Abstract Quantum mechanics admits two distinct evolutions: deterministic unitary dynamics governed by the Schrödinger equation and the probabilistic collapse of the wave function. We show that the continuous collapse of a quantum state under measurement can, on a trajectory-by-trajectory basis, be equivalently described as unitary evolution generated by a time- and state-dependent Hermitian Hamiltonian with stochastic parameters. While the ensemble dynamics remains non-unitary, each individual trajectory thus admits a unitary representation. We derive explicit forms of such Hamiltonians for projective measurements on arbitrary n -level systems and for continuous position measurements of a harmonic oscillator, and we propose experimental schemes to test these predictions. Our framework provides a new approach to modeling and controlling continuously monitored quantum systems using only state-dependent unitary resources.
Abstract The problem of how measurement in quantum mechanics takes place has existed since its formulation. Von Neumann proposed a scheme where he treated measurement as a two-part process— a unitary evolution in the full system-ancilla space and then a projection onto one of the pointer states of the ancilla (representing the ‘collapse’ of the wavefunction). The Lindblad master equation, which has been extensively used to explain dissipative quantum phenomena in the presence of an environment, can effectively describe the first part of the von Neumann measurement scheme when the jump operators in the master equation are Hermitian. We have proposed a non-Hermitian Hamiltonian formalism to emulate the first part of the von Neumann measurement scheme. We have used the embedding protocol to dilate a non-Hermitian Hamiltonian that governs the dynamics in the system subspace into a higher-dimensional Hermitian Hamiltonian that evolves the full space unitarily. We have obtained the various constraints and the required dimensionality of the ancilla Hilbert space in order to achieve the required embedding. Using this particular embedding and a specific projection operator, one obtains non-Hermitian dynamics in the system subspace that closely follow the Lindblad master equation. This work lends a new perspective to the measurement problem by employing non-Hermitian Hamiltonians.
Ever since the formulation of quantum mechanics, there is very little understanding of the process of the collapse of a wavefunction. We have proposed a dynamical model to emulate the measurement postulates of quantum mechanics. We postulate that a non-Hermitian Hamiltonian operates during the process of measurement, which evolves any state to an attracting equilibrium state, thus, mimicking a "collapse". We demonstrate this using a 2-level system and then extend it to an N-level system. For a 2-level system, we also demonstrate that the dynamics generated by the Lindblad master equation can be replicated as an incoherent sum of the evolution by two separate non-Hermitian Hamiltonians.
In the context of the measurement problem, we propose to model the interaction between a quantum particle and an "apparatus" through a non-Hermitian Hamiltonian term. We simulate the time evolution of a normalized quantum state split into two spin components (via a Stern-Gerlach experiment) and that undergoes a wave-function collapse driven by a non-Hermitian Hatano-Nelson Hamiltonian. We further analyze how the strength and other parameters of the non-Hermitian perturbation influence the time-to-collapse of the wave function obtained under a Schrödinger-type evolution. We finally discuss a thought experiment where manipulation of the apparatus could challenge standard quantum mechanics predictions.
Abstract In the context of the measurement problem, we propose to model the interaction between a quantum particle and an ‘apparatus’ through a non-Hermitian Hamiltonian term. We simulate the time evolution of a normalized quantum state split into two spin components (via a Stern–Gerlach experiment) and that undergoes a wavefunction collapse driven by a non-Hermitian Hatano-Nelson Hamiltonian. We further analyze how the strength and other parameters of the non-Hermitian perturbation influence the time-to-collapse of the wave function obtained under a Schödinger-type evolution. We finally discuss a thought experiment where manipulation of the apparatus could challenge standard quantum mechanics predictions.
In this paper, we examine measurement using the Q-based, objective-field model for quantum mechanics. Schrodinger considered a microscopic system prepared in a superposition of states which is then coupled to a macroscopic meter. We analyze the entangled meter and system, and measurements on it, by solving forward-backward stochastic differential equations for real amplitudes
x(t) and
p(t) that correspond to the phase-space variables of the Q function of the system at a time
t. We model the system and meter as single-mode fields, and measurement of
x^ by amplification of the amplitude
x(t). Our conclusion is that the outcome for the measurement is determined at (or by) the time
tm, when the coupling to the meter is complete, the meter states being macroscopically distinguishable. There is consistency with macroscopic realism. By evaluating the distribution of the amplitudes
x and
p postselected on a given outcome of the meter, we show how the
Q-based model represents a more complete description of quantum mechanics: The variances associated with amplitudes
x and
p are too narrow to comply with the uncertainty principle, ruling out that the distribution represents a quantum state. We conclude that the collapse of the wavefunction occurs as a two-stage process: First there is an amplification that creates branches of amplitudes
x(t) of the meter, associated with distinct eigenvalues. The outcome of measurement is determined by
x(t) once amplified, explaining Born's rule. Second, the distribution that determines the final collapse is the state inferred for the system conditioned on the outcome of the meter: information is lost about the meter, in particular, about the complementary variable
p.