The typicality approach and the Hilbert space averaging method as its technical manifestation are important concepts of quantum statistical mechanics. Extensively used for expectation values we extend them in this paper to transition probabilities. In this context we also find that the transition probability of two random uniformly distributed states is connected to the spectral statistics of the considered operator. Furthermore, within our approach we are capable to consider distributions of matrix elements between states, that are not orthogonal. We will demonstrate our quite general result numerically for a kicked spin chain in the integrable resp. chaotic regime.
We perform the analysis of probabilistic assumptions of Bell's approach. We emphasize that J. Bell wrote about probability without to specify the concrete axiomatics of probability theory. The careful analysis demonstrated that (surprisingly) J. Bell did not apply the classical probability model (Kolmogorov) to describe ``classical physical framework.'' In fact, he created his own probabilistic model and compared it with the quantum one. The crucial point is that J. Bell did not pay attention to {\it conditional probabilities.} We show that conditional probability in his model cannot be define…
The Horodecki family employed the Jaynes maximum-entropy principle, fitting the mean (b_{1}) of the Bell-CHSH observable (B). This model was extended by Rajagopal by incorporating the dispersion (σ_{1}^2) of the observable, and by Canosa and Rossignoli, by generalizing the observable (B_α). We further extend the Horodecki one-parameter model in both these manners, obtaining a three-parameter (b_{1},σ_{1}^2,α) two-qubit model, for which we find a highly interesting/intricate continuum (-\infty < α< \infty) of Hilbert-Schmidt (HS) separability probabilities -- in which, the golden ratio is featu…
We investigate convergence properties of discrete-time semigroup quantum dynamics, including asymptotic stability, probability and speed of convergence to pure states and subspaces. These properties are of interest in both the analysis of uncontrolled evolutions and the engineering of controlled dynamics for quantum information processing. Our results include two Hilbert space decompositions that allow for deciding the stability of the subspace of interest and for estimating of the speed of convergence, as well as a formula to obtain the limit probability distribution for a set of orthogonal i…
Jakobczyk and Siennicki studied two-dimensional sections of a set of (generalized) Bloch vectors corresponding to n x n density matrices of two-qubit systems (that is, the case n = 4). They found essentially five different types of (nontrivial) separability regimes. We compute the Euclidean/Hilbert-Schmidt (HS) separability probabilities assigned to these regimes, and conduct parallel two-dimensional sectional analyses for the higher-level cases n=6,8,9 and 10. Making use of the newly-introduced capability for integration over implicitly defined regions of version 5.1 of Mathematica -- also fr…
We significantly advance the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A function P(alpha), incorporating a family of six hypergeometric functions, all with argument 27/64 =(3/4)^3, is obtained here. It reproduces a series, alpha = 1/2, 1, 3/2, 2,...,32$, of sixty-four Hilbert-Schmidt generic 2 x 2 generalized separability probabilities, advanced on the basis of systematic high-accuracy probability-distribution-reconstruction computations, employing 7,501 determinant…
As it is known, Gleason's theorem is not applicable for a two-dimensional Hilbert space since in this situation Gleason's axioms are not strong enough to imply Born's rule thus leaving room for a dispersion-free probability measure i.e., one that has only values 0 and 1. To strengthen Gleason's axioms one must add at least one more assumption. But, as it is argued in the present paper, alternatively one can give up the lattice condition lying in the foundation of Gleason's theorem. Particularly, the lattice structure based on the closed linear subspaces in the Hilbert space could be weakened b…
Firstly, we reduce the long-standing problem of ascertaining the Hilbert-Schmidt probability that a generic pair of qubits is separable to that of determining the specific nature of a one-dimensional (separability) function of the radial coordinate (r) of the unit ball in 15-dimensional Euclidean space, and similarly for a generic pair of rebits, using the 9-dimensional unit ball. Separability probabilities, could, then, be directly obtained by integrating the products of these functions (which we numerically estimate and plot) with jacobian factors of r^m over r in [0,1], with m=17 for the tw…
A confluence of numerical and theoretical results leads us to conjecture that the Hilbert-Schmidt separability probabilities of the 15- and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 x 4 density matrices rho) are 8/33 and 8/17, respectively. Central to our reasoning are the modifications of two ansatze, recently advanced (quant-ph/0609006), involving incomplete beta functions B_{nu}(a,b), where nu= (rho_{11} rho_{44})/(rho_{22} rho_{33}). We, now, set the separability function S_{real}(nu) propto B_{nu}(nu,1/2},2) =(2/3) (3-nu) sqrt{nu}. Then, in the com…
Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state ($ρ$) is separable/disentangled is $\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal $4F3$-hypergeometric moment formula (Appendix A), we reach, {\it via} density-approximation procedures, the conclusion that one-half ($\frac{4}{33}$) of this probability arises when the determinantal inequality $|ρ^{PT}|>|ρ|$, where $PT$ denotes the partial transpose, is satisfied, and, the oth…
We report major advances in the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A highly succinct separability probability function P(alpha) is put forth, yielding for generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit systems, P(1) = 8/33 and (27-dimensional) two-quater(nionic)bit systems, P(2)=26/323. This particular form of P(alpha) was obtained by Qing-Hu Hou and colleagues by applying Zeilberger's algorithm ("creative telescoping") t…
This Chapter develops a realist information-theoretic interpretation of the nonclassical features of quantum probabilities. On this view, what is fundamental in the transition from classical to quantum physics is the recognition that \emph{information in the physical sense has new structural features}, just as the transition from classical to relativistic physics rests on the recognition that space-time is structurally different than we thought. Hilbert space, the event space of quantum systems, is interpreted as a kinematic (i.e., pre-dynamic) framework for an indeterministic physics, in the …
One-dimensional problem for quantum harmonic oscillator with "regular+random" frequency subjected to the external "regular+random" force is considered. Averaged transition probabilities are found.
The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities. In our approach it is forbidden to consider abstract context independent probabilities: ``first context and then probability.'' We start with the conventional formula…
We analyze closed one-dimensional chains of weakly coupled many level systems, by means of the so-called Hilbert space average method (HAM). Subject to some concrete conditions on the Hamiltonian of the system, our theory predicts energy diffusion with respect to a coarse-grained description for almost all initial states. Close to the respective equilibrium we investigate this behavior in terms of heat transport and derive the heat conduction coefficient. Thus, we are able to show that both heat (energy) diffusive behavior as well as Fourier's law follows from and is compatible with a reversib…
We use the fact that some linear Hamiltonian systems can be considered as ``finite level'' quantum systems, and the description of quantum mechanics in terms of probabilities, to associate probability distributions with this particular class of linear Hamiltonian systems.
Quantum theory does not only predict probabilities, but also relative phases for any experiment, that involves measurements of an ensemble of systems at different moments of time. We argue, that any operational formulation of quantum theory needs an algebra of observables and an object that incorporates the information about relative phases and probabilities. The latter is the (de)coherence functional, introduced by the consistent histories approach to quantum theory. The acceptance of relative phases as a primitive ingredient of any quantum theory, liberates us from the need to use a Hilbert …
We extend the findings and analyses of our two recent studies (Phys. Rev. A, 75, 032326 [2007] and arXiv:0704.3723) by, first, obtaining numerical estimates of the separability function based on the (Euclidean, flat) Hilbert-Schmidt (HS) metric for the 27-dimensional convex set of quaternionic two-qubit systems. The estimated function appears to be strongly consistent with our previously-formulated Dyson-index (beta = 1, 2, 4) ansatz, dictating that the quaternionic (beta=4) separability function should be exactly proportional to the square of the separability function for the 15-dimensional c…
We use Bub's (2016) correlation arrays and Pitowksy's (1989b) correlation polytopes to analyze an experimental setup due to Mermin (1981) for measurements on the singlet state of a pair of spin-$\frac12$ particles. The class of correlations allowed by quantum mechanics in this setup is represented by an elliptope inscribed in a non-signaling cube. The class of correlations allowed by local hidden-variable theories is represented by a tetrahedron inscribed in this elliptope. We extend this analysis to pairs of particles of arbitrary spin. The class of correlations allowed by quantum mechanics i…
We study the possibility to describe pure quantum states and evens with classical probability distributions and conditional probabilities and show that the distributions and/or conditional probabilities have to assume negative values, except for a simple model whose realistic space dimension is not smaller than the Hilbert space dimension of the quantum system. This gives a negative answer to a question proposed by Montina [Phys.Rev.Lett.{\bf 97}, 180401 (2006)] whether or not does there exist a classical theory whose phase-space dimension is much smaller than the Hilbert space dimension for a…
We study a set of truncated matrices, given by Smith~\cite{Smith2005}, in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert's tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable choice of some parameter it can always be removed. We also argue that it is only an artefact of the truncation of the underlying Hilbert spaces, through showing its sensitivity to different boundary conditions available for such a truncation. It is maintained that the criterion, in general, shou…
