Reasonable conditions for joint probabilities of non-commuting observables

Hofmann, Holger F.

doi:10.1007/s40509-014-0010-x

“…Instead, there are additional correlations associated with the discrepancy given by Eq. (25). It is worth considering the role of this discrepancy in more detail.…”

Section: Elements Of Causalitymentioning

confidence: 97%

“…As explained in [25], an additional requirement that uniquely identifies the Dirac distribution as the natural joint probability of two non-commuting observables is the condition that the quasi-probability P (a, b|ψ) should be zero for any state | ψ that is orthogonal to either | a or | b . Since this requirement can only be satisfied by multiplying the state with a projector, the operator of the quasi-probability is constructed from the product of the two projectors,…”

Section: Statistical Moments and Quasi-probabilitiesmentioning

confidence: 99%

What does the operator algebra of quantum statistics tell us about the objective causes of observable effects?

Hofmann

2020

Preprint

0

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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.

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“…As explained in [ 25 ], an additional requirement that uniquely identifies the Dirac distribution as the natural joint probability of two non-commuting observables is the condition that the quasi-probability

should be zero for any state

that is orthogonal to either

or

. Since this requirement can only be satisfied by multiplying the state with a projector, the operator of the quasi-probability is constructed from the product of the two projectors,

…”

Section: Statistical Moments and Quasi-probabilitiesmentioning

confidence: 99%

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

Hofmann

¹

2020

Entropy

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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.

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“…Namely, the topology induced from the total variation norm is too strong (fine) for our convenience. A fundamental cure for this would thus be to equip the space with a weaker (coarser) topology on M C (B) such that, at least, it may allow us to construct sufficiently abundant sequences (or nets, in general) of the 'inputs' in L 1 (B) that converges towards δ 0 , and that the sequence of the resulting 'outputs' (i.e., the multiplicative product (3.109)) would subsequently converge towards the desired solution in the limit 13 .…”

Section: Preliminary Observationsmentioning

confidence: 99%

Quasi-probabilities in conditioned quantum measurement and a geometric/statistical interpretation of Aharonov’s weak value

Lee

¹

,

Tsutsui

²

2017

Progress of Theoretical and Experimental Physics

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We show that the joint behaviour of an arbitrary pair of (generally non-commuting) quantum observables can be described by quasi-probabilities, which are an extended version of the standard probabilities used for describing the outcome of measurement for a single observable. The physical situations that require these quasi-probabilities arise when one considers quantum measurement of an observable conditioned by some other variable, with the notable example being the weak measurement employed to obtain Aharonov's weak value. Specifically, we present a general prescription for the construction of quasi-joint-probability (QJP) distributions associated with a given combination of observables. These QJP distributions are introduced in two complementary approaches: one from a bottom-up, strictly operational construction realised by examining the mathematical framework of the conditioned measurement scheme, and the other from a top-down viewpoint realised by applying the results of spectral theorem for normal operators and its Fourier transforms. It is then revealed that, for a pair of simultaneously measurable observables, the QJP distribution reduces to the unique standard joint-probability distribution of the pair, whereas for a non-commuting pair there exists an inherent indefiniteness in the choice of such QJP distributions, admitting a multitude of candidates that may equally be used for describing the joint behaviour of the pair. In the course of our argument, we find that the QJP distributions furnish the space of operators in the underlying Hilbert space with their characteristic geometric structures such that the orthogonal projections and inner products of observables can, respectively, be given statistical interpretations as 'conditionings' and 'correlations'. The weak value A w for an observable A is then given a geometric/statistical interpretation as either the orthogonal projection of A onto the subspace generated by another observable B, or equivalently, as the conditioning of A given B with respect to the QJP distribution under consideration.

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Reasonable conditions for joint probabilities of non-commuting observables

Cited by 14 publications

References 16 publications

What does the operator algebra of quantum statistics tell us about the objective causes of observable effects?

What does the operator algebra of quantum statistics tell us about the objective causes of observable effects?

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

Quasi-probabilities in conditioned quantum measurement and a geometric/statistical interpretation of Aharonov’s weak value

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