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In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park,^[1] in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by Wootters and Zurek^[2] as well as Dieks^[3] the same year). The aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem (as generally understood) concerns only pure states whereas the generalized statement regarding mixed states is known as the no-broadcast theorem.

In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by Wootters and Zurek as well as Dieks the same year). The aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem (as generally understood) concerns only pure states whereas the generalized statement regarding mixed states is known as the no-broadcast theorem.

在物理学中，不可克隆原理指出，不可能创造一个独立的、完全相同的任意未知量子态的副本，这一论断在量子计算领域及其他领域有着深远的意义。这个定理是詹姆斯 · 帕克在1970年提出的禁止入内定理的演变，在这个定理中，他证明了一个既简单又完美的非干扰性测量方案是不可能存在的(同样的结果在1982年由伍特斯和祖雷克以及迪克斯独立地推导出来)。上述定理并不排除一个系统的状态与另一个系统的状态纠缠在一起，因为克隆特指创造具有相同因子的可分离状态。例如，我们可以使用受控的非门和 Walsh-Hadamard 门来纠缠两个量子位而不违反不可克隆原理，因为没有明确定义的状态可以用纠缠态的子系统来定义。不可克隆原理定理(一般理解)只涉及纯态，而关于混合态的广义陈述被称为不广播定理。

The no-cloning theorem has a time-reversed dual, the no-deleting theorem. Together, these underpin the interpretation of quantum mechanics in terms of category theory, and, in particular, as a dagger compact category.^[4]^[5] This formulation, known as categorical quantum mechanics, allows, in turn, a connection to be made from quantum mechanics to linear logic as the logic of quantum information theory (in the same sense that intuitionistic logic arises from Cartesian closed categories).

The no-cloning theorem has a time-reversed dual, the no-deleting theorem. Together, these underpin the interpretation of quantum mechanics in terms of category theory, and, in particular, as a dagger compact category. This formulation, known as categorical quantum mechanics, allows, in turn, a connection to be made from quantum mechanics to linear logic as the logic of quantum information theory (in the same sense that intuitionistic logic arises from Cartesian closed categories).

不可克隆原理有一个时间反转的对偶，即不删除定理。总之，这些支持了范畴理论对量子力学的解释，特别是作为一个匕首紧凑范畴。这个公式，被称为范畴量子力学，反过来，允许从量子力学到线性逻辑的联系，就像量子信息论的逻辑一样(在同样的意义上，直觉主义逻辑起源于笛卡儿闭范畴范畴)。

History

= 历史 =

According to Asher Peres^[6] and David Kaiser,^[7] the publication of the 1982 proof of the no-cloning theorem by Wootters and Zurek^[2] and by Dieks^[3] was prompted by a proposal of Nick Herbert^[8] for a superluminal communication device using quantum entanglement, and Giancarlo Ghirardi^[9] had proven the theorem 18 months prior to the published proof by Wootters and Zurek in his referee report to said proposal (as evidenced by a letter from the editor^[9]). However, Ortigoso^[10] pointed out in 2018 that a complete proof along with an interpretation in terms of the lack of simple nondisturbing measurements in quantum mechanics was already delivered by Park in 1970.^[1]

According to Asher Peres and David Kaiser, the publication of the 1982 proof of the no-cloning theorem by Wootters and Zurek and by Dieks was prompted by a proposal of Nick Herbert for a superluminal communication device using quantum entanglement, and Giancarlo Ghirardi had proven the theorem 18 months prior to the published proof by Wootters and Zurek in his referee report to said proposal (as evidenced by a letter from the editor). However, Ortigoso pointed out in 2018 that a complete proof along with an interpretation in terms of the lack of simple nondisturbing measurements in quantum mechanics was already delivered by Park in 1970.

