Epistemic Restriction Research Articles

A quantum prediction as a collection of epistemically restricted classical predictions

Spekkens has introduced an epistemically restricted classical theory of discrete systems, based on discrete phase space. The theory manifests a number of quantum-like properties but cannot fully imitate quantum theory because it is noncontextual. In this paper we show how, for a certain class of quantum systems, the quantum description of an experiment can be decomposed into classical descriptions that are epistemically restricted, though in a different sense than in Spekkens' work. For each aspect of the experiment—the preparation, the transformations, and the measurement—the epistemic restriction limits the form of the probability distribution an imagined classical observer may use. There are also global constraints that the whole collection of classical descriptions must satisfy. Each classical description generates its own prediction regarding the outcome of the experiment. One recovers the quantum prediction via a simple but highly nonclassical rule: the "nonrandom part" of the predicted quantum probabilities is obtained by summing the nonrandom parts of the classically predicted probabilities. By "nonrandom part" we mean the deviation from complete randomness, that is, from what one would expect upon measuring the fully mixed state.

Contextuality, Fine-Tuning and Teleological Explanation

I assess various proposals for the source of the intuition that there is something problematic about contextuality, ultimately concluding that contextuality is best thought of in terms of fine-tuning. I then argue that as with other fine-tuning problems in quantum mechanics, this behaviour can be understood as a manifestation of teleological features of physics. Finally I discuss several formal mathematical frameworks that have been used to analyse contextuality and consider how their results should be interpreted by scientific realists. In the course of this discussion I obtain several new mathematical results - I demonstrate that preparation contextuality is a form of fine-tuning, I show that measurement contextuality can be explained by appeal to a global constraint forbidding closed causal loops, and I demonstrate how negative probabilities can arise from a classical ontological model together with an epistemic restriction.

Metaphilosophy and Argument: The Case of the Justification of Abduction

This paper is an essay on metaphilosophy that reviews, describes, categorises, and discusses different ways philosophers have approached the justification of abduction as a mode of reasoning and arguing. Advocating an argumentative approach to abduction, I model the philosophical debate over its justification as the critical assessment of a warrant-establishing argument allowing “H explains D” to be used as a reason for “H can be inferred from D.” Philosophers have discussed the conditions under which such kind of generic argument can be accepted, and I identify five kinds of such conditions, namely: a) dialectical/procedural restriction; b) claim restriction; c) restriction over acceptable explanatory principles; d) balancing restriction; and e) epistemic restriction.

Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable

What does it take for real-deterministic $c$-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic $c$-valued variables out of the weak values obtained via a nonperturbing weak measurement of quantum operators with a postselection over a complete set of state vectors basis, which always satisfies the Kennard-Robertson-Schr\"odinger uncertainty relation. First, we introduce an auxiliary global random variable and couple it to the imaginary part of the weak value to convert the incompatibility between the quantum operator and the basis into the fluctuation of an ``error term,'' and then superimpose it onto the real part of the weak value. We show that this class of ``$c$-valued physical quantities'' provides a real-deterministic contextual hidden-variable model for the quantum expectation value of a certain class of operators. We then show that the Schr\"odinger and the Kennard-Robertson lower bounds can be obtained separately by imposing the classical uncertainty relation to the $c$-valued physical quantities associated with a pair of Hermitian operators. Within the representation, the complementarity between two incompatible quantum observables manifests the absence of a basis wherein the error terms of the associated two $c$-valued physical quantities simultaneously vanish. Furthermore, quantum uncertainty principle is captured by a specific irreducible epistemic restriction between the two $c$-valued physical quantities, foreign in classical mechanics, constraining the allowed form of their joint distribution. We then suggest an epistemic interpretation of the two terms decomposing the $c$-valued physical quantity as the optimal estimate under the epistemic restriction and the associated estimation error, and discuss the classical limit.

Derivation of Pauli Equation and its Bipartite Form based on Budiyono-Rohrlich Ontic Extension and Epistemic Restriction of Statistical Model of Quantum Mechanics

In this report, we discuss a possible way to derive the Pauli equation of a spin−1/2 particle based on Budiyono-Rohrlich ontic extension and epistemic restriction axioms of a statistical model of quantum mechanics. Applying the conservation of energy and probability continuity equation, we show that for a pure quantum mechanical particle spin, the corresponding equation can be reconstructed by assuming the existence of separable probability distribution functions of spin. The associated particle spin is still considered as a pure quantum phenomenon which has no classical counterpart. We also discuss the possibility of constructing a bipartite Pauli equation of nonlocal interacting two spin−1/2 particles. We examine the corresponding necessary conditions for the separability of the Bell-like-states and Werner-like-states through the Peres–Horodecki positive partial transpose criterion.

