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The no-Cloning Theorem Based on a Statistical Model

Koji Nagata, Tadao Nakamura

Published Sep 1, 2018• 0 citations
OpenAlex

Abstract

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.

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Theoretical Study of the no-Cloning Theorem

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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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No-cloning of nonorthogonal states does not require inner product preserving

Dafa Li, Xiangrong Li, Hongtao Huang +1 more

The no-cloning theorem says there is no quantum copy machine which can copy any one-qubit state. Inner product preserving was always used to prove the no-cloning of nonorthogonal states. In this paper we show that the no-cloning of nonorthogonal states does not require inner product preserving and discuss the minimal properties which a linear operator possesses to copy two different states at the same device. In this paper, we obtain the following necessary and sufficient condition. For any two different states ∣ψ⟩=a∣0⟩+b∣1⟩ and ∣φ⟩=c∣0⟩+d∣1⟩, assume that a linear operator L can copy them, that is, L(∣ψ,0⟩)=∣ψ,ψ⟩ and L(∣φ,0⟩)=∣φ,φ⟩. Then the two states are orthogonal if and only if L(∣0,0⟩) and L(∣1,0⟩) are unit length states. Thus we only need linearity and that L(∣0,0⟩) and L(∣1,0⟩) are unit length states to prove the no-cloning of nonorthogonal states. It implies that inner product preserving is not necessary for the no-cloning of nonorthogonal states.

Journal of Mathematical Physics•Jul 28, 2005•2 citations

A stronger no-cloning theorem

Richard Jozsa

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.

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No-cloning theorem for a single observable

Alexey E. Rastegin

Cloning of statistics of general quantum measurement is discussed. The presented approach is connected with the known concept of observable cloning, but differs in some essential respects. The reasons are illustrated within some variety of B92 protocol. As it is shown, there exist pairs of states such that the perfect cloning of given POVM is not possible. We discuss some properties of these intolerant sets. An example allowing the perfect cloning is presented as well.

arXiv:0907.4598•Jul 27, 2009•0 citations
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.
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.
The no-cloning theorem says there is no quantum copy machine which can copy any one-qubit state. Inner product preserving was always used to prove the no-cloning of nonorthogonal states. In this paper we show that the no-cloning of nonorthogonal states does not require inner product preserving and discuss the minimal properties which a linear operator possesses to copy two different states at the same device. In this paper, we obtain the following necessary and sufficient condition. For any two different states ∣ψ⟩=a∣0⟩+b∣1⟩ and ∣φ⟩=c∣0⟩+d∣1⟩, assume that a linear operator L can copy them, that is, L(∣ψ,0⟩)=∣ψ,ψ⟩ and L(∣φ,0⟩)=∣φ,φ⟩. Then the two states are orthogonal if and only if L(∣0,0⟩) and L(∣1,0⟩) are unit length states. Thus we only need linearity and that L(∣0,0⟩) and L(∣1,0⟩) are unit length states to prove the no-cloning of nonorthogonal states. It implies that inner product preserving is not necessary for the no-cloning of nonorthogonal states.
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.