanics which are independent of any specific ontological picture. I then give an operational formulation of the Choi-Jamiołkowski isomorphism and show that in an operational theory which exhibits thi…
Quantum information science is a source of task-related axioms whose consequences can be explored in general settings encompassing quantum mechanics, classical theory, and more. Quantum states are compendia of probabilities for the outcomes of possible operations we may perform a system: ``operational states.'' I discuss general frameworks for ``operational theories'' (sets of possible operational states of a system), in which convexity plays key role. The main technical content of the paper is in a theorem that any such theory naturally gives rise to a ``weak effect algebra'' when outcomes ha…
An operational time of arrival is introduced using a realistic position and momentum measurement scheme. The phase space measurement involves the dynamics of a quantum particle probed by a measuring device. For such a measurement an operational positive operator valued measure in phase space is introduced and investigated. In such an operational formalism a quantum mechanical time operator is constructed and analyzed. A phase space time and energy uncertainty relation is derived.
We introduce a general and systematic theoretical framework for Operational Dynamic Modeling (ODM) by combining a kinematic description of a model with the evolution of the dynamical average values. The kinematics includes the algebra of the observables and their defined averages. The evolution of the average values is drawn in the form of Ehrenfest-like theorems. We show that ODM is capable of encompassing wide ranging dynamics from classical non-relativistic mechanics to quantum field theory. The generality of ODM should provide a basis for formulating novel theories.
The operational meaning of some measures of noise and disturbance in measurements is analyzed and their limitations are pointed out. The cases of minimal noise and least disturbance are characterized.
In this paper, I propose a project of enlisting quantum information science as a source of task-oriented axioms for use in the investigation of operational theories in a general framework capable of encompassing quantum mechanics, classical theory, and more. Whatever else they may be, quantum states of systems are compendia of probabilities for the outcomes of possible operations we may perform on the systems: ``operational theories.'' I discuss appropriate general frameworks for such theories, in which convexity plays a key role. Such frameworks are appropriate for investigating what things l…
This talk is a survey of the question of joint measurability of coexistent observables and its is based on the monograph Operational Quantum Physics [1] and on the papers [2,3,4].
We give a unitary irreducible representation of the proper Poincaré group that leads to an operational version of the classical relativistic dynamics of a massive spinless particle. Unlike quantum mechanics, in this operational theory there is no uncertainty principle between position and momentum. It will be shown that the theory contains the Koopman-von Neumann formalism as a particular case, and a explicit connection with relativistic Hamiltonian mechanics will be given.
Proposals for nonlinear extenstions of quantum mechanics are discussed. Two different concepts of "mixed state" for any nonlinear version of quantum theory are introduced: (i) >genuine mixture< corresponds to operational "mixing" of different ensembles, and (ii) a mixture described by single density matrix without having a canonical operational possibility to pick out its specific convex decomposition is called here an >elementary mixture<. Time evolution of a class of nonlinear extensions of quantum mechanics is introduced. Evolution of an elementary mixture cannot be generally given by evolu…
Quantum Mechanics (QM) is a very special probabilistic theory, yet we don't know which operational principles make it so. All axiomatization attempts suffer at least one postulate of a mathematical nature. Here I will analyze the possibility of deriving QM as the mathematical representation of a "fair operational framework", i.e. a set of rules which allows the experimenter to make predictions on future "events" on the basis of suitable "tests", e.g. without interference from uncontrollable sources. Two postulates need to be satisfied by any fair operational framework: NSF: "no-signaling from …
We consider a model of quantum measurement built on an ideal operational amplifier operating in the limit of infinite gain, infinite input impedance and null output impedance and with a feddback loop. We evaluate the intensity and voltage noises which have to be added to the classical amplification equations in order to fulfill the requirements of quantum mechanics. We give a description of this measurement device as a quantum network scattering quantum fluctuations from input to output ports.
Joint, or simultaneous, measurements of non-commuting observables are possible within quantum mechanics, if one accepts an increase in the variances of the jointly measured observables. In this paper, we discuss joint measurements of a spin 1/2 particle along any two directions. Starting from an operational locality principle, it is shown how to obtain a bound on how sharp the joint measurement can be. We give a direct interpretation of this bound in terms of an uncertainty relation.
