The dynamical equation of hybrid systems, being the combination of Schrödinger and Liouville equations, produces noncausal evolution when the initial state of interacting quantum and classical mechanical systems is as it is demanded in discussions regarding the problem of measurement. It is found that state of quantum mechanical system instantaneously collapses due to the non-negativity of probabilities.
We present a method for estimating the probabilities of outcomes of a quantum circuit using Monte Carlo sampling techniques applied to a quasiprobability representation. Our estimate converges to the true quantum probability at a rate determined by the total negativity in the circuit, using a measure of negativity based on the 1-norm of the quasiprobability. If the negativity grows at most polynomially in the size of the circuit, our estimator converges efficiently. These results highlight the role of negativity as a measure of non-classical resources in quantum computation.
Any bipartite quantum state has quasi-probability representations in terms of separable states. For entangled states these quasi-probabilities necessarily exhibit negativities. Based on the general structure of composite quantum states, one may reconstruct such quasi-propabilities from experimental data. Because of ambiguity, the quasi-probabilities obtained by the bare reconstruction are insufficient to identify entanglement. An optimization procedure is introduced to derive quasi-probabilities with a minimal amount of negativity. Negativities of optimized quasi-probabilities unambiguously pr…
We give a general proof that Hughston's stochastic extension of the Schrödinger equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We also show that for a system composed of independent subsystems, Hughston's equation separates into similar independent equations for the each of the subsystems, correlated only through the common Wiener process that drives the state reduction.
We show that including both the system and the apparatus in the quantum description of the measurement process, and using the concept of conditional probabilities, it is possible to deduce the statistical operator of the system after a measurement with a given result, which gives the probability distribution for all possible consecutive measurements on the system. This statistical operator, representing the state of the system after the first measurement, is in general not the same that would be obtained using the postulate of collapse.
We propose Bell inequalities for discrete or continuous quantum systems which test the compatibility of quantum physics with an interpretation in terms of deterministic hidden-variable theories. The wave function collapse that occurs in a sequence of quantum measurements enters the upper bound via the concept of quantum conditional probabilities. The resulting hidden-variable inequality is applicable to an arbitrary observable that is decomposable into a weighted sum of non-commuting projectors. We present local and non-local examples of violation of generalized Bell inequalities in phase spac…
Quantum computers will provide considerable speedups with respect to their classical counterparts. However, the identification of the innately quantum features that enable these speedups is challenging. In the continuous-variable setting - a promising paradigm for the realisation of universal, scalable, and fault-tolerant quantum computing - contextuality and Wigner negativity have been perceived as two such distinct resources. Here we show that they are in fact equivalent for the standard models of continuous-variable quantum computing. While our results provide a unifying picture of continuo…
In a recent article [Phys. Rev. A 94, 052128 (2016)], the authors compute the predictions of two collapse models on the transition probabilities of neutral mesons. Notably, they claim to find an influence on the decay rates and attempt to prove that a new parameter $θ(0)$ is required to fully characterize the noise of standard collapse models. These two claims are incorrect and motivated by flawed computations. This comment derives the correct transition probabilities exactly from the master equation, explains how they could be computed perturbatively in a safe way and finally shows where the …
We discuss the role of contextuality within quantum fluctuation theorems, in the light of a recent no-go result by Perarnau \emph{et al}. We show that any fluctuation theorem reproducing the two-point-measurement scheme for classical states either admits a notion of work quasi-probability or fails to describe protocols exhibiting contextuality. Conversely, we describe a protocol that smoothly interpolates between the two-point measurement work distribution for projective measurements and Allahverdyan's work quasi-probability for weak measurements, and show that the negativity of the latter is …
Zurek claims to have derived Born's rule noncircularly in the context of an ontological no-collapse interpretation of quantum states, without any "deus ex machina imposition of the symptoms of classicality." After a brief review of Zurek's derivation it is argued that this claim is exaggerated if not wholly unjustified. In order to demonstrate that Born's rule arises noncircularly from deterministically evolving quantum states, it is not sufficient to assume that quantum states are somehow associated with probabilities and then prove that these probabilities are given by Born's rule. One has t…
Is wave function collapse a prediction of the Schrödinger equation? This unusual problem is explored in an enlarged framework of interpretation, where quantum dynamics is considered exact and its interpretation is extended to include local entanglement of two systems, including a macroscopic one. This property of local entanglement, which results directly from the Schrödinger equation but is unrelated with observables, is measured by local probabilities, fundamentally distinct from quantum probabilities and evolving nonlinearly. When applied to a macroscopic system and the fluctuations in its …
We analyze the objective meaning of probabilities in the context of the many-worlds interpretation of Everett. For this purpose we study in details the weak law of large numbers and the role of typicality and universally negligible probabilities (through the works of Cournot and Borel). We demonstrate that Everett's theory doesn't provide any clue for fixing a probability rule and therefore contradicts irrevocably empirical facts and Born's law.