We employ a quasirandom methodology, recently developed by Martin Roberts, to estimate the separability probabilities, with respect to the Bures (minimal monotone/statistical distinguishability) measure, of generic two-qubit and two-rebit states. This procedure, based on generalized properties of the golden ratio, yielded, in the course of almost seventeen billion iterations (recorded at intervals of five million), two-qubit estimates repeatedly close to nine decimal places to $\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.073313783$. Howeer, despite the use of over twenty-three billion it…
In quantum logic, i.e., within the structure of the Hilbert lattice imposed on all closed linear subspaces of a Hilbert space, the assignment of truth values to quantum propositions (i.e., experimentally verifiable propositions relating to a quantum system) is unambiguously determined by the state of the system. So, if only pure states of the system are considered, can a probability measure mapping the probability space for truth values to the unit interval be assigned to quantum propositions? In other words, is a probability concept contingent or emergent in the logic of quantum propositions?…
We reexamine a recent analysis in which, using the volume of the associated quantum steering ellipsoid (QES) as a measure, we sought to estimate the probability that a two-qubit state is separable. In the estimation process, we, in effect, sought to attach to states random with respect to Hilbert-Schmidt (HS) measure, the corresponding QES volumes. However, a study of the relations between HS and Bures ensembles and their well-supported separability probabilities of $\frac{8}{33}$ and $\frac{25}{341}$, respectively, now lead us to explore as a possible alternative measure, the QES volume divid…
We construct, for any finite dimension $n$, a new hidden measurement model for quantum mechanics based on representing quantum transition probabilities by the volume of regions in projective Hilbert space. For $n=2$ our model is equivalent to the Aerts sphere model and serves as a generalization of it for dimensions $n \geq 3$. We also show how to construct a hidden variables scheme based on hidden measurements and we discuss how joint distributions arise in our hidden variables scheme and their relationship with the results of Fine.
We consider discrete quantum systems coupled to finite environments which may possibly consist of only one particle in contrast to the standard baths which usually consist of continua of oscillators, spins, etc. We find that such finite environments may, nevertheless, act as thermostats, i.e., equilibrate the system though not necessarily in the way predicted by standard open system techniques. Thus, we apply a novel technique called the Hilbert space Average Method (HAM) and verify its results numerically.
We examine the notion of Haldane's dimension and the corresponding statistics in a probabilistic spirit. Motivated by the example of dimensional-regularization we define the dimension of a space as the trace of a diagonal `unit operator', where the diagonal matrix elements are not, in general, unity but are probabilities to place the system into a given state. These probabilities are uniquely defined by the rules of Haldane's statistics. We calculate the second virial coefficient for our system and demonstrate agreement with Murthy and Shankar's calculation. The partition function for an ideal…
The extended semantic realism (ESR) model recently worked out by one of the authors embodies the mathematical formalism of standard (Hilbert space) quantum mechanics in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide here a Hilbert space representation of the generalized observables introduced by the ESR model that satisfy a simple physical condition, propose a generalization of the projection postulate, and suggest a possible mathematical description of the measurement process in terms of evolution of the compound system made up o…
We discuss some aspects related to the so-called Hilbert space Average Method, as an alternative to describe the dynamics of open quantum systems. First we present a derivation of the method which does not make use of the algebra satisfied by the operators involved in the dynamics, and extend the method to systems subject to a Hamiltonian that changes with time. Next we examine the performance of the adiabatic quantum search algorithm with a particular model for the environment. We relate our results to the criteria discussed in the literature for the validity of the above-mentioned method for…
We establish a necessary and sufficient condition for averages over complex valued weight functions on R^N to be represented as statistical averages over real, non-negative probability weights on C^N. Using this result, we show that many path-integrals for time-ordered expectation values of bosonic degrees of freedom in real-valued time can be expressed as statistical averages over ensembles of paths with complex-valued coordinates, and then speculate on possible consequences of this result for the relation between quantum and classical mechanics.