A new measure of information in quantum mechanics is proposed which takes into account that for quantum systems the only feature known before an experiment is performed are the probabilities for various events to occur. The sum of the individual measures of information for mutually complementary observations is invariant under the choice of the particular set of complementary observations and conserved if there is no information exchange with an environment. That operational quantum information invariant results in N bits of information for a system consisting of N qubits.
The concept and the formalization of the arrival time in quantum mechanics are discussed. Different approaches based on trajectories, quantization rules, time operators, phase space techniques, renewal equations or operational procedures are reviewed or proposed. Open questions and loose ends are pointed out.
We establish an operational characterization of general convex resource theories -- describing the resource content of not only states, but also measurements and channels, both within quantum mechanics and in general probabilistic theories (GPTs) -- in the context of state and channel discrimination. We find that discrimination tasks provide a unified operational description for quantification and manipulation of resources by showing that the family of robustness measures can be understood as the maximum advantage provided by any physical resource in several different discrimination tasks, as …
A qualitative but formalized representation of microstates is first established quite independently of the quantum mechanical mathematical formalism, exclusively under epistemological-operational-methodological constraints. Then, using this representation as a reference-and-imbedding-structure, the foundations of an intelligible reconstruction of the Hilbert-Dirac formulation of Quantum Mechanics is developed. Inside this reconstruction the measurement problem as well as the other major problems raised by the quantum mechanical formalism, dissolve.
Bohmian mechanics (BM) is a popular interpretation of quantum mechanics in which particles have real positions. The velocity of a point x in configuration space is defined as the standard probability current j(x) divided by the probability density P(x). However, this ``standard'' j is in fact only one of infinitely many that transform correctly and satisfy \dot P + \del . j=0. In this article I show that there is a unique j that can be determined experimentally as a weak value using techniques that would make sense to a classical physicist. Moreover, this operationally defined j equals the sta…
The analysis of the model quantum clocks proposed by Aharonov et al. [Phys. Rev. A 57 (1998) 4130 - quant-ph/9709031] requires considering evanescent components, previously ignored. We also clarify the meaning of the operational time of arrival distribution which had been investigated.
A rigorous theory of quantum state reduction, the state change of the measured system caused by a measurement conditional upon the outcome of measurement, is developed fully within quantum mechanics without leading to the vicious circle relative to the von Neumann chain. For the basis of the theory, the local measurement theorem provides the joint probability distribution for the outcomes of local successive measurements on a noninteracting entangled system without assuming the projection postulate, and the quantum Bayes principle enables us to determine operationally the quantum state from a …
In order to understand the source and extent of the greater-than-classical information processing power of quantum systems, one wants to characterize both classical and quantum mechanics as points in a broader space of possible theories. One approach to doing this, pioneered by Abramsky and Coecke, is to abstract the essential categorical features of classical and quantum mechanics that support various information-theoretic constraints and possibilities, e.g., the impossibility of cloning in the latter, and the possibility of teleportation in both. Another approach, pursued by the authors and …
The wave function in quantum mechanics presents an interesting challenge to our understanding of the physical world. In this paper, I show that the wave function can be understood as four intrinsic relations on physical space. My account has three desirable features that the standard account lacks: (1) it does not refer to any abstract mathematical objects, (2) it is free from the usual arbitrary conventions, and (3) it explains why the wave function has its gauge degrees of freedom, something that are usually put into the theory by hand. Hence, this account has implications for debates in phi…
It is commonly believed that the most general type of a quantum-mechanical measurement is one described by a positive-operator valued measure (POVM). In the present paper, this statement is proven for any measurements on quantum systems with a finite-dimensional state space. The proof of POVM nature of an arbitrary measurement is carried out using a purely operational approach, which is fully ignorant about what is inside a measurement apparatus. The suggested approach gives also an opportunity to derive the Born rule.