根据 Asher Peres 和 David Kaiser 的说法，1982年 wootter 和 Zurek 以及 Dieks 发表的不可克隆原理定理的证明，是由 Nick Herbert 提出的一个使用量子纠缠的超光速通讯设备的提议促成的，Giancarlo Ghirardi 在 wooters 和 Zurek 发表的证明报告之前18个月就已经证明了这个定理(编辑的一封信证明了这一点)。然而，ortiz 在2018年指出，Park 在1970年就已经提供了一个完整的证据，以及一个缺乏简单的非干扰性测量的解释，在21量子力学。

Theorem and proof

Suppose we have two quantum systems A and B with a common Hilbert space [math]\displaystyle{ H = H_A = H_B }[/math]. Suppose we want to have a procedure to copy the state [math]\displaystyle{ |\phi\rangle_A }[/math] of quantum system A, over the state [math]\displaystyle{ |e\rangle_B }[/math] of quantum system B, for any original state [math]\displaystyle{ |\phi\rangle_A }[/math] (see bra–ket notation). That is, beginning with the state [math]\displaystyle{ |\phi\rangle_A \otimes |e\rangle_B }[/math], we want to end up with the state [math]\displaystyle{ |\phi\rangle_A \otimes |\phi\rangle_B }[/math]. To make a "copy" of the state A, we combine it with system B in some unknown initial, or blank, state [math]\displaystyle{ |e\rangle_B }[/math] independent of [math]\displaystyle{ |\phi\rangle_A }[/math], of which we have no prior knowledge.

Suppose we have two quantum systems A and B with a common Hilbert space H = H_A = H_B. Suppose we want to have a procedure to copy the state |\phi\rangle_A of quantum system A, over the state |e\rangle_B of quantum system B, for any original state |\phi\rangle_A (see bra–ket notation). That is, beginning with the state |\phi\rangle_A \otimes |e\rangle_B , we want to end up with the state |\phi\rangle_A \otimes |\phi\rangle_B . To make a "copy" of the state A, we combine it with system B in some unknown initial, or blank, state |e\rangle_B independent of |\phi\rangle_A, of which we have no prior knowledge.

The state of the initial composite system is then described by the following tensor product:

然后用下列张量积描述初始合成系统的状态:

[math]\displaystyle{ |\phi\rangle_A \otimes |e\rangle_B. }[/math]

|\phi\rangle_A \otimes |e\rangle_B.

| phi rangle a o times | e rangle b.

(in the following we will omit the [math]\displaystyle{ \otimes }[/math] symbol and keep it implicit).

(in the following we will omit the \otimes symbol and keep it implicit).

(在下面我们将省略 otimes 符号并保持它的隐含)。

There are only two permissible quantum operations with which we may manipulate the composite system:

只有两种允许的量子操作可以用来操纵复合系统:

We can perform an observation, which irreversibly collapses the system into some eigenstate of an observable, corrupting the information contained in the qubit(s). This is obviously not what we want.
Alternatively, we could control the Hamiltonian of the combined system, and thus the time-evolution operator U(t), e.g. for a time-independent Hamiltonian, [math]\displaystyle{ U(t) = e^{-iHt/\hbar} }[/math]. Evolving up to some fixed time [math]\displaystyle{ t_0 }[/math] yields a unitary operator U on [math]\displaystyle{ H \otimes H }[/math], the Hilbert space of the combined system. However, no such unitary operator U can clone all states.

We can perform an observation, which irreversibly collapses the system into some eigenstate of an observable, corrupting the information contained in the qubit(s). This is obviously not what we want.
Alternatively, we could control the Hamiltonian of the combined system, and thus the time-evolution operator U(t), e.g. for a time-independent Hamiltonian, U(t) = e^{-iHt/\hbar}. Evolving up to some fixed time t_0 yields a unitary operator U on H \otimes H, the Hilbert space of the combined system. However, no such unitary operator U can clone all states.

我们可以进行一个观测，这个观测不可逆地将系统折叠成一个可观测的特征态，腐蚀量子比特中包含的信息。这显然不是我们想要的。
或者，我们可以控制组合系统的哈密顿量，从而控制时间演化算符 u (t) ，例如。对于与时间无关的哈密顿函数 u (t) = e ^ {-iHt/hbar }。向上演化到某个固定时间 t _ 0，得到组合系统的 Hilbert 空间 h 上的幺正算符 u。然而，没有这样的幺正算符可以克隆所有的状态。

The no-cloning theorem answers the following question in the negative: Is it possible to construct a unitary operator U, acting on [math]\displaystyle{ H_A \otimes H_B = H \otimes H }[/math], under which the state the system B is in always evolves into the state the system A is in, regardless of the state system A is in?

The no-cloning theorem answers the following question in the negative: Is it possible to construct a unitary operator U, acting on H_A \otimes H_B = H \otimes H, under which the state the system B is in always evolves into the state the system A is in, regardless of the state system A is in?