In the quantum mechanical Hilbert space formalism, the probabilistic interpretation is a later ad-hoc add-on, more or less enforced by the experimental evidence, but not motivated by the mathematical model itself. A model involving a clear probabilistic interpretation from the very beginning is provided by the quantum logics with unique conditional probabilities. It includes the projection lattices in von Neumann algebras and here probability conditionalization becomes identical with the state transition of the Lueders - von Neumann measurement process. This motivates the definition of a hiera…
We consider the space of probabilities {P(x)}, where the x are coordinates of a configuration space. Under the action of the translation group there is a natural metric over the space of parameters of the group given by the Fisher-Rao metric. This metric induces a metric over the space of probabilities. Our next step is to set the probabilities in motion. To do this, we introduce a canonically conjugate field S and a symplectic structure; this gives us Hamiltonian equations of motion. We show that it is possible to extend the metric structure to the full space of the {P,S} and this leads in a …
We study the average probability that a discrete-time quantum walk finds a marked vertex on a graph. We first show that, for a regular graph, the spectrum of the transition matrix is determined by the weighted adjacency matrix of an augmented graph. We then consider the average search probability on a distance regular graph, and find a formula in terms of the adjacency matrix of its vertex-deleted subgraph. In particular, for any family of (1) complete graphs, or (2) strongly regular graphs, or (3) distance regular graphs of a fixed parameter $d$, varying valency $k$ and varying size $n$, such…
It $d-$pends. Wigner's symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive $\varepsilon$ in a separable Hilbert space admits a weak linear approximation, i.e. one relative to any fixed observable. This implies the existence of a linear approximation that is $4\sqrt{\varepsilon} d$-close in Hilbert-Schmidt norm, with $d$ the Hilbert sp…
Employing Hilbert-Schmidt measure, we explicitly compute and analyze a number of determinantal product (bivariate) moments |rho|^k |rho^{PT}|^n, k,n=0,1,2,3,..., PT denoting partial transpose, for both generic (9-dimensional) two-rebit (alpha = 1/2) and generic (15-dimensional) two-qubit (alpha=1) density matrices rho. The results are, then, incorporated by Dunkl into a general formula (Appendix D6), parameterized by k, n and alpha, with the case alpha=2, presumptively corresponding to generic (27-dimensional) quaternionic systems. Holding the Dyson-index-like parameter alpha fixed, the induce…
Max Born's statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. While the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable …
We conduct a pair of quasirandom estimations of the separability probabilities with respect to ten measures on the 15-dimensional convex set of two-qubit states, using its Euler-angle parameterization. The measures include the (non-monotone) Hilbert-Schmidt one, plus nine others based on operator monotone functions. Our results are supportive of previous assertions that the Hilbert-Schmidt and Bures (minimal monotone) separability probabilities are $\frac{8}{33} \approx 0.242424$ and $\frac{25}{341} \approx 0.0733138$, respectively, as well as suggestive of the Wigner-Yanase counterpart being …
We constructed the representation of contextual probabilistic dynamics in the complex Hilbert space. Thus dynamics of the wave function can be considered as Hilbert space projections of realistic dynamics in a ``prespace''. The basic condition for representing of the prespace-dynamics is the law of statistical conservation of energy -- conservation of probabilities. Construction of the dynamical representation is an important step in the development of contextual statistical viewpoint to quantum processes. But the contextual statistical model is essentially more general than the quantum one. T…
Hughston has recently proposed a stochastic extension of the Schrödinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston's equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston's equation to earlier work on norm-preserving stochastic equations, and show that Hughston's equation can be written as a manifest…