A natural generalization of the CSL (Continuous Spontaneous Localization) theory of dynamical collapse is applied to a relativistic quantum scalar field $ϕ({\bf x},t)$. It is shown that the modified Schrödinger equation is relativistically invariant, that the probabilities associated to all possible values of the classical scalar random field $w({\bf x},t)$ (which determines the eventual state of collapse) add up to 1, that there is no energy production out of the vacuum and, in the limit of large time, the collapse is toward eigenstates of $ϕ({\bf x},0)$.
Many of the conceptual problems students have in understanding quantum mechanics arise from the way probabilities are introduced in standard (textbook) quantum theory through the use of measurements. Introducing consistent microscopic probabilities in quantum theory requires setting up appropriate sample spaces taking proper account of quantum incompatibility. When this is done the Schrodinger equation can be used to calculate probabilities independent of whether a system is or is not being measured, and the results usually ascribed to wave function collapse are obtained in a less misleading w…
We set up a general formalism for models of spontaneous wave function collapse with dynamics represented by a stochastic differential equation driven by general Gaussian noises, not necessarily white in time. In particular, we show that the non-Schrodinger terms of the equation induce the collapse of the wave function to one of the common eigenstates of the collapsing operators, and that the collapse occurs with the correct quantum probabilities. We also develop a perturbation expansion of the solution of the equation with respect to the parameter which sets the strength of the collapse proces…
It is argued that Feynman's rules for evaluating probabilities, combined with von Neumann's principle of psycho-physical parallelism, help avoid inconsistencies, often associated with quantum theory. The former allows one to assign probabilities to entire sequences of hypothetical Observers' experiences, without mentioning the problem of wave function collapse.The latter limits the Observer's (e.g., Wigner's friend's) participation in a measurement to the changes produced in material objects,thus leaving his/her consciousness outside the picture.
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…
We analyze three important experimental domains (SQUIDs, molecular interferometry, and Bose-Einstein condensation) as well as quantum-biophysical studies of the neuronal apparatus to argue that (i) the universal validity of unitary dynamics and the superposition principle has been confirmed far into the mesoscopic and macroscopic realm in all experiments conducted thus far; (ii) all observed "restrictions" can be correctly and completely accounted for by taking into account environmental decoherence effects; (iii) no positive experimental evidence exists for physical state-vector collapse; (iv…
The quantum measurement problem may have a resolution in de Broglie-Bohm theory in which measurements lead to dynamical wavefunction collapse. We study the collapse in a simple setup and find that there may be slight differences between probabilities derived from standard quantum mechanics versus those from de Broglie-Bohm theory in certain situations, possibly paving the way for an experimental test.
We believe that the hypothesis `it from bit' originates from the assumption that probabilities have a fundamental, irremovable status in quantum theory. We argue against this assumption and highlight four well-known reformulations / modifications of the theory in which probabilities and the measuring apparatus do not play a fundamental role. These are: Bohmian Mechanics, Dynamical Collapse Models, Trace Dynamics, and Quantum Theory without Classical Time. Here the `it' is primary and the `bit' is derived from the `it'.