Efficient classical computation of expectation values in a class of quantum circuits with an epistemically restricted phase space representation

We devise a classical algorithm which efficiently computes the quantum expectation values arising in a class of continuous variable quantum circuits wherein the final quantum observable—after the Heisenberg evolution associated with the circuits—is at most second order in momentum. The classical computational algorithm exploits a specific epistemic restriction in classical phase space which directly captures the quantum uncertainty relation, to transform the quantum circuits in the complex Hilbert space into classical albeit unconventional stochastic processes in the phase space. The resulting multidimensional integral is then evaluated using the Monte Carlo sampling method. The convergence rate of the classical sampling algorithm is determined by the variance of the classical physical quantity over the epistemically restricted phase space distribution. The work shows that for the specific class of computational schemes, Wigner negativity is not a sufficient resource for quantum speedup. It highlights the potential role of the epistemic restriction as an intuitive conceptual tool which may be used to study the boundary between quantum and classical computations.

Nonlinear Schrödinger equations and generalized Heisenberg uncertainty principle from estimation schemes violating the principle of estimation independence

One of the advantages of a reconstruction of quantum mechanics based on transparent physical axioms is that it may offer insight to naturally generalize quantum mechanics by relaxing the axioms. Here, we discuss possible extensions of quantum mechanics within a general epistemic framework based on an operational scheme of estimation of momentum given positions under epistemic restriction. The epistemic restriction is parameterized by a global-nonseparable random variable on the order of Planck constant, an ontic extension to the separable classical phase space variables. Within the estimation scheme, the canonical quantum laws is reconstructed for a specific estimator and estimation error. In the present work, keeping the Born's quadratic law intact, we construct a class of nonlinear variants of Schr\"odinger equation and generalized Heisenberg uncertainty principle within the estimation scheme by assuming a more general class of estimation errors. The nonlinearity of the Schr\"odinger equation and the deviation from the Heisenberg uncertainty principle thus have a common transparent operational origin in terms of generalizations of estimation errors. We then argue that a broad class of nonlinearities and deviations from Heisenberg uncertainty principle arise from estimation errors violating a plausible inferential-causality principle of estimation independence which is respected by the standard quantum mechanics. This result therefore constrains possible extensions of quantum mechanics, and suggests directions to generalize quantum mechanics which comply with the principle of estimation independence.

Estimation independence as an axiom for quantum uncertainty

Quantum uncertainty is the cornerstone of quantum mechanics which underlies many counterintuitive nonclassical phenomena. Recent studies remarkably showed that it also fundamentally limits nonclassical correlation, and crucially, a deviation from its exact form may lead to a violation of the second law of thermodynamics. Are there deep and natural principles which uniquely determines its form? Here we work within a general epistemic framework for a class of nonclassical theories, introducing an epistemic restriction to an otherwise classical theory, so that the distributions of positions are irreducibly parameterized by the underlying momentum fields. It was recently shown that the mathematics of quantum mechanics formally arises within an operational scheme, wherein an agent makes a specific estimation of the momentum given information on the positions and the experimental settings. Moreover, quantum uncertainty can be traced back to the `specific' choice of estimator and the associated estimation error. In the present work, we show that a plausible principle of estimation independence, which requires that the estimation of momentum of one system must be independent of the position of another system independently prepared of the first, singles out the specific forms of the estimator, and especially the estimation error up to its strength given by a global-nonseparable random variable on the order of Planck constant.

Quantum mechanics as a calculus for estimation under epistemic restriction

Consider a statistical model with an epistemic restriction such that, unlike in classical mechanics, the allowed distribution of positions is fundamentally restricted by the form of an underlying momentum field. Assume an agent (observer) who wishes to estimate the momentum field given information on the conjugate positions. We discuss a classically consistent, weakly unbiased, best estimation of the momentum field minimizing the mean squared error, based on which the abstract mathematical rules of quantum mechanics can be derived. The results suggest that quantum wave function is not an objective agent-independent attribute of reality, but represents the agent's best estimation of the momentum, given the positions, under epistemic restriction. Quantum uncertainty and complementarity between momentum and position find their epistemic origin from the trade-off between the mean squared errors of simultaneous estimations of momentum field and mean position, with the Gaussian wave function represents the simultaneous efficient estimations, achieving the Cram\'er-Rao bounds of the associated mean squared errors. We then argue that unitary time evolution and wave function collapse in measurement are normative rules for an agent to update her/his estimation given information on the experimental settings.

