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Journal of Physics A Mathematical and Theoretical

Time-symmetrized description of nonunitary time asymmetric quantum evolution

Igor Prlina, Nataša N. Nedeljković

Published Dec 17, 2015• 2 citations
Publisher

Abstract

Abstract We discuss how systems which evolve manifestly asymmetrically in time can be described within the framework of the time-symmetrized quantum mechanics. An obvious case of asymmetry arises when a pure state evolves into a mixed state via effectively non-unitary evolution. A two-state method for finding the intermediate probability in postselected systems under such evolution is developed and the time-symmetry aspects of the method are explicitly considered. A specific feature is the existence of the so-called second scenario in which the state originating from the postselection measurement evolves under different evolution superoperator than the state from the preselection measurement. The evolution of the second scenario is explicitly defined. We illustrate the method with two characteristic examples: the spontaneous deexcitation of atoms and the systems approaching thermal equilibrium. We consider the systems with two energy levels and calculate the time-symmetrized probability of finding the system in excited state, under general preselection and postselection conditions. The consequences of the asymmetry of the time evolution on this probability are discussed. It is demonstrated that the arrow of time can be reconstructed in some special cases of postselected systems, while, for a general system, this is not the case.

DOI
10.1088/1751-8113/49/3/035301

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Abstract We discuss how systems which evolve manifestly asymmetrically in time can be described within the framework of the time-symmetrized quantum mechanics. An obvious case of asymmetry arises when a pure state evolves into a mixed state via effectively non-unitary evolution. A two-state method for finding the intermediate probability in postselected systems under such evolution is developed and the time-symmetry aspects of the method are explicitly considered. A specific feature is the existence of the so-called second scenario in which the state originating from the postselection measurement evolves under different evolution superoperator than the state from the preselection measurement. The evolution of the second scenario is explicitly defined. We illustrate the method with two characteristic examples: the spontaneous deexcitation of atoms and the systems approaching thermal equilibrium. We consider the systems with two energy levels and calculate the time-symmetrized probability of finding the system in excited state, under general preselection and postselection conditions. The consequences of the asymmetry of the time evolution on this probability are discussed. It is demonstrated that the arrow of time can be reconstructed in some special cases of postselected systems, while, for a general system, this is not the case.
We investigate the origin of the arrow of time in quantum mechanics in the context of quantum cosmology. The ``Copenhagen'' quantum mechanics of measured subsystems incorporates a fundamental arrow of time. Extending discussions of Aharonov, Bergmann and Lebovitz, Griffiths, and others we investigate a generalized quantum mechanics for cosmology that utilizes both an initial and a final density matrix to give a time-neutral formulation without a fundamental arrow of time. Time asymmetries can arise for particular universes from differences between their initial and final conditions. Theories for both would be a goal of quantum cosmology. A special initial condition and a final condition of indifference would be sufficient to explain the observed time asymmetries of the universe. In this essay we ask under what circumstances a completely time symmetric universe, with T-symmetric initial and final condition, could be consistent with the time asymmetries of the limited domain of our experience. We discuss the approach to equilibrium, the electromagnetic system, and T-violation in the weak interactions in such universes. The decoherence of alternatives that is necessary for the prediction of probabilities in the quantum mechanics of closed systems is the origin of a fundamental limitation on quantum cosmologies with initial and final conditions. The initial and final density matrices cannot both be pure; if they are there is no decoherence and no prediction.
Time asymmetry and irreversibility are signal features of our world. They are the reason of our aging and the basis for our belief that effects are preceded by causes. These features have many manifestations called arrows of time. In classical physics, some of these arrows are described by the increase of entropy or probability, and others by time‐asymmetric boundary conditions of time‐symmetric equations (e.g., Maxwell or Einstein). However, there is some controversy over whether probability or boundary conditions are more fundamental. For quantum systems, entropy increase is usually associated with the effects of an environment or measurement apparatus on a quantum system and is described by the von Neumann‐Liouville equation. But since the traditional (von Neumann) axioms of quantum mechanics do not allow time‐asymmetric boundary conditions for the dynamical differential equations (Schrödinger or Heisenberg), there is no quantum analogue of the radiation arrow of time. In this paper, we review consequences of a modification of a fundamental axiom of quantum mechanics. The new quantum theory is time asymmetric and accommodates an irreversible time evolution of isolated quantum systems.