A superqubit, belonging to a $(2|1)$-dimensional super-Hilbert space, constitutes the minimal supersymmetric extension of the conventional qubit. In order to see whether superqubits are more nonlocal than ordinary qubits, we construct a class of two-superqubit entangled states as a nonlocal resource in the CHSH game. Since super Hilbert space amplitudes are Grassmann numbers, the result depends on how we extract real probabilities and we examine three choices of map: (1) DeWitt (2) Trigonometric (3) Modified Rogers. In cases (1) and (2) the winning probability reaches the Tsirelson bound $p_{w…
Any attempt to introduce probabilities into quantum mechanics faces difficulties due to the mathematical structure of Hilbert space, as reflected in Birkhoff and von Neumann's proposal for a quantum logic. The (consistent or decoherent) histories solution is provided by its single framework rule, an approach that includes conventional (Copenhagen) quantum theory as a special case. Mermin's Ithaca interpretation addresses the same problem by defining probabilities which make no reference to a sample space or event algebra (``correlations without correlata''). But this leads to severe conceptual…
We consider the hardness of computing additive approximations to output probabilities of random quantum circuits. We consider three random circuit families, namely, Haar random, $p=1$ QAOA, and random IQP circuits. Our results are as follows. For Haar random circuits with $m$ gates, we improve on prior results by showing $\mathsf{coC_=P}$ hardness of average-case additive approximations to an imprecision of $2^{-O(m)}$. Efficient classical simulation of such problems would imply the collapse of the polynomial hierarchy. For constant depth circuits i.e., when $m=O(n)$, this linear scaling in th…
After a review of the arrows of time, we describe the possibilities of a time-asymmetry in quantum theory. Whereas Hilbert space quantum mechanics is time-symmetric, the rigged Hilbert space formulation, which arose from Dirac's bra-ket formalism, allows the choice of asymmetric boundary conditions analogous to the retarded solutions of the Maxwell equations for the radiation arrow of time. This led to irreversibility on the microphysical level as exemplified by decaying states or resonances. Resonances are mathematically represented by Gamow kets, functionals over a space of very well-behaved…
The practically useful criteria of separable states $ρ=\sum_{k}w_{k}ρ_{k}$ in $d=2\times2$ are discussed. The equality $G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0$ for any two projection operators $P({\bf a})$ and $P({\bf b})$ provides a necessary and sufficient separability criterion in the case of a separable pure state $ρ=|ψ\rangle\langleψ|$. We propose the separability criteria of mixed states, which are given by ${\rm Tr}ρ\{{\bf a}\cdot {\bf σ}\otimes {\bf b}\cdot {\bf σ}\…
Milz and Strunz recently reported substantial evidence to further support the previously conjectured separability probability of $\frac{8}{33}$ for two-qubit systems ($ρ$) endowed with Hilbert-Schmidt measure. Additionally, they found that along the radius ($r$) of the Bloch ball representing either of the two single-qubit subsystems, this value appeared constant (but jumping to unity at the locus of the pure states, $r=1$). Further, they also observed (personal communication) such separability probability $r$-invariance, when using, more broadly, random induced measure ($K=3,4,5,\ldots$), wit…
We develop a general framework for the construction of probabilities for the time of arrival in quantum systems. The time of arrival is identified with the time instant when a transition in the detector's degrees of freedom takes place. Thus, its definition is embedded within the larger issue of defining probabilities with respect to time for general quantum transitions. The key point in our analysis is that we manage to reduce the problem of defining a quantum time observable to a mathematical model where time is associated to a transition from a subspace of the Hilbert space of the total sys…
A SIC consists of N^2 equiangular unit vectors in an N dimensional Hilbert space. The frame potential is a function of N^2 unit vectors. It has a unique global minimum if the vectors form a SIC, and this property has been made use of in numerical searches for SICs. When the vectors form an orbit of the Heisenberg group the frame potential becomes a function of a single fiducial vector. We analytically compute the average of this function over Hilbert space. We also compute averages when the fiducial vector is placed in certain special subspaces defined by the Clifford group.
