Unsolvability of the Halting Problem in Quantum Dynamics

Abstract

It is shown that the halting problem cannot be solved consistently in both the Schrödinger 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. PACS numbers:With the construction of universal quantum Turing machine, Deutsch proposed [1] a quantum version of the halting problem first proved by Alan Turing in 1936 [2]. In recent years, a lot of interest has been focused on quantum computation [3], and the discussion of the halting problem using a quantum computer has also received attention. Myers argued [4] that due to entanglement between a halt qubit and a system, it may be difficult to measure the halt qubit, which may spoil the computation. Subsequent discussions on the halting problem with a quantum computer have mainly focused on the superposition and entanglement of the halt qubit [5,6,7]. In this paper, we approach the halting scheme differently and use two pictures of quantum dynamics, i.e., the Schrödinger and Heisenberg pictures. Schrödinger's wave mechanics and Heisenberg's matrix mechanics were formulated in the early twentieth century and have been considered to be equivalent, i.e., two different ways of describing the same physical phenomenon that we observe. Therefore, in order to consider a halting scheme for a quantum system, we need to examine whether the procedure is consistent in both the Schrödinger and Heisenberg pictures. We will give an example in quantum dynamics that shows this cannot be achieved. We will then argue that it is the Heisenberg picture, rather than both pictures, that yields the correct description that not only does not run into the inconsistency shown through the halting scheme but also is physically sensible.In order to discuss the halting problem, we first wish to define notations to be used. In particular we will follow a similar notation used in [8,9] such that it is convenient in both Schrödinger and Heisenberg pictures. A qubit, a basic unit of quantum information, is a two-level quantum system written as |ψ = a|0 + b|1 . Using a Bloch sphere notation, i.e., with a = exp(−iφ/2) cos(θ/2) and b = exp(iφ/2) sin(θ/2), a qubit in a density matrix form can be written as |ψ ψ| = 1 2 (1 +v · σ) where (v x , v y , v z ) = (sin θ cos φ, sin θ sin φ, cos θ) and σ = (σ x , σ y , σ z ) with σ x = |0 1| + |1 0|, σ y = −i|0 1| + i|1 0|, and σ z = |0 0| − |1 1|. Therefore a qubit, |ψ ψ|, can be represented as a unit vectorv = (v x , v y , v z ) pointing in (θ, φ) of a sphere with 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π. A unitary transformation of a qubit in the unit vector notationv can be obtained by applying U to σ i for the corresponding ith component of the vectorv, i.e., v i , where i = x, y, z (al...

Remarks on Nondeterministic Computation, Choices, and Formal Language

“…This advancement, which exhibited subjectivity, was not easily accepted by many researchers at the time (Einstein et al, 1935). In particular, it was argued (Song, 2008) that one may consider the observable as the frame of reference of the observer, when observing the given quantum system. This postulate leads to an asymmetry between the Schrödinger and Heisenberg pictures, i.e., active and passive transformations, respectively, when the very object being observed is the frame of reference itself -a phenomenon that only occurs in consciousness (Figure 1).…”

“…In this paper, we have provided a more specific model of the subjective universe model proposed in (Song, 2008;, i.e., the observer and the object are not separable. It was discussed that the choice of the observer in classical spacetime with continuity fills up the universe as negative sea.…”

Notes on Information and Choices

“…Motivated by the assumption that physics provides a description between the observer and the object, it was argued (Song, 2008) that the observable should be identified as a reference frame when observing the state vector. In particular, it was noted that just as the state vector was an exact and ultimate mathematical description of the object, i.e., when we accept the noncausal probabilistic nature at the fundamental level, the observable should also be an exact and ultimate mathematical description of the observer.…”

“…In (Song, 2008;, the concept of the subjective universe model has been proposed as a way out of the dilemma of describing consciousness, as the observer and the observed cannot be separated. The description provided in this paper suggests that the reality should correspond to the observer's choice, therefore, subjective reality.…”

“…First, in (Song, 2008), after showing the symmetry breakdown between the Schrödinger and the Heisenberg pictures, a possible resolution to the dilemma of describing self-observation of the reference frame was studied. In particular, it was discussed that, based on the postulate that what the observer observes is time forwarding, the Heisenberg picture with time going backwards should be the appropriate description of nature.…”