A slight modification of one axiom of quantum theory changes a reversible theory into a time asymmetric theory. Whereas the standard Hilbert space axiom does not distinguish mathematically between the space of states (in-states of scattering theory) and the space of observables (out-“states ” of scattering theory) the new axiom associates states and observables to two different Hardy subspaces which are dense in the same Hilbert space and analytic in the lower and upper complex energy plane, respectively. As a consequence of this new axiom the dynamical equations (Schrödinger or Heisenberg) integrate to a semigroup evolution. Extending this new Hardy space axiom to a relativistic theory provides a relativistic theory of resonance scattering and decay with Born probablilities that fulfill Einstein causality and the exponential decay law. 1 Introduction—Time Asymmetry Time asymmetry, irreversibility, time reversal non-invariance are different concepts and they are (probably) not (all) related to each other [1], [2], [3]. These concepts are usually called arrows-of-time. Time asymmetry comes from time asymmetric boundary conditions of time symmetric equations; its most prominent consequence is causality. The radiation arrow of time is its example from classical physics. Irreversibility
Here we present the time-bidirectional state formalism (TBSF) unifying in a general manner the standard quantum mechanical formalism with no postselection and the time-symmetrized two-state (density) vector formalism, which deals with postselected states. In the proposed approach, a quantum particle's state, called a time-bidirectional state, is equivalent to a joined state of two particles propagating in opposite time directions. For a general time-bidirectional state, we derive outcome probabilities of generalized measurements, as well as mean and weak values of Hermitian observables. We also show how the obtained expressions reduce to known ones in the special cases of no postselection and generalized two-state (density) vectors. Then we develop tomography protocols based on mutually unbiased bases and a symmetric informationally complete positive operator-valued measure, allowing experimental reconstruction of an unknown single-qubit time-bidirectional state. Finally, we employ the developed techniques for tracking of a qubit's time-reversal journey in a quantum teleportation protocol realized with a cloud-accessible noisy superconducting quantum processor. The obtained results justify an existence of a postselection-induced qubit's proper time arrow, which is different from the time arrow of a classical observer, and demonstrate capabilities of the TBSF for exploring quantum phenomena brought forth by a postselection in the presence of noise.
A quantum theory that applies to the (closed) universe needs to be time asymmetric because of the cosmological arrow of time. The preparation $\ensuremath{\Rightarrow}$ registration arrow of time (a state must be prepared before an observable can be detected in it) of the quantum mechanics of measured systems provides a phenomenological reason for an asymmetric semigroup time evolution. The standard theory in the Hilbert space (HS) is inadequate for either since the mathematics of the HS allows only reversible unitary group evolution and time symmetric boundary conditions. The mathematical theory that describes time-asymmetric quantum physics in addition to providing the mathematics for the Dirac kets is the rigged Hilbert space (RHS) theory. It uses a pair of RHS's of Hardy class with complementary analyticity property, one for the prepared states (``in states'') and the other for the registered observables (``out states''). The RHS's contain Gamow kets which have all the properties needed to represent decaying states and resonances. Gamow kets have only asymmetric time evolution. The neutral kaon system is used to show that quasistationary microphysical systems can be experimentally isolated if their time of preparation can be accurately identified. The theoretical predictions for a Gamow ket have the same features as the observed decay probabilities, including the time ordering. This time ordering is the same as the time ordering in the probabilities of histories for the quantum universe. The fundamental quantum mechanical arrow of time represented by the semigroup in the RHS is therefore the same as the cosmological arrow of time, assuming that the universe can be considered a closed quantum system.
The standard approach to quantum measurements is to assume that they lead to effectively instantaneous collapse of the quantum state. However, if we assume that we are unable to enforce at what exact moment of time the measurement occurs due to a finite resolution of any time measurement device, at the level of the ensemble, the measurement would lead to an effectively nonunitary evolution involving a mixed state. Each individual ensemble member would face an instantaneous collapse at different moments of time. This process is completely indistinguishable from fundamental nonunitary evolution at the level of each individual ensemble member, within the framework of strong projective measurements. In this paper, we show that weak postselected measurements can distinguish these two types of evolution. An experimental protocol for determining the nature of quantum collapse is described, and the example of a hydrogen atom is analyzed in detail.