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

arXiv:2303.15640QuantaFeb 6, 20241 citation

Measurement and the Observer

Herbert S. Green

Springer eBooksJan 1, 20003 citations

Quantum Interpretation of the Proton Anomalous Magnetic Moment

Michaele Suisse, P. Cameron

A century’s proliferation of Quantum Interpretations has been driven by an incomplete understanding of the measurement problem, by an incomplete understanding of the unobservable wavefunction and its unobservable interactions. Yet it remains that the observer effect - observing a phenomenon necessarily changes that phenomenon - is at the historical foundation of quantum mechanics. The semantic confusion, that wavefunction interactions are unobservable yet the observer effect is irrefutable, can be resolved by considering the difference between simple passive measurement and active transfer function measurement. In the first, one registers whatever lump of energy lands upon a sensor, and there is no observer effect. In the second, one excites the system of interest and measures the response, and there is an observer effect. Magnetic moment measurements are transfer function measurements. To measure an amplitude one must align the spin, applying a magnetic field to separate the energy eigenstates, and excite the system to measure their difference. A paper presented to Spin 2016 suggests that the measured anomaly is not an intrinsic property of the fermionic proton, but rather an observer effect. In our presentation to Spin 2018 we propose to extend the Spin 2016 interpretation to the neutron anomalous moment.A century’s proliferation of Quantum Interpretations has been driven by an incomplete understanding of the measurement problem, by an incomplete understanding of the unobservable wavefunction and its unobservable interactions. Yet it remains that the observer effect - observing a phenomenon necessarily changes that phenomenon - is at the historical foundation of quantum mechanics. The semantic confusion, that wavefunction interactions are unobservable yet the observer effect is irrefutable, can be resolved by considering the difference between simple passive measurement and active transfer function measurement. In the first, one registers whatever lump of energy lands upon a sensor, and there is no observer effect. In the second, one excites the system of interest and measures the response, and there is an observer effect. Magnetic moment measurements are transfer function measurements. To measure an amplitude one must align the spin, applying a magnetic field to separate the energy eigenstates, and excite the system to measure their difference. A paper presented to Spin 2016 suggests that the measured anomaly is not an intrinsic property of the fermionic proton, but rather an observer effect. In our presentation to Spin 2018 we propose to extend the Spin 2016 interpretation to the neutron anomalous moment.

Feb 1, 20170 citations

Measurement as Constraint: Reframing the Observer Problem Without Consciousness

Sabrina Dawn Palmer

Jan 1, 20260 citations

The Theory of Measurement

Roger Balian

The frontiers collectionJan 1, 20173 citations

Effective theory approach to the measurement problem

William Icefield

One understanding of the Niels Bohr's interpretation of quantum mechanics suggests that the measurement problem arises because an observer description of reality is incomplete due to lack of complete access to an entire system of observed and observer. In general, an additional and external measuring observer is required to access the entire system. Inevitable entanglement between observers, measurement apparatus and measured objects is key to such an understanding of the measurement problem. Quantum mechanics only provides a complete `incomplete' description of reality. Nevertheless, some issues, especially providing an account of why a preferred measurement basis arises, remain. Replacing relations between observed and observer in relational quantum mechanics with those between effective (approximate) theories, one arrives at a more clear understanding of why the measurement problem arises and how a preferred measurement basis is determined. The measurement problem exists because of measurement apparatus limitations, which only allow us to formulate and verify effective theories that inevitably neglect some details of nature.One understanding of the Niels Bohr's interpretation of quantum mechanics suggests that the measurement problem arises because an observer description of reality is incomplete due to lack of complete access to an entire system of observed and observer. In general, an additional and external measuring observer is required to access the entire system. Inevitable entanglement between observers, measurement apparatus and measured objects is key to such an understanding of the measurement problem. Quantum mechanics only provides a complete `incomplete' description of reality. Nevertheless, some issues, especially providing an account of why a preferred measurement basis arises, remain. Replacing relations between observed and observer in relational quantum mechanics with those between effective (approximate) theories, one arrives at a more clear understanding of why the measurement problem arises and how a preferred measurement basis is determined. The measurement problem exists because of measurement apparatus limitations, which only allow us to formulate and verify effective theories that inevitably neglect some details of nature.

May 13, 20200 citations

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.