This paper offers a brief introduction to the framework of "general probabilistic theories", otherwise known as the "convex-operational" approach the foundations of quantum mechanics. Broadly speaking, the goal of research in this vein is to locate quantum mechanics within a very much more general, but conceptually very straightforward, generalization of classical probability theory. The hope is that, by viewing quantum mechanics "from the outside", we may be able better to understand it. We illustrate several respects in which this has proved to be the case, reviewing work on cloning and broa…
We undertake a reconstruction of the epistemic significance of research on operational theories in quantum foundations. We suggest that the space of operational theories is analogous to the space of possible worlds employed in the possible world semantics for modal logic, so research of this sort can be understood as probing modal structure. Thus we argue that operational axiomatisations of quantum mechanics may be interpreted as a novel form of structural realism; we discuss the consequences of this interpretation for the philosophy of structural realism and the future of operational theories…
The complementarity between signaling and randomness in any communicated resource that can simulate singlet statistics is generalized by relaxing the assumption of free will in the choice of measurement settings. We show how to construct an ontological extension for quantum mechanics (QM) through the oblivious embedding of a sound simulation protocol in a Newtonian spacetime. Minkowski or other intermediate spacetimes are ruled out as the locus of the embedding by virtue of hidden influence inequalities. The complementarity transferred from a simulation to the extension unifies a number of res…
The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. In this paper, based on the toolkit of algorithmic randomness, we present a refinement of the Born rule, as an alternative rule to it, for specifying the property of the results of quantum measurements in an operational way. Algorithmic ran…
A new realist interpretation of quantum mechanics is introduced. Quantum systems are shown to have two kinds of properties: the usual ones described by values of quantum observables, which are called extrinsic, and those that can be attributed to individual quantum systems without violating standard quantum mechanics, which are called intrinsic. The intrinsic properties are classified into structural and conditional. A systematic and self-consistent account is given. Much more statements become meaningful than any version of Copenhagen interpretation would allow. A new approach to classical pr…
Following Frauchiger and Renner's discovery of a conflict between quantum mechanics and certain commonsense reasoning axioms, much work has gone into finding alternative axiomatizations that can avoid the conflict. However, this body of work is largely based on specific interpretations of quantum mechanics, and at times employs specialized formalism that may be inaccessible to mainstream quantum information experts. Taking an interpretation-agnostic approach, we propose a simple operational principle called superpositional solipsism to aid rational agents in making situational inferences. We s…
The Born's rule introduces intrinsic randomness to the outcomes of a measurement performed on a quantum mechanical system. But, if the system is prepared in the eigenstate of an observable then the measurement outcome of that observable is completely predictable and hence there is no intrinsic randomness. On the other hand, if two incompatible observables are measured (either sequentially on a particle or simultaneously on two identical copies of the particle) then uncertainty principle guarantees intrinsic randomness in the subsequent outcomes independent of the preparation state of the syste…
A formulation of quantum mechanics based on an operational definition of state is presented. This formulation, which includes explicitly the macroscopic systems, assumes the probabilistic interpretation and is nevertheless objective. The classical paradoxes of quantum mechanics are analyzed and their origin is found to be the fictitious properties that are usually attributed to quantum-mechanical states. The hypothesis that any mixed state can always be considered as an incoherent superposition of pure states is found to contradict quantum mechanics. A solution of EPR paradox is proposed. It i…
Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems, their dynamics, and interaction. Since the inception of quantum theory, it has been debated whether complex numbers are actually essential, or whether an alternative consistent formulation is possible using real numbers only. Here, we attack this long-standing problem both theoretically and experimentally, using the powerful tools of quantum resource theories. We show that - under reasonable assumptions - quantum states are easier to create…
In order to analyze non-Markovianity of tripartite quantum states from a resource theoretical viewpoint, we introduce a class of quantum operations performed by three distant parties, and investigate an operational resource theory (ORT) induced it. A tripartite state is a free state if and only if it is a quantum Markov chain. We prove monotonicity of functions such as the conditional quantum mutual information, intrinsic information, squashed entanglement, a generalization of entanglement of purification, and the relative entropy of recovery. The ORT has five bound sets, each of which corresp…
