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arXiv:quant-ph/0610171 · Physica Scripta

Probabilistic exact deletion and probabilistic no-signalling

Indranil Chakrabarty, Satyabrata Adhikari, Binayak Samadder Choudhury

Published Oct 11, 2006• 2 citations
arXivHTMLPDF

Abstract

In this study, we show that one cannot use non-local resources for probabilistic signalling even if one can delete a quantum state with the help of a probabilistic quantum deletion machine (PQDM). Here, we find that PQDM is not going to help us in identifying two statistical mixtures at a remote location. Also we derive the bound on deletion probability from a no-signalling condition.

DOI
10.1088/0031-8949/74/5/012

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In this study, we show that one cannot use non-local resources for probabilistic signalling even if one can delete a quantum state with the help of a probabilistic quantum deletion machine (PQDM). Here, we find that PQDM is not going to help us in identifying two statistical mixtures at a remote location. Also we derive the bound on deletion probability from a no-signalling condition.
We show that non-local resources cannot be used for probabilistic signalling even if one can produce exact clones with the help of a probabilistic quantum cloning machine (PQCM). We show that PQCM cannot help to distinguish two statistical mixtures at a remote location. Thus quantum theory rules out the possibility of sending superluminal signals not only deterministically but also probabilistically. We give a bound on the success probability of producing multiple clones in an entangled system. All the useful information that can be transmitted has a universal speed limit, namely, the speed of light. In quantum mechanics the situation seems to be changed when Einstein, Podolsky and Rosen put forward their famous thought experiment on two systems that had interacted in the past but are no longer in direct contact [1]. These kind of systems are described by entangled states which show non-local correlations that cannot be explained by any local hidden variable theory [2]. In [1] they proved that the measurement outcome of one system can instantaneously affect the result of the other. This suggested that one can
In this paper, the probabilistic quantum deletion machine constructed from the NB-deletion machine, controlled by the deletion probability p and the error probability q. It is observed that, under specific conditions, controlled errors can enhance the deletion process: when p is minimal and q is maximal, both deletion and retention fidelities reach their optimal values. Compared to the original NB-machine, this probabilistic machine performs better in both deleting and preserving states. Although the output states do not violate Bell’s inequalities, they still achieve teleportation fidelities above the classical limit. The entanglement of machine’s output state appears only when p > 0, and the mixedness increases with p. Altogether, these results indicate that controlled error can enhance certain non-local features of a probabilistic deletion machine.
We construct a probabilistic cloning and deleting machine which, taking several copies of an input quantum state, can output a linear superposition of multiple cloning and deleting states. Since the machine can perform cloning and deleting in a single unitary evolution, the probabilistic cloning and other cloning machines proposed in the previous literature can be thought of as special cases of our machine. A sufficient and necessary condition for successful cloning and deleting is presented, and it requires that the copies of an arbitrarily presumed number of the input states are linearly independent. This simply generalizes some results for cloning. We also derive an upper bound for the success probability of the cloning and deleting machine.
Unmeasureability of a quantum state has important consequences in practical implementation of quantum computers. Like copying, deleting of an unknown state from among several copies is prohibited. This is called no-deletion prinicple. Here, we present a no deleting principle for qudits. We obtain a bound on NNN-to-MMM deleting and show that the quality of deletion drops exponentially with the number of copies to be deleted. In addition, we investigate conditional, state-dependent and approximate quantum deleting of unknown states. We prove that unitarity does not allow us to delete copies from an alphabet of two non-orthogonal states exactly. Further, we show that no-deleting principle is consistent with no-signalling.
One of the fundamental restrictions that quantum mechanics imposes is the "No deletion Theorem" which tells us that given two identical unknown quantum states, it is impossible to delete one of them. But nevertheless if not perfect, people have tried to delete it approximately. In these approximate deleting processes our basic target is to delete one of the two identical copies as much as possible while preserving the other copy. In this brief report, by using the No communication theorem (NCT) (impossibility of sending signal faster than light using a quantum resource) as a guiding principle, we obtain a bound on the sum of the fidelity of deletion and the fidelity of preservation. Our result not only brings out the complementary relation between these two fidelities but also predicts the optimal value of the fidelity of deletion achievable for a given fidelity of preservation under no signaling constraint. This work eventually saturates the quest for finding out the optimal value of deletion within the NCT framework.
We propose a scenario to increase the probability of probabilistic quantum deletion and to enhance the fidelity of approximate quantum deletion for two non-orthogonal states via weak measurement. More interestingly, by pretreating the given non-orthogonal states, the probability of probabilistic quantum deletion and fidelity of approximate quantum deletion can reach 1. Since outcomes of the weak measurement that we required are probabilistic, we perform the subsequent deleting process only when the outcome of weak measurement is “yes”. Remarkably, we find that our scenario has better performance in quantum information process; for example, it costs less quantum resources and time.