arXiv:2208.07923 · Quanta

Dual Instruments and Sequential Products of Observables

Stanley Gudder

Published Aug 30, 2022• 3 citations

Abstract

We first show that every operation possesses an unique dual operation and measures an unique effect. If a and b are effects and J is an operation that measures a, we define the sequential product of a then b relative to J. Properties of the sequential product are derived and are illustrated in terms of Lüders and Holevo operations. We next extend this work to the theory of instruments and observables. We also define the concept of an instrument (observable) conditioned by another instrument (observable). Identity, state-constant and repeatable instruments are considered. Sequential products of finite observables relative to Lüders and Holevo instruments are studied. Quanta 2022; 11: 15–27.

DOI: 10.12743/quanta.v11i1.197

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arXiv:2208.07923 · Quanta

Dual Instruments and Sequential Products of Observables

Stanley Gudder

Published Aug 30, 2022• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.12743/quanta.v11i1.197

Related papers

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arXiv:2305.17584QuantaJun 6, 20232 citations

Our basic concept is the set

\mathcal{E}(H)

of effects on a finite dimensional complex Hilbert space

H

. If

a,b\in\mathcal{E}(H)

, we define the sequential product

a[\mathcal{I}]b

a

then

b

. The sequential product depends on the operation

\mathcal{I}

used to measure

a

. We begin by studying the properties of this sequential product. It is observed that

b\mapsto a[\mathcal{I}]b

is an additive, convex morphism and we show by examples that

a\mapsto a[\mathcal{I}]b

enjoys very few conditions. This is because a measurement of

a

can interfere with a later measurement of

b

. We study sequential products relative to Kraus, Lüders and Holevo operations and find properties that characterize these operations. We consider repeatable effects and conditions on

a[\mathcal{I}]b

that imply commutativity. We introduce the concept of an effect

b

given an effect

a

and study its properties. We next extend the sequential product to observables and instruments and develop statistical properties of real-valued observables. This is accomplished by employing corresponding stochastic operators. Finally, we introduce an uncertainty principle for conditioned observables.

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.

This article introduces the concepts of multi-observables and multi-instruments in quantum mechanics. A multi-observable

A

(multi-instrument

\mathcal{I}

) has an outcome space of the form

Ω=Ω_1\times\cdots\timesΩ_n

and is denoted by

A_{x_1\cdots x_n}

(

\mathcal{I}_{x_1\cdots x_n}

) where

(x_1,\ldots ,x_n)\inΩ

. We also call

A

(

\mathcal{I}

) an

n

-observable (

n

-instrument) and when

n=2

we call

A

(

\mathcal{I}

) a bi-observable (bi-instrument). We point out that bi-observables

A

(

\mathcal{I}

) and bi-instruments have been considered in past literature, but the more general case appears to be new. In particular, two observables (instruments) have been defined to coexist or be compatible if they possess a joint bi-observable (bi-instrument). We extend this definition to

n

observables and

n

instruments by considering joint marginals of

n

-observables and joint reduced marginals of

n

-instruments. We show that a

n

-instrument measures a unique

n

-observable and if a finite umber of instruments coexist, then their measured observables coexist. We prove that there is a close relationship between a nontrivial

n

-observable and its parts. Moreover, a similar result holds for instruments. We next show that a natural definition for the tensor product of a finite number of instruments exist and possess reasonable properties. We then discuss sequential products of a finite number of observables and instruments. We present various examples such as Kraus, Holevo and Lüders instruments.