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arXiv:1907.08561

Dynamics of Intermediate Measurements in Quantum Systems

Hollis Daniel Williams

Published Jul 19, 2019• 0 citations
arXivHTMLPDF

Abstract

We study the dynamics of a quantum system in which an intermediate property $m$ is measured in between initial and final measurements of two different non-commuting properties $a$ and $b$. Since this intermediate measurement must involve an interaction, we use this case to explore in more detail the dynamics of measurement and propose that the physics of this change is described by a unitary transformation parametrised by a random parameter, with the selection of $m$ described by a superposition of these unitaries. We prove a set of conditions which must hold in order for this superposition to remain unitary and then argue that in the case where $U$ is a matrix, it must be a random unitary matrix. We outline a numerical example with matrices and make some predictions based on the conditions we have found which are capable of experimental verification. We finish with a brief discussion of the complex phases which appear in the relations between non-commuting properties and their relevance for the dynamics associated with measurement.

DOI
10.48550/arxiv.1907.08561

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We study the dynamics of a quantum system in which an intermediate property mmm is measured in between initial and final measurements of two different non-commuting properties aaa and bbb. Since this intermediate measurement must involve an interaction, we use this case to explore in more detail the dynamics of measurement and propose that the physics of this change is described by a unitary transformation parametrised by a random parameter, with the selection of mmm described by a superposition of these unitaries. We prove a set of conditions which must hold in order for this superposition to remain unitary and then argue that in the case where UUU is a matrix, it must be a random unitary matrix. We outline a numerical example with matrices and make some predictions based on the conditions we have found which are capable of experimental verification. We finish with a brief discussion of the complex phases which appear in the relations between non-commuting properties and their relevance for the dynamics associated with measurement.
Experimentally, the imaginary parts of complex weak values are obtained from the response of the system to small unitary phase shifts generated by the target observable. The complex conditional probabilities obtained from weak measurements can therefore be explained in terms of transformation dynamics. Specifically, the complex phase of weak conditional probabilities provides a complete description of the transformation dynamics between the initial and the final state generated by the intermediate states. The result is a measure of quantum state overlap that relates quantum statistical properties directly to the dynamical action of unitary transformations.
Quantum mechanics does not permit joint measurements of non-commuting observables.However, it is possible to measure the weak value of a projection operator, followed by the precise measurement of a different property.The results can be interpreted as complex joint probabilities of the two non-commuting measurement outcomes.Significantly, it is possible to predict the outcome of completely different measurements by combining the joint probabilities of the initial state with complex conditional probabilities relating the new measurement to the possible combinations of measurement outcomes used in the characterization of the quantum state.We can therefore conclude that the complex conditional probabilities observed in weak measurements describe fundamental state-independent relations between non-commuting properties that represent the most fundamental form of universal laws in quantum physics.
Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.
We present a graduate course on ideal measurements, analyzed as dynamical processes of interaction between the tested system S and an apparatus A, described by quantum statistical mechanics. The apparatus A = M + B involves a macroscopic measuring device M and a bath B. The requirements for ideality of the measurement allow us to specify the Hamiltonian of the isolated compound system S + M + B . The resulting dynamical equations may be solved for simple models. Conservation laws are shown to entail two independent relaxation mechanisms: truncation and registration. Approximations, justified by the large size of M and of B, are needed. The final density matrix 𝒟 ^ ( t f ) of S + A has an equilibrium form. It describes globally the outcome of a large set of runs of the measurement. The measurement problem, i.e., extracting physical properties of individual runs from 𝒟 ^ ( t f ), then arises due to the ambiguity of its splitting into parts associated with subsets of runs. To deal with this ambiguity, we postulate that each run ends up with a distinct pointer value A i of the macroscopic M. This is compatible with the principles of quantum mechanics. Born’s rule then arises from the conservation law for the tested observable; it expresses the frequency of occurrence of the final indications A i of M in terms of the initial state of S. Von Neumann’s reduction amounts to updating of information due to selection of A i . We advocate the terms q -probabilities and q -correlations when analyzing measurements of non-commuting observables. These ideas may be adapted to different types of courses.
Abstract I employ random-matrix methods to set up and solve statistical models of noisy nonunitary dynamics that appear in the context of monitored quantum systems. The models cover a range of scenarios combining random dynamics and measurements of variable strength of one or several qubits. The combined dynamics drive the system into states whose statistics reflect the competition of randomizing unitary evolution and the measurement-induced backaction collapsing the state. These effects are mediated by entanglement, as I describe in detail by analytical results. For the paradigmatic case of monitoring via a single designated qubit, this reveals a simple statistical mechanism, in which the monitoring conditions the state of the monitored qubit, which then imposes statistical constraints on the remaining quantities of the system. For the case of monitoring several qubits with prescribed strength, the developed formalism allows one to set up the statistical description and solve it numerically. Finally, I also compare the analytical results to the monitored dynamics of a quantum kicked top, revealing two regimes where the statistical model either describes the full stationary dynamics, or resolves time scales during particular parts of the evolution.