arXiv:2101.10690 · Entropy

Conditional Action and Imperfect Erasure of Qubits

Heinz–Jürgen Schmidt

Published Feb 26, 2021• 2 citations

Abstract

We consider state changes in quantum theory due to "conditional action" and relate these to the discussion of entropy decrease due to interventions of "intelligent beings" and the principles of Szilard and Landauer/Bennett. The mathematical theory of conditional actions is a special case of the theory of "instruments", which describes changes of state due to general measurements and will therefore be briefly outlined in the present paper. As a detailed example, we consider the imperfect erasure of a qubit that can also be viewed as a conditional action and will be realized by the coupling of a spin to another small spin system in its ground state.

DOI: 10.3390/e23030289

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arXiv:2101.10690 · Entropy

Conditional Action and Imperfect Erasure of Qubits

Heinz–Jürgen Schmidt

Published Feb 26, 2021• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.3390/e23030289

Related papers

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We begin with a study of operations and the effects they measure. We define the probability that an effect

a

occurs when the system is in a state

\rho

P_{\rho}(a)=\textrm{Tr}(\rho a)

. If

P_{\rho}(a)\ne0

and

\mathcal{I}

is an operation that measures

a

, we define the conditional probability of an effect

b

given

a

relative to

\mathcal{I}

P_{\rho}(b\mid a)=\textrm{Tr}[\mathcal{I}(\rho)b]/P_{\rho}(a)

. We characterize when Bayes' quantum second rule

P_{\rho}(b\mid a)=\frac{P_{\rho}(b)}{P_{\rho}(a)}\,P_{\rho}(a\mid b)

holds. We then consider Lüders and Holevo operations. We next discuss instruments and the observables they measure. If

A

and

B

are observables and an instrument

\mathcal{I}

measures

A

, we define the observable

B

conditioned on

A

relative to

\mathcal{I}

and denote it by

(B\mid A)

. Using these concepts, we introduce Bayes' quantum first rule. We observe that this is the same as the classical Bayes' first rule, except it depends on the instrument used to measure

A

. We then extend this to Bayes' quantum first rule for expectations. We show that two observables

B

and

C

are jointly commuting if and only if there exists an atomic observable

A

such that

B=(B\mid A)

and

C=(C\mid A)

. We next obtain a general uncertainty principle for conditioned observables. Finally, we discuss observable conditioned quantum entropies. The theory is illustrated with many examples. Quanta 2024; 13: 1–10.