Epistemically restricted phase-space representation, weak momentum value, and reconstruction of the quantum wave function

A phase space distribution associated with a quantum state was previously proposed, which incorporates a specific epistemic restriction parameterized by a global random variable on the order of Planck constant, transparently manifesting quantum uncertainty in phase space. Here we show that the epistemically restricted phase space (ERPS) distribution can be determined via weak measurement of momentum followed by post-selection on position. In the ERPS representation, the phase and amplitude of the wave function are neatly captured respectively by the position-dependent (conditional) average and variance of the epistemically restricted momentum fluctuation. They are in turn respectively determined by the real and imaginary parts of the weak momentum value, permitting a reconstruction of wave function using weak momentum value measurement, and an interpretation of momentum weak value in term of epistemically restricted momentum fluctuations. The ERPS representation thus provides a transparent and rich framework to study the deep conceptual links between quantum uncertainty embodied in epistemic restriction, quantum wave function, and weak momentum measurement with position post-selection, which may offer useful insight to better understand their meaning.

State-injection schemes of quantum computation in Spekkens' toy theory

Spekkens' toy theory is a non-contextual hidden variable model with an epistemic restriction, a constraint on what the observer can know about the reality. It has been shown in [3] that for qudits of odd dimensions it is operationally equivalent to stabiliser quantum mechanics by making use of Gross' theory of discrete Wigner functions. This result does not hold in the case of qubits, because of the unavoidable negativity of any Wigner function representation of qubit stabiliser quantum mechanics. In this work we define and characterise the subtheories of Spekkens' theory that are operationally equivalent to subtheories of stabiliser quantum mechanics. We use these Spekkens' subtheories as a unifying framework for the known examples of state-injection schemes where contextuality is an injected resource to reach universal quantum computation. In addition, we prove that, in the case of qubits, stabiliser quantum mechanics can be reduced to a Spekkens' subtheory in the sense that all its objects that do not belong to the Spekkens' subtheory, namely non-covariant Clifford gates, can be injected. This shows that within Spekkens' subtheories we possess the toolbox to perform state-injection of every object outside of them and it suggests that there is no need to use bigger subtheories to reach universal quantum computation via state-injection. We conclude with a novel scheme of computation suggested by our approach which is based on the injection of CCZ states and we also relate different proofs of contextuality to different state injections of non-covariant gates.

Interferometric Computation Beyond Quantum Theory

There are quantum solutions for computational problems that make use of interference at some stage in the algorithm. These stages can be mapped into the physical setting of a single particle travelling through a many-armed interferometer. There has been recent foundational interest in theories beyond quantum theory. Here, we present a generalized formulation of computation in the context of a many-armed interferometer, and explore how theories can differ from quantum theory and still perform distributed calculations in this set-up. We shall see that quaternionic quantum theory proves a suitable candidate, whereas box-world does not. We also find that a classical hidden variable model first presented by Spekkens [Phys. Rev. A, 75:3:32100, 2007] can also be used for this type of computation due to the epistemic restriction placed on the hidden variable.

Quantum mechanics as classical statistical mechanics with an ontic extension and an epistemic restriction

Where does quantum mechanics part ways with classical mechanics? How does quantum randomness differ fundamentally from classical randomness? We cannot fully explain how the theories differ until we can derive them within a single axiomatic framework, allowing an unambiguous account of how one theory is the limit of the other. Here we derive non-relativistic quantum mechanics and classical statistical mechanics within a common framework. The common axioms include conservation of average energy and conservation of probability current. But two axioms distinguish quantum mechanics from classical statistical mechanics: an “ontic extension” defines a nonseparable (global) random variable that generates physical correlations, and an “epistemic restriction” constrains allowed phase space distributions. The ontic extension and epistemic restriction, with strength on the order of Planck’s constant, imply quantum entanglement and uncertainty relations. This framework suggests that the wave function is epistemic, yet it does not provide an ontic dynamics for individual systems.

