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arXiv:1804.09248 · Progress of Theoretical and Experimental Physics

On statistical independence and no-correlation for a pair of random variables taking two values: Classical and quantum

Toru Ohira

Published Jul 10, 2018• 4 citations
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Abstract

It is well known that when a pair of random variables is statistically independent, it has no-correlation (zero covariance, |$E[XY] - E[X]E[Y] = 0$|⁠), and that the converse is not true. However, if both of these random variables take only two values, no-correlation entails statistical independence. We provide here a general proof. We subsequently examine whether this equivalence property carries over to quantum mechanical systems. A counter-example is explicitly constructed to show that it does not. This observation provides yet another simple theorem separating classical and quantum theories.

DOI
10.1093/ptep/pty086

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It is well known that when a pair of random variables is statistically independent, it has no-correlation (zero covariance, |E[XY]−E[X]E[Y]=0E[XY] - E[X]E[Y] = 0E[XY]−E[X]E[Y]=0|⁠), and that the converse is not true. However, if both of these random variables take only two values, no-correlation entails statistical independence. We provide here a general proof. We subsequently examine whether this equivalence property carries over to quantum mechanical systems. A counter-example is explicitly constructed to show that it does not. This observation provides yet another simple theorem separating classical and quantum theories.
Abstract We consider a quantum entangled state for two particles, each particle having two basis states, which includes an entangled pair of spin 1/2 particles. We show that, for any quantum entangled state vectors of such systems, one can always find a pair of observable operators X,Y\mathcal{X}, \mathcal{Y}X,Y with zero correlations (⟨ψ∣XY∣ψ⟩−⟨ψ∣X∣ψ⟩⟨ψ∣Y∣ψ⟩=0\langle{\psi}|\mathcal{X}\mathcal{Y}|{\psi}\rangle - \langle{\psi}|\mathcal{X}|{\psi}\rangle\langle{\psi}|\mathcal{Y}|{\psi}\rangle= 0⟨ψ∣XY∣ψ⟩−⟨ψ∣X∣ψ⟩⟨ψ∣Y∣ψ⟩=0). At the same time, if we consider the analogous classical system of a “classically entangled” (statistically non-independent) pair of random variables taking two values, one can never have zero correlations (zero covariance, E[XY]−E[X]E[Y]=0E[XY] - E[X]E[Y] = 0E[XY]−E[X]E[Y]=0). We provide a general proof to illustrate the different nature of entanglements in classical and quantum theories.
This paper is concerned with inferences from phenomenological variables to hidden causes or hidden variables. A number of theorems of a general sort are stated. The paper concludes with a treatment of Bell's inequalities and their generalization to more than four observables. INTRODUCTION In this paper we are concerned to present a number of theorems about inferences from phenomenological correlations to hidden causes. In other terms, the theorems are mainly theorems about hidden variables. Most of the proofs will not be given but references will be cited where they may be found. To emphasize conceptual matters and to keep technical simplicity in the forefront, we consider only two-valued random variables taking the values ±1. We shall also assume symmetry for these random variables in that their expectations will be zero and thus they will each have a positive variance of one. For emphasis we state: GENERAL ASSUMPTION The phenomenological random variables X 1 , …, X N have possible values ±1, with means E(X i ) = 0, 1 ≤ i ≤ N . We also use the notation X, Y , and Z for phenomenological random variables. We use the notation E(XY) for covariance which for these symmetric random variables is also the same as their correlation ρ( X, Y ).
We present a simple proof of the strong converse for identification via discrete memoryless quantum channels, based on a novel covering lemma. The new method is a generalization to quantum communication channels of Ahlswede's (1979, 1992) approach to classical channels. It involves a development of explicit large deviation estimates to the case of random variables taking values in self-adjoint operators on a Hilbert space. This theory is presented separately in an appendix, and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems.
We investigate the no-cloning theorem that relies on the properties of a statistical model. Usually, the no-cloning theorem implies that two quantum states are identical or orthogonal if we allow a cloning to be on the two quantum states. Here, we rely on a statistical model. We may result in the fact that the two quantum states under consideration could not be orthogonal if we accept the statistical model. The no-cloning theorem may imply that the two quantum states under consideration may be identical if we accept the statistical model. The no-cloning theorem itself has this character.
The term Bell's theorem refers to a set of closely related results which imply that quantum mechanics is incompatible with local hidden variable theories. Bell's inequality is the statement that if measurements are performed independently on two space-like separated particles of an entangled pair, the assumption that outcomes depend on hidden variables implies an upper bound on the correlations between the outcomes. Quantum mechanics predicts correlations which violate this upper bound. The CHSH inequality is a specific Bell inequality in which classical correlation (i.e. if local hidden variables exist) can take the maximum value of 2. Quantum mechanics violates this bound, allowing for a higher bound on the correlation, which can take the maximum value 222\sqrt{2}22​, known as the Tsirelson bound. Popescu and Rohrlich showed that the assumption of relativistic causality allows for an even higher bound on the CHSH correlation, this value being 4. Why is the bound coming from causality higher than the Tsirelson bound? Are there relativistic causal dynamical theories which violate the Tsirelson bound? In the present paper we answer this question in the affirmative. We show that the pre-quantum theory of trace dynamics, from which quantum theory is emergent as a thermodynamic approximation, permits the CHSH correlation to take values higher than 222\sqrt{2}22​. We interpret our findings to suggest that quantum theory is approximate, and emergent from the more general theory of trace dynamics.