Center Leo Apostel for Interdisciplinary Studies, Brussels Free University（布鲁塞尔自由大学利奥·阿波斯泰尔跨学科研究中心）；Laboratorio di Autoricerca di Base（基础自主研究实验室）；University of Leicester（莱斯特大学）
The research on human cognition has recently benefited from the use of the mathematical formalism of quantum theory in Hilbert space. However, cognitive situations exist which indicate that the Hilbert space structure, and the associated Born rule, would be insufficient to provide a satisfactory modeling of the collected data, so that one needs to go beyond Hilbert space. In Part I of this paper we follow this direction and present a general tension-reduction (GTR) model, in the ambit of an operational and realistic framework for human cognition. In this Part II we apply this non-Hilbertian qu…
We detect a certain pattern of behavior of separability probabilities $p(r_A,r_B)$ for two-qubit systems endowed with Hilbert-Schmidt, and more generally, random induced measures, where $r_A$ and $r_B$ are the Bloch radii ($0 \leq r_A,r_B \leq 1$) of the qubit reduced states ($A,B$). We observe a relative repulsion of radii effect, that is $p(r_A,r_A) < p(r_A,1-r_A)$, except for rather narrow "crossover" intervals $[\tilde{r}_A,\frac{1}{2}]$. Among the seven specific cases we study are, firstly, the "toy" seven-dimensional $X$-states model and, then, the fifteen-dimensional two-qubit states ob…
We investigate the lower bound of the amount of entanglement for faithfully teleporting a quantum state belonging to a subset of the whole Hilbert space. Moreover, when the quantum state belongs to a set composed of two states, a probabilistic teleportation scheme is presented using a non-maximally entangled state as the quantum channel. We also calculate the average transmission efficiency of this scheme.
The observed probabilities of quantum mechanics possess a time asymmetry which is based on the truism that a state must be prepared before an observable can be measured in it. While Hilbert space quantum theory cannot incorporate this arrow of time, the Rigged Hilbert Space (RHS) formulation of quantum mechanics provides a theory of time symmetric as well as time asymmetric quantum physics.
The general problem is studied on a simple example. A quantum particle in an infinite one-dimensional well potential is considered. Let the boundaries of well changes in a finite time $T$. The standard methods for calculating probability of transition from an initial to the final state are in general inapplicable since the states of different wells belong to different Hilbert spaces. If the final well covers only a part of the initial well (and, possibly, some outer part of the configuration space), the total probability of the transition from any stationary state of the initial well into {\bf…
In this paper we elaborate on the structure of the continuous-time histories description of quantum theory, which stems from the consistent histories scheme. In particular, we examine the construction of history Hilbert space, the identification of history observables and the form of the decoherence functional (the object that contains the probability information). It is shown how the latter is equivalent to the closed-time-path (CTP) generating functional. We also study the phase space structure of the theory first through the construction of general representations of the history group (the …
We prove the existence for each Hilbert space of the two new quasi hidden variable (qHV) models, statistically noncontextual and context-invariant, reproducing all the von Neumann joint probabilities via nonnegative values of real-valued measures and all the quantum product expectations -- via the qHV (classical-like) average of the product of the corresponding random variables. In a context-invariant model, a quantum observable X can be represented by a variety of random variables satisfying the functional condition required in quantum foundations but each of these random variables equivalent…
Quantum mechanics introduces the concept of probability at the fundamental level, yielding the measurement problem. On the other hand, recent progress in cosmology has led to the "multiverse" picture, in which our observed universe is only one of the many, bringing an apparent arbitrariness in defining probabilities, called the measure problem. In this paper, we discuss how these two problems are intimately related with each other, developing a complete picture for quantum measurement and cosmological histories in the quantum mechanical universe. On one hand, quantum mechanics eliminates the a…
