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Quantum Measurement Theory Improves the no-Cloning Theorem

Koji Nagata, Tadao Nakamura

Published Jul 1, 2013• 0 citations
OpenAlex

Abstract

We improve the no-cloning theorem that relies on the property of the quantum theory. Usually, the no-cloning theorem allows for a cloning two orthogonal quantum states, simultaneously. Here we take into account specific quantum measurement theory. We result in the fact that we cannot allow for a cloning two orthogonal quantum states, simultaneously. Especially, we systematically describe our assertion based on more mathematical analysis using raw data.

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We review the no-cloning theorem that relies on the properties of the quantum theory. Usually, the no-cloning theorem implies that two quantum states are identical or orthogonal if we allow a cloning to be on the two quantum states. Here, we rely on the maximum value of the square of an expected value. We may result in the fact that the two quantum states under consideration could not be orthogonal if we consider the maximum value of the square of the expected value. The no-cloning theorem may imply that the two quantum states under consideration may be identical if we consider the maximum value of the square of the expected value. The nocloning theorem itself has this character.

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The no-cloning theorem has always been a hot topic in quantum theorem: As an impossibility result, this theorem precludes attempts to communicated faster than light and reinforces the security of several quantum cryptography protocol. In this survey, I will briefly introduce the history of quantum no-cloning theorem–how the thesis was invented, how did it get its name; And i would also discuss the hitherto progress on no-cloning theorem, several approach of none-perfect cloning on non-orthogonal states as well as the application of quantum no-cloning theorem in quantum cryptography and quantum information. This survey incorporates the lecture ”Quantum Information Amplifiers Beyond the No-Cloning Theorem” given by Prof. Giulio Chiribella and ”Quantum cryptography” given by Prof. Xiongfeng Ma.

•Jan 1, 2014•0 citations
We improve the no-cloning theorem that relies on the property of the quantum theory. Usually, the no-cloning theorem allows for a cloning two orthogonal quantum states, simultaneously. Here we take into account specific quantum measurement theory. We result in the fact that we cannot allow for a cloning two orthogonal quantum states, simultaneously. Especially, we systematically describe our assertion based on more mathematical analysis using raw data.
We review the no-cloning theorem that relies on the properties of the quantum theory. Usually, the no-cloning theorem implies that two quantum states are identical or orthogonal if we allow a cloning to be on the two quantum states. Here, we rely on the maximum value of the square of an expected value. We may result in the fact that the two quantum states under consideration could not be orthogonal if we consider the maximum value of the square of the expected value. The no-cloning theorem may imply that the two quantum states under consideration may be identical if we consider the maximum value of the square of the expected value. The nocloning theorem itself has this character.
We investigate the no-cloning theorem that relies on the properties of a statistical model. Usually, the no-cloning theorem implies that two quantum states are identical or orthogonal if we allow a cloning to be on the two quantum states. Here, we rely on a statistical model. We may result in the fact that the two quantum states under consideration could not be orthogonal if we accept the statistical model. The no-cloning theorem may imply that the two quantum states under consideration may be identical if we accept the statistical model. The no-cloning theorem itself has this character.
Modern quantum physics now enables control of quantum systems at the level of individual trajectories, opening a new frontier that links quantum information theory, quantum many-body physics, and quantum thermodynamics, and uncovers novel non-equilibrium phenomena such as deep thermalization and measurement-induced entanglement. However, a central challenge remains: their characterization relies on measuring nonlinear properties of individual quantum states, a task tantamount to fine-grained cloning of a quantum ensemble. Here, the fundamental laws governing the cloning of quantum ensembles are investigated. First, a general no-cloning theorem for arbitrary ensembles is established from an information-theoretic perspective, even assuming multiple copies of the ensemble's purification. It is then shown that this barrier can be unexpectedly circumvented for physical ensembles generated by finite-time evolutions. Nevertheless, these tasks are proven to remain computationally intractable, even when the full circuit description of state preparation is known. This stands in sharp contrast to the conventional no-cloning theorem, which relies on the state being unknown. Together, these results establish new fundamental principles of quantum mechanics, reveal intrinsic trade-offs among sample complexity, computational complexity, and quantum measurements, and highlight the necessity of problem-specific strategies for probing measurement-induced quantum phenomena.
The inherent limitations of physical processes prevent the copying of arbitrary quantum states. Furthermore, even if we only aim to clone two distinct quantum states, it remains impossible unless they are mutually orthogonal. To overcome this limitation, we propose a virtual-cloning protocol that bypasses the restrictions imposed by the quantum no-cloning theorem. Specifically, we begin by outlining the general framework for virtual cloning and deriving a necessary and sufficient criterion for the existence of a virtual operation capable of simultaneously cloning a set of states. Subsequently, through an analysis of the simulation cost of the virtual-cloning process, we demonstrate that the problem of identifying an optimal virtual-cloning protocol can be cast as a semidefinite programming problem. Finally, we establish a connection between virtual cloning and state discrimination, from which universal bounds on the optimal cloning cost are derived.
It is well known that (non-orthogonal) pure states cannot be cloned so one may ask: how much or what kind of additional (quantum) information is needed to supplement one copy of a quantum state in order to be able to produce two copies of that state by a physical operation? For classical information, no supplementary information is required. However for pure quantum (non-orthogonal) states, we show that the supplementary information must always be as large as it can possibly be i.e. the clone must be able to be generated from the additional information alone, independently of the first (given) copy.
The no-cloning theorem has always been a hot topic in quantum theorem: As an impossibility result, this theorem precludes attempts to communicated faster than light and reinforces the security of several quantum cryptography protocol. In this survey, I will briefly introduce the history of quantum no-cloning theorem–how the thesis was invented, how did it get its name; And i would also discuss the hitherto progress on no-cloning theorem, several approach of none-perfect cloning on non-orthogonal states as well as the application of quantum no-cloning theorem in quantum cryptography and quantum information. This survey incorporates the lecture ”Quantum Information Amplifiers Beyond the No-Cloning Theorem” given by Prof. Giulio Chiribella and ”Quantum cryptography” given by Prof. Xiongfeng Ma.