What Does the Operator Algebra of Quantum Statistics Tell Us about the Objective Causes of Observable Effects?

Abstract

Quantum physics can only make statistical predictions about possible measurement outcomes, and these predictions originate from an operator algebra that is fundamentally different from the conventional definition of probability as a subjective lack of information regarding the physical reality of the system. In the present paper, I explore how the operator formalism accommodates the vast number of possible states and measurements by characterizing its essential function as a description of causality relations between initial conditions and subsequent observations. It is shown that any complete description of causality must involve non-positive statistical elements that cannot be associated with any directly observable effects. The necessity of non-positive elements is demonstrated by the uniquely defined mathematical description of ideal correlations which explains the physics of maximally entangled states, quantum teleportation and quantum cloning. The operator formalism thus modifies the concept of causality by providing a universally valid description of deterministic relations between initial states and subsequent observations that cannot be expressed in terms of directly observable measurement outcomes. Instead, the identifiable elements of causality are necessarily non-positive and hence unobservable. The validity of the operator algebra therefore indicates that a consistent explanation of the various uncertainty limited phenomena associated with physical objects is only possible if we learn to accept the fact that the elements of causality cannot be reconciled with a continuation of observable reality in the physical object.

Statistical Signatures of Quantum Contextuality

“…As explained in previous work, Kirkwood-Dirac distributions are the most consistent representation of quantum statistics, accurately reproducing the relations between non-orthogonal states in the Hilbert space formalism [28][29][30][31][32][33][34][35]. Such relations represent the non-classical structure of operator statistics and ensure that quantum theory cannot be reconciled with any classical statistical theory [14,31].…”

“…Not surprisingly, the results show that the quantum formalism introduces non-positive quasi probabilities into the statistics, which explains the differences between the predictions of quantum mechanics and the non-contextual hidden variable theories proposed as an alternative. The problem is that there is a wide range of possible quasi probabilities based on different selections of measurements or different operator expansions [14]. It may therefore be more promising to investigate the relation between quantum states and experimentally observable statistics in more detail.…”

Contextuality of quantum fluctuations characterized by conditional weak values of entangled states

“…The quantum fluctuations of a physical property  are not just observed in direct measurements of that property, but also in measurements of any other orthogonal basis {|m }. The specific outcome |m is then decided by a function of the initial conditions expressed by the state |ψ and the physical property given by the operator  [11][12][13]. It is therefore possible to assign values of A(m) to each measurement outcome by combining the information contained in the initial state |ψ with the information gained from the measurement outcome |m .…”

“…All remaining physical properties can be expressed as functions of the physical properties determined in this direct manner [18,22]. The deviations between weak values and eigenvalues are a necessary consequence of the relation between physical properties described by the operator formalism [11][12][13]. The correct explanation for this deviation is the contextuality of quantum fluctuations.…”

Contextuality of quantum fluctuations characterized by conditional weak values of entangled states

“…All remaining physical properties can be expressed as functions of the physical properties determined in this direct manner [14,18]. The deviations between weak values and eigenvalues are a necessary consequence of the relation between physical properties described by the operator formalism [19,20]. The correct explanation for this deviation is the contextuality of quantum fluctuations.…”