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Quantum Measurement Theory for an Implementation of Deutsch's Algorithm

Koji Nagata, Tadao Nakamura

Published Jul 1, 2013• 0 citations
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Abstract

We discuss projective measurement theory does not meet easy detector model for Pauli observable. We propose a solution of the problem by changing the value of the result of quantum measurements and by considering macroscopic system. We discuss how our solution is used in an implementation of Deutsch's algorithm. Especially, we systematically describe our assertion based on more mathematical analysis using raw data.

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•Jun 1, 2016•0 citations
We discuss projective measurement theory does not meet easy detector model for Pauli observable. We propose a solution of the problem by changing the value of the result of quantum measurements and by considering macroscopic system. We discuss how our solution is used in an implementation of Deutsch's algorithm. Especially, we systematically describe our assertion based on more mathematical analysis using raw data.
We discuss the fact that there is a crucial contradiction within Von Neumann’s theory. We derive a proposition concerning a quantum expected value under an assumption of the existence of the orientation of reference frames in N spin-1/2 systems (1 ≤ N to a new constant . It may be said that a new type of the quantum theory early approaches Newton’s theory in the macroscopic scale than the old quantum theory does. We discuss how our solution is used in an implementation of Deutsch’s algorithm.
We consider the two quantum measurement theories for measuring a single Pauli observable. We assume also the existence of a classical probability space for the two measurement theories. We cannot avoid the Kochen-Specker (KS) contradiction when we measure the Pauli observable by using the projective measurement theory if we introduce a classical probability space. The results of measurement are either +1+1+1 or −1-1−1 (in ℏ/2\hbar/2ℏ/2 unit) when we consider a spin-1/2 system. The projective measurement theory does not accept a classical probability space when we measure the Pauli observable. We propose a new measurement theory based on the truth values, i.e., the truth T (1) for true and the falsity F (0) for false. The results of measurement are either +1+1+1 or 0 (in ℏ/2\hbar/2ℏ/2 unit). We avoid the KS contradiction when we measure the Pauli observable by using the new measurement theory if we introduce a classical probability space. The new measurement theory accepts a classical probability space when we measure the Pauli observable.
We discuss the fact that there is a crucial contradiction within Von Neumann's theory. We derive a proposition concerning a quantum expected value under an assumption of the existence of the orientation of reference frames in NNN spin-1/2 systems (1≤N<+∞1 \leq N < +\infty1≤N<+∞). This assumption intuitively depictures our physical world. However, the quantum predictions within the formalism of Von Neumann's projective measurement violate the proposition with a magnitude that grows exponentially with the number of particles. We have to give up either the existence of the directions or the formalism of Von Neumann's projective measurement. Therefore, Von Neumann's theory cannot depicture our physical world with a violation factor that grows exponentially with the number of particles. The theoretical formalism of the implementation of the Deutsch-Jozsa algorithm relies on Von Neumann's theory. We investigate whether Von Neumann's theory meets the Deutsch-Jozsa algorithm. We discuss the fact that the crucial contradiction makes the quantum-theoretical formulation of Deutsch-Jozsa algorithm questionable. Further, we discuss the fact that projective measurement theory does not meet an easy detector model for a single Pauli observable. Especially, we systematically describe our assertion based on more mathematical analysis using raw data. We propose a solution of the problem. Our solution is equivalent to changing Planck's constant (ℏ\hbarℏ) to a new constant (ℏ/2\hbar/\sqrt{2}ℏ/2​). It may be said that a new type of the quantum theory early approaches Newton's theory in the macroscopic scale than the old quantum theory does. We discuss how our solution is used in an implementation of Deutsch's algorithm.
Deutsch's problem is the simplest and most frequently examined example of computational problem used to demonstrate the superiority of quantum computing over classical computing. Deutsch's quantum algorithm has been claimed to be faster than any classical ones solving the same problem, only to be discovered later that this was not the case. Various de-quantised solutions for Deutsch's quantum algorithm—classical solutions which are as efficient as the quantum one—have been proposed in the literature. These solutions are based on the possibility of classically simulating "superpositions", a key ingredient of quantum algorithms, in particular, Deutsch's algorithm. The de-quantisation proposed in this note is based on a classical simulation of the quantum measurement achieved with a model of observed system. As in some previous de-quantisations of Deutsch's quantum algorithm, the resulting de-quantised algorithm is deterministic. Finally, we classify observers (as finite state automata) that can solve the problem in terms of their "observational power".
Recently, a new measurement theory based on the truth values is proposed \cite{NN1}. The results of measurements are either 0 or 1. The measurement theory accepts a hidden variables model for a single Pauli observable. Therefore we can introduce a classical probability space for the measurement theory in this case. On the other hand, we discuss the fact that the projective measurement theory (the results of measurements are either +1+1+1 or −1-1−1) does not meet a hidden variables model for a single Pauli observable. Hence we cannot introduce a classical probability space for the projective measurement theory in this case. Our discussion provides new insight to formulate quantum measurement theory, by using the measurement theory based on the truth values.