We consider quantum computations comprising only commuting gates, known as IQP computations, and provide compelling evidence that the task of sampling their output probability distributions is unlikely to be achievable by any efficient classical means. More specifically we introduce the class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection, and prove first that post-IQP equals the classical class PP. Using this result we show that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, even u…
Two categories of results regarding quantum measurements are derived in this work and applied to the problem of collapse. The first category is concerned with local and transient features of the entanglement between a macroscopic measuring system and a measured one. These properties result directly from the Schrödinger equation. They cannot be formulated in terms of observables, do not affect the wave functions themselves but express their history in an irreversible way. They carry a specific new kind of local probabilities, which evolve with a finite velocity under nonlinear wave equations. T…
The dynamics of a quantum system with internal degrees of freedom undergoing spontaneous collapse in the position basis are analysed; e.g., neutral mesons or neutrinos. Surprisingly, the value of the Heaviside function $θ(x)$ at $x=0$ that can in general be chosen in the interval [0,1] leads to different physical predictions. For the QMUPL (Quantum Mechanics with Universal Position Localization) model only a single value leads to probabilities conserving the particle number. Herewith the physical properties of the noise field can be constrained. This opens a road to study the physical properti…
We study the Two Photon Quantum Rabi Model by way of its spectral functions and survival probabilities. This approach allows numerical precision with large truncation numbers, and thus exploration of the spectral collapse. We provide independent checks and calibration of the numerical results by studying an exactly solvable case and comparing the essential qualitative structure of the spectral functions. We stress that the large time limit of the survival probability provides us with an indicator of spectral collapse, and propose a technique for the detection of this signal in the current and …
Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to within $\exp(-Ω(m\log m))$ additive error is hard for any classical computer, where $m$ is the number of gates in the quantum computation. More precisely, we show that the above problem is $\#\mathsf{P}$-hard under $\mathsf{BPP}^{\mathsf{NP}}$ reduction. I…
This paper is a comparison of two theories of the probability of a history in quantum mechanics. One is derived from Copenhagen quantum mechanics using the projection postulate and is the basis of the "consistent histories" interpretation; the other is based on a proposal by Bell, originally for the "pilot state" theory but here applied to pure unitary quantum mechanics. The first can be used for a wider class of histories but depends on the projection postulate, or "collapse", which is widely held to be an unsatisfactory feature of the theory; the second can be used in a theory of the univers…
We analyze in mathematical detail, within the framework of the QMUPL model of spontaneous wave function collapse, the von Neumann measurement scheme for the measurement of a 1/2 spin particle. We prove that, according to the equation of the model: i) throughout the whole measurement process, the pointer of the measuring device is always perfectly well localized in space; ii) the probabilities for the possible outcomes are distributed in agreement with the Born probability rule; iii) at the end of the measurement the state of the microscopic system has collapsed to the eigenstate corresponding …
An experiment for checking the Dynamical Spacetime approach to wavefunction collapse is proposed. The Dynamical Spacetime approach predicts deviations from Born's rule, when a solid evolves into a three-state superposition, and when the displacement between the superposed states is, at the reduction point in time, significantly larger than the spatial variation of the solids nuclei, being typically on the order of a tenth of an Angstroem. The solid is brought into the three-state superposition by splitting a photon into three beams and by detecting it in each beam by avalanche photodiodes, whi…
A new formulation of quantum mechanics is developed which does not require the concept of the wave-particle duality. Rather than assigning probabilities to outcomes, probabilities are instead assigned to entire fine-grained histories. The formulation is fully relativistic and applicable to multi-particle systems. It shall be shown that this new formulation makes the same experimental predictions as quantum field theory, but without having to rely upon the notion of a system evolving in a superposition of quantum states until collapsed by an observation. It is thus free from the problem of deci…
In this paper we introduce the three main notions of probability used by physicists and discuss how these are to be used when invoking spacelike separated observers in a relativistic format. We discuss a standard EPRB experiment and concentrate upon problems of the interpretation of probabilities. We promote a particularly conservative interpretation of this experiment (which need not invoke an objective notion of collapse) where probabilities are, tentatively, passively Lorentz invariant. We also argue that the Heisenberg picture is preferable in relativistic situations due to a conflict betw…
In this work we revisit collapse and revival oscillations in superfluids suddenly quenched by strong local interactions for the case of a one-dimensional Bose-Hubbard model. As the main result we identify the inherent nonequilibrium quantum many-body character of these oscillations by revealing that they are controlled by a sequence of underlying dynamical quantum phase transitions in the real-time evolution after the quench, which manifest as temporal nonanalyticities in return probabilities or Loschmidt echos. Specifically, we find that the time scale of the collapse and revival oscillations…
Collapse models including some external noise of unknown origin are routinely used to describe phenomena on the quantum-classical border; in particular, quantum measurement. Although containing nonlinear dynamics and thereby exposed to the possibility of superluminal signaling in individual events, such models are widely accepted on the basis of fully reproducing the non-signaling statistical predictions of quantum mechanics. Here we present a deterministic nonlinear model without any external noise, in which randomness - instead of being universally present - emerges in the measurement proces…
