It is shown that the halting problem cannot be solved consistently in both the Schrodinger and Heisenberg pictures of quantum dynamics. The existence of the halting machine, which is assumed from quantum theory, leads into a contradiction when we consider the case when the observer's reference frame is the system that is to be evolved in both pictures. We then show that in order to include the evolution of observer's reference frame in a physically sensible way, the Heisenberg picture with time going backwards yields a correct description.
It is shown that the halting problem cannot be solved consistently in both the Schrodinger and Heisenberg pictures of quantum dynamics. The existence of the halting machine, which is assumed from quantum theory, leads into a contradiction when we consider the case when the observer's reference frame is the system that is to be evolved in both pictures. We then show that in order to include the evolution of observer's reference frame in a physically sensible way, the Heisenberg picture with time going backwards yields a correct description.
The halting problem is a decision problem first posed and proved by Alan Turing in 1936. With the recent surge of interest in quantum computation, one is led to ask if the problem can also be considered for a quantum computer. It is reported that the halting problem may not be solved consistently in both the Schrodinger and Heisenberg pictures of quantum dynamics. The assumption of the existence of the quantum halting machine leads to a contradiction when a vector representing an observable is the system that is to be unitarily evolved in both pictures.
In [arXiv:0706.4180], a certain natural phenomenon of self-observing consciousness was shown to be incompatible with the standard axioms of quantum theory. In order to consider a possibility of removing this inconsistency, which arises due to the separation between an observing party and a state vector being observed stated in the quantum theory axioms, we provide a conjecture where ‘being ’ is identified with the conscious experience of observing the state vector. That is, the observer is not observing the state vector in terms of the relative difference between reference frames, instead, the conscious experience of observing the state vector should define the observer, or being. It is shown that the Heisenberg picture provides a natural description of this conjecture. 1
In non-relativistic quantum mechanics, there exist several different ways of describing the time dependence of dynamics of physical systems. Of particular importance for quantum scattering theory are the Schrodinger, Heisenberg and Dirac pictures. For establishing a connection between quantum mechanical formalisms and classical mechanics, as well as for investigating relativistic solutions, the Heisenberg picture appears to be the most convenient. This picture of quantum mechanics is particularly adapted for the examination of free fields. The customary wave-mechanical version of quantum theory is obtained if we choose, e.g., in the framework of the Schrodinger picture to work with the coordinate representation. Possible results of measurements on a quantum system are called eigenvalues. To each of the latter values corresponds one or more eigenstates. An eigenstate is degenerate when more than one eigenstate correspond to a single eigenvalue.
Quantum non-Markovianity has been widely studied and connected to the existence of memory effects in the dynamics of open systems. Surprisingly, working in the Schrödinger or in the Heisenberg picture can provide inequivalent description non-Markovianity: a process can appear to be memoryless in one picture, while displaying memory effects in the other. Here, we investigate which kind of memory is relevant for different quantum information tasks. Some of them, such as sending information via a noisy channel, require memory in both pictures in order to exhibit revivals in the task performance. For others, only one type of memory is sufficient. We also provide necessary conditions for non-Markovianity in both pictures by only considering the dynamics in one picture, showing for instance that the previously considered witness of Schrödinger non-Markovianity in terms of the volume of accessible states does indeed witness non-Markovianity in both pictures at the same time.
The description of the time evolution of systems is technically known as the "Schrodinger picture." An important alternative description is the "Heisenberg picture," which indeed was the one Heisenberg used when he first formulated quantum mechanics. The Schrodinger and Heisenberg pictures represent two different but equivalent sets of rules. In other words, when applied to a given physical situation they lead to the same observable predictions. In some respects, the Heisenberg picture presents definite technical advantages. It is particularly convenient, for instance, in quantum field theory. The chapter discusses work consistently with the equivalent Schrodinger set of rules. For this reason, it describes briefly the Heisenberg formulation. The chapter shows that the expressions used in the Schrodinger picture can be carried over to the Heisenberg picture. It implies, however, that the kets associated with the possible values that can be found upon measurement of an observable are themselves time-dependent kets.
The Unitary Dynamical Group in Schrödinger's Picture Up to now we have been concerned with the description of a physical system at a fixed instant of time. The time evolution of quantum systems constitutes a many-faced, wide issue which, however, is somewhat tangential to the main aim of this volume. We shall touch upon it only briefly, restricting ourselves to the more familiar aspects of the problem. We assume, to begin with, that the physical system has no superselection rules, so that states and physical quantities correspond one-to-one to density operators and self-adjoint operators, respectively, on the Hilbert space. To describe the dynamical evolution of the system one has to specify the way the representatives of states and physical quantities evolve in, which is fixed in time, since the physical system is supposed to preserve its identity. There is an old tenet according to which there are two kinds of time evolution: the discontinuous, nondeterministic evolution undergone by the quantum system when a measurement is made on it, and the continuous, deterministic evolution caused by the interaction with external forces or with other quantum systems. The first kind of evolution will be dealt with in Chapter 8, where we shall find it less alarming than is often claimed.