Spekkens’ toy model in all dimensions and its relationship with stabiliser quantum mechanics

Spekkens’ toy model is a non-contextual hidden variable model with an epistemic restriction, a constraint on what an observer can know about reality. The aim of the model, developed for continuous and discrete prime degrees of freedom, is to advocate the epistemic view of quantum theory, where quantum states are states of incomplete knowledge about a deeper underlying reality. Many aspects of quantum mechanics and protocols from quantum information can be reproduced in the model. In spite of its significance, a number of aspects of Spekkens’ model remained incomplete. Formal rules for the update of states after measurement had not been written down, and the theory had only been constructed for prime-dimensional and infinite dimensional systems. In this work, we remedy this, by deriving measurement update rules and extending the framework to derive models in all dimensions, both prime and non-prime. Stabiliser quantum mechanics (SQM) is a sub-theory of quantum mechanics with restricted states, transformations and measurements. First derived for the purpose of constructing error correcting codes, it now plays a role in many areas of quantum information theory. Previously, it had been shown that Spekkens’ model was operationally equivalent to SQM in the case of odd prime dimensions. Here, exploiting known results on Wigner functions, we extend this to show that Spekkens’ model is equivalent to SQM in all odd dimensions, prime and non-prime. This equivalence provides new technical tools for the study of technically difficult compound-dimensional SQM.

THE NEUROSCIENCE OF WESLEYAN SOTERIOLOGY: THE DYNAMIC OF BOTH INSTANTANEOUS AND GRADUAL CHANGE

AbstractIn his work Rewired: Exploring Religious Conversion, dealing with Wesleyan soteriology and neuroscience, Paul Markham claims that when one incorporates biology as an epistemic restriction in theologies of conversion, doctrines of instantaneous conversion are invalidated. He asserts that conversion must always be gradual, because the mechanism by which the brain changes in response to experience does not occur instantaneously; rather change is initiated and consolidated over an often lengthy span of time. I argue, however, that doctrines of instantaneous conversion are maintained when taking neuroscience into account. First, for doctrines of conversion that hold to the imputation of Christ's righteousness, neuroscience is irrelevant, because statements of instantaneous change are in terms of a relational status and not biological. Rapid conversion is maintained as a metaphysical position. Second, an embodied and neurologically realized change is expected in theologies of conversion that hold to impartation and, contrary to Markham, immediate change is neurologically possible in a variety of ways.

Weak values in a classical theory with an epistemic restriction

Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value: a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in our theory of classical mechanics with an epistemic restriction. By using this epistemically restricted theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimized and the limits of weak measurement.

A System-Level Game Semantics

Game semantics is a trace-like denotational semantics for programming languages where the notion of legal observable behaviour of a term is defined combinatorially, by means of rules of a game between the term (the Proponent) and its context (the Opponent). In general, the richer the computational features a language has the less constrained the rules of the semantic game. In this paper we consider the consequences of taking this relaxation of rules to the limit, by granting the Opponent omnipotence, that is, permission to play any move without combinatorial restrictions. However, we impose an epistemic restriction by not granting Opponent omniscience, so that Proponent can have undisclosed secret moves. We introduce a basic C-like programming language and we define such a semantic model for it. We argue that the resulting semantics is an appealingly simple combination of operational and game semantics and we show how certain traces explain system-level attacks, i.e. plausible attacks that are realisable outside of the programming language itself. We also show how allowing Proponent to have secrets ensures that some desirable equivalences in the programming language are preserved.

Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction

How would the world appear to us if its ontology was that of classical mechanics but every agent faced a restriction on how much they could come to know about the classical state? We show that in most respects, it would appear to us as quantum. The statistical theory of classical mechanics, which specifies how probability distributions over phase space evolve under Hamiltonian evolution and under measurements, is typically called Liouville mechanics, so the theory we explore here is Liouville mechanics with an epistemic restriction. The particular epistemic restriction we posit as our foundational postulate specifies two constraints. The first constraint is a classical analogue of Heisenberg's uncertainty principle -- the second-order moments of position and momentum defined by the phase-space distribution that characterizes an agent's knowledge are required to satisfy the same constraints as are satisfied by the moments of position and momentum observables for a quantum state. The second constraint is that the distribution should have maximal entropy for the given moments. Starting from this postulate, we derive the allowed preparations, measurements and transformations and demonstrate that they are isomorphic to those allowed in Gaussian quantum mechanics and generate the same experimental statistics. We argue that this reconstruction of Gaussian quantum mechanics constitutes additional evidence in favour of a research program wherein quantum states are interpreted as states of incomplete knowledge, and that the phenomena that do not arise in Gaussian quantum mechanics provide the best clues for how one might reconstruct the full quantum theory.