In this treatise I introduce the time dependent Generalized Born's Rule for the probabilities of quantum events, including conditional and consecutive probabilities, as the unique fundamental time evolution equation of quantum theory. Then these probabilities, computed from states and events, are to be compared with relative frequencies of observations. Schrodinger's equation still is valid in one model of the axioms of quantum theory, which I call the Schrodinger model. However, the role of Schrodinger's equation is auxiliary, since it serves to help compute the continuous temporal evolution …
The quantum measurement problem can be regarded as the tension between the two alternative dynamics prescribed by quantum mechanics: the unitary evolution of the wave function and the state-update rule (or "collapse") at the instant a measurement takes place. The notorious Wigner's friend gedankenexperiment constitutes the paradoxical scenario in which different observers (one of whom is observed by the other) describe one and the same interaction differently, one --the Friend-- via state-update and the other --Wigner-- unitarily. This can lead to Wigner and his friend assigning different prob…
One attractive interpretation of quantum mechanics is the ensemble interpretation, where Quantum Mechanics merely describes a statistical ensemble of objects and not individual objects. But this interpretation does not address why the wave-function plays a central role in the calculations of probabilities, unlike most other interpretations of quantum mechanics. On the other hand, Classical Statistical Field Theory suffers from severe mathematical inconsistencies (specially for Hamiltonians which are non-polynomial in the fields, e.g. General relativistic statistical field theory). We claim tha…
Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi-Rimini-Weber (GRW) model of spontaneous wavefunction collapse. The GRW model was proposed as a solution of the measurement problem of quantum mechanics and involves a stochastic and nonlinear modification of the Schrödinger equation. It deviates very little from the Schrödinger equation for microscopic systems but efficiently suppresses, for macroscopic systems, superpositions of macroscopically different states. As suggested by Bell, we take the primitive ontology, or local be…
Quantum mechanics may be formulated as SENSIBLE QUANTUM MECHANICS (SQM) so that it contains nothing probabilistic, except, in a certain frequency sense, conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued AWARENESS OPERATORS in a quantum state of the universe which never jumps or collapses. Ratios of the measures for these sets of perceptions can be interpreted as frequency-type probabilities for many actually existing sets rather than as propensities for potentialities to be actualized, so the…
We examine the time evolution of a two level ion interacting with a light field in harmonic oscillator trap and in a trap with anharmonicities. The anharmonicities of the trap are quantified in terms of the deformation parameter $τ$ characterizing the q-analog of the harmonic oscillator trap. Initially the ion is prepared in a Schrödinger cat state. The entanglement of the center of mass motional states and the internal degrees of freedom of the ion results in characteristic collapse and revival pattern. We calculate numerically the population inversion I(t), quasi-probabilities $Q(t),$ and pa…
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…
A geometric approach to quantum mechanics with unitary evolution and non-unitary collapse processes is developed. In this approach the Schrodinger evolution of a quantum system is a geodesic motion on the space of states of the system furnished with an appropriate Riemannian metric. The measuring device is modeled by a perturbation of the metric. The process of measurement is identified with a geodesic motion of state of the system in the perturbed metric. Under the assumption of random fluctuations of the perturbed metric, the Born rule for probabilities of collapse is derived. The approach i…
It is convenient to describe a quantum system at all times by means of a "history operator" $C$, encoding measurements and unitary time evolution between measurements. These operators naturally arise when computing the probability of measurement sequences, and generalize the "sum over position histories " of the Feynman path-integral. As we argue in the present note, this description has some computational advantages over the usual state vector description, and may help to clarify some issues regarding nonlocality of quantum correlations and collapse. A measurement on a system described by $C$…
We outline the rationale and preliminary results of using the State Context Property (SCOP) formalism, originally developed as a generalization of quantum mechanics, to describe the contextual manner in which concepts are evoked, used, and combined to generate meaning. The quantum formalism was developed to cope with problems arising in the description of (1) the measurement process, and (2) the generation of new states with new properties when particles become entangled. Similar problems arising with concepts motivated the formal treatment introduced here. Concepts are viewed not as fixed rep…
We outline the rationale and preliminary results of using the state context property (SCOP) formalism, originally developed as a generalization of quantum mechanics, to describe the contextual manner in which concepts are evoked, used and combined to generate meaning. The quantum formalism was developed to cope with problems arising in the description of (i) the measurement process, and (ii) the generation of new states with new properties when particles become entangled. Similar problems arising with concepts motivated the formal treatment introduced here. Concepts are viewed not as fixed rep…
I argue that background freedom in quantum gravity automatically leads to a dissociation of the quantum state into states having a classical space. That is, interference is not completely well-defined for states with different space geometries, even if their linear combination is. Interference of states with different space geometry is still allowed at small scales, but precluded at macro-scales. Macrostates, including measuring devices, appear classical. The distribution of space geometries automatically gives the Born rule. The dissociation entails a kind of absolute decoherence, making the …
Quantum mechanics may be formulated as Sensible Quantum Mechanics (SQM) so that it contains nothing probabilistic, except, in a certain frequency sense, conscious perceptions. Sets of these perceptions can be deterministically realized with measures given by expectation values of positive-operator-valued awareness operators in a quantum state of the universe which never jumps or collapses. Ratios of the measures for these sets of perceptions can be interpreted as frequency-type probabilities for many actually existing sets rather than as propensities for potentialities to be actualized, so the…
