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A mathematical model for measurements

Akitaka Kishimoto

Published Mar 31, 2013• 0 citations
OpenAlex

Abstract

We will give a new model for measurements of a quantum system, expressed as the compact operators on a Hilbert space, such that the measuring apparatuses are described by a unital separable non-type I nuclear simple C$^*$-algebra equipped with certain unital endomorphisms and pure states. An interaction between the quantum system and the apparatus is specified by a unitary associated with the combined system as before. Magnifying to the classical level some aspects of the quantum system so captured in the apparatus is explicitly done by applying the endomorphism; then the resulting state is the superposition of {\em phases} with weights. Nature will then choose each phase according to the probability prescribed by the weights just as does one when multiple phases appear as in phase transition. Thus in our model state-reduction (or collapse of the wave function) is a primary event; whether this corresponds to the measurement of an observable or which one if it does is another matter.

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We will give a new model for measurements of a quantum system, expressed as the compact operators on a Hilbert space, such that the measuring apparatuses are described by a unital separable non-type I nuclear simple C∗^*∗-algebra equipped with certain unital endomorphisms and pure states. An interaction between the quantum system and the apparatus is specified by a unitary associated with the combined system as before. Magnifying to the classical level some aspects of the quantum system so captured in the apparatus is explicitly done by applying the endomorphism; then the resulting state is the superposition of {\em phases} with weights. Nature will then choose each phase according to the probability prescribed by the weights just as does one when multiple phases appear as in phase transition. Thus in our model state-reduction (or collapse of the wave function) is a primary event; whether this corresponds to the measurement of an observable or which one if it does is another matter.
We will give a new model for measurements of a quantum system such that the measuring apparatuses are described by a unital separable non-type I nuclear simple C∗^*∗-algebra equipped with certain unital endomorphisms and pure states. An interaction between the quantum system and the apparatus is specified by a unitary associated with the combined system as before. Magnifying to the classical level some aspects of the quantum system so captured in the apparatus is explicitly done by applying the endomorphism; then the resulting state is the superposition of {\em phases} with weights. Nature will then choose each phase according to the probability prescribed by the weights just as does one when multiple phases appear as in phase transition. Thus in our model state-reduction (or collapse of the wave function) is a primary event; whether this corresponds to the measurement of an observable or which one if it does is another matter.
We present a new interpretation of the terms superposition, entanglement, and measurement that appear in quantum mechanics. We hypothesize that the structure of the wave function for a quantum system at the sub-Planck scale has a deterministic cyclic structure. Each cycle comprises a sequential succession of the eigenstates that comprise a given wave function. Between unitary operations or measurements on the wave function, the sequential arrangement of the current eigenstates chosen by the system is immaterial, but once chosen it remains fixed until another unitary operation or measurement changes the wave function. The probabilistic aspect of quantum mechanics is interpreted by hypothesizing a measurement mechanism which acts instantaneously but the instant of measurement is chosen randomly by the classical measuring apparatus over a small but finite interval from the time the measurement apparatus is activated. At the instant the measurement is made, the wave function irrevocably collapses to a new state (erasing some of the past quantum information) and continues from thereon in that state till changed by a unitary operation or a new measurement.
We outline the Minimalistic Measurement Scheme (MMS) compatible with regular unitary evolution of a closed quantum system. Within this approach, a part of the system becomes informationally isolated (restricted) which leads to a natural emergence of the classical domain. This measurement scenario is a simpler alternative to environment-induced decoherence. In its basic version, MMS involves two ancilla qubits, AAA and XXX, entangled with each other and with the System SSS. Informational or thermodynamic cost of measurement is represented by XXX-qubit being isolated, i.e. becoming unavailable for future interactions with the rest of the system. Conditional upon this isolation, AAA-qubit, that plays the role of an Apparatus, becomes classical and records the outcome of the measurement. The procedure may be used to perform von Neumann-style projective measurements or generalized ones, that corresponds to Positive-Operator Value Measure (POVM). By repeating the same generalized measurement multiple times with different AAA- and XXX-qubits, one asymptotically approaches the wave function collapse in the basis determined by the premeasurement process. We present a simple result for the total information extracted after NNN such weak measurements. Building upon MMS, we propose a construction that maps a history of a quantum system onto a set of AAA-qubits. It resembles the Consistent History (CH) formulation of Quantum Mechanics (QM), but is distinct from it, and is built entirely within the conventional QM. In particular, consistency postulate of CH formalism is not automatically satisfied, but rather is an emerging property. Namely, each measurement event corresponds to the branching of mutually exclusive classical realities whose probabilities are additive. In a general case, however, the superposition between different histories is determined by the history density matrix.
We describe the scheme of quantum mechanics, in which the Hilbert space and linear operators are not primary notions of the theory. Instead of this we consider some variant of the algebraic approach. As primary notions we use elements of thenoncommutative algebra (observable) and functionals on this algebra (elementary states), which we associate with results of individual measurements. Such a scheme allows us to use, on the one hand, the apparatus of classical (Kolmogorov’s) probability theory, and on the other hand, to reproduce the standard mathematical apparatus of quantum mechanics and to specify borders of its applicability.
A transition would lead the system from its original superposition state to another state, in which the system and the measuring apparatus acquire definite properties. The problem of measurement is approached from the theory of environment-induced decoherence. This theoretical program, based on the study of the effects of the interaction between a quantum system, is considered as an open system and its environment. In turn, even the measurement problem has been reformulated, expanding its scope. As Schlosshauer points out, the conceptual difficulties of quantum measurement can now be concentrated around two cases: The definite outcome, which consists of the traditional question about quantum measurement. The problem of the preferred basis: since there is a theoretical ambiguity in the definition of the observable measured due to the mathematical possibility of a base change in the expression of the system state.
Let V=CNV=\mathbb{C}^NV=CN, and HHH (an observable) a Hermitian linear operator on VVV. Let v1,...,vnv_1,..., v_nv1​,...,vn​ be an orthonormal basis for VVV. Let M\mathcal{M}M be a measurement apparatus prepared to measure a state of an observed system and collapses the state to one of the vjv_jvj​'s. Here we propose a simple model which explains the Born rule and is compatible with entanglement.