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Chinese Physics Letters

Quantum Deletion of Copies of Two Non-orthogonal Quantum States via Weak Measurement*

Wei-Min Shang, Jie Zhou, Hui-Xian Meng, Jing‐Ling Chen

Published May 1, 2020• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0256-307x/37/5/050302

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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.
We propose a scheme to enhance the fidelity of a symmetric quantum cloning machine via weak measurements. By adjusting the intensity of the weak measurement parameter ppp, we obtain copies of initial states with different values of fidelity. Choosing a proper value of ppp, we can obtain perfect copies. In this paper, we focus on $1\ensuremath{-}2$ quantum cloning for two nonorthogonal states. Sets containing more than two linear independent states are also briefly discussed. Due to the probabilistic nature of weak measurements, we obtain high-fidelity copies probabilistically. If the weak measurement is a success, we do subsequent operations to obtain high-fidelity copies; otherwise, the cloning process fails and we quit. From this perspective, the scheme we propose is economical for saving quantum resources and time. Our scheme may be very useful in quantum information processing.
Suppose we are given two identical copies of an unknown quantum state and we wish to delete one copy from among the given two copies. The quantum no-deletion principle restricts us from perfectly deleting a copy but it does not prohibit us from deleting a copy approximately. Here we construct two types of a ``universal quantum deletion machine'' which approximately deletes a copy such that the fidelity of deletion does not depend on the input state. The two types of universal quantum deletion machines are (1) a conventional deletion machine described by one unitary operator and (2) a modified deletion machine described by two unitary operators. Here it is shown that the modified deletion machine deletes a qubit with fidelity 3∕43∕43∕4, which is the maximum limit for deleting an unknown quantum state. In addition to this we also show that the modified deletion machine retains the qubit in the first mode with average fidelity 0.77 (approx.) which is slightly greater than the fidelity of measurement for two given identical states, showing how precisely one can determine its state [S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 (1995)]. We also show that the deletion machine itself is input state independent, i.e., the information is not hidden in the deleting machine, and hence we can delete the information completely from the deletion machine.
We further verify the impossibility of deleting an arbitrary unknown quantum state, and also show it is impossible to delete two nonorthogonal quantum states as a consequence of unitarity of quantum mechanics. A quantum approximate (deterministic) deleting machine and a probabilistic (exact) deleting machine are constructed. The estimation for the global fidelity characterizing the efficiency of the quantum approximate deleting is given. We then demonstrate that unknown nonorthogonal states chosen from a set with their multiple copies can evolve into a linear superposition of multiple deletions and failure branches by a unitary process if and only if the states are linearly independent. It is notable that the proof for necessity is somewhat different from Pati's [Phys. Rev. Lett. 83, 2849 (1999)]. Another deleting machine for the input states that are unnecessarily linearly independent is also presented. The bounds on the success probabilities of these deleting machines are derived. So we expound some preliminary analogies between quantum cloning and deleting.
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.
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.