不可克隆原理回答了以下问题: 是否有可能构造一个幺正算符 u，作用于 h a o 乘以 h b = h o 乘以 h，在这个状态下系统 b 总是进化到系统 a 所处的状态，而不管系统 a 处于何种状态？

Theorem: There is no unitary operator U on [math]\displaystyle{ H \otimes H }[/math] such that for all normalised states [math]\displaystyle{ |\phi \rangle_A }[/math] and [math]\displaystyle{ |e\rangle_B }[/math] in [math]\displaystyle{ H }[/math]

Theorem: There is no unitary operator U on H \otimes H such that for all normalised states |\phi \rangle_A and |e\rangle_B in H

定理: h 上没有幺正算符 u，所以对于所有的正规化状态，h 中的 | phi rangle a 和 | e rangle b 都是不存在的

[math]\displaystyle{ U(|\phi\rangle_A |e\rangle_B) = e^{i \alpha(\phi,e)} |\phi\rangle_A |\phi\rangle_B }[/math]

U(|\phi\rangle_A |e\rangle_B) = e^{i \alpha(\phi,e)} |\phi\rangle_A |\phi\rangle_B

u (| phi rangle a | e rangle b) = e ^ { i alpha (phi，e)} | phi rangle a | phi rangle b

for some real number [math]\displaystyle{ \alpha }[/math] depending on [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ e }[/math].

for some real number \alpha depending on \phi and e.

取决于 φ 和 e 的 α 值。

The extra phase factor expresses the fact that a quantum-mechanical state defines a normalised vector in Hilbert space only up to a phase factor i.e. as an element of projectivised Hilbert space.

额外相位因子表示量子力学态在希尔伯特空间中只定义一个相位因子，即一个量子力学态。作为射影 Hilbert 空间的一个元素。

To prove the theorem, we select an arbitrary pair of states [math]\displaystyle{ |\phi\rangle_A }[/math] and [math]\displaystyle{ |\psi\rangle_A }[/math] in the Hilbert space [math]\displaystyle{ H }[/math]. Because U is supposed to be unitary, we would have

To prove the theorem, we select an arbitrary pair of states |\phi\rangle_A and |\psi\rangle_A in the Hilbert space H. Because U is supposed to be unitary, we would have

为了证明这个定理，我们在希尔伯特空间 h 中选择一对任意的状态 | phi rangle a 和 | psi rangle a，因为 u 是幺正的，所以我们有

[math]\displaystyle{ \langle \phi| \psi\rangle \langle e | e \rangle \equiv \langle \phi|_A \langle e|_B |\psi\rangle_A |e\rangle_B = \langle \phi|_A \langle e|_B U^\dagger U |\psi\rangle_A |e\rangle_B = e^{-i(\alpha(\phi, e) - \alpha(\psi, e))} \langle \phi|_A \langle \phi|_B |\psi\rangle_A |\psi\rangle_B \equiv e^{-i(\alpha(\phi, e) - \alpha(\psi, e))} \langle \phi |\psi\rangle^2. }[/math]

Since the quantum state [math]\displaystyle{ |e\rangle }[/math] is assumed to be normalized, we thus get

\langle \phi| \psi\rangle \langle e | e \rangle \equiv
\langle \phi|_A \langle e|_B |\psi\rangle_A |e\rangle_B =
\langle \phi|_A \langle e|_B U^\dagger U |\psi\rangle_A |e\rangle_B =
e^{-i(\alpha(\phi, e) - \alpha(\psi, e))} \langle \phi|_A \langle \phi|_B  |\psi\rangle_A |\psi\rangle_B \equiv
e^{-i(\alpha(\phi, e) - \alpha(\psi, e))} \langle \phi |\psi\rangle^2.

Since the quantum state |e\rangle is assumed to be normalized, we thus get

Langle phi | [咒语]E rangle equiv langle phi | 一个小角度 | 2012年10月15日2.1.2E rangle b = langle phi | 一个小角度 | 2. 把你的手放在你的脚上2.1.2E rangle _ b = e ^ {-i (alpha (phi，e)-alpha (psi，e))} langle phi | 一个 langle phi | 2012年10月15日2.1.2Psi rangle _ b equiv e ^ {-i (alpha (phi，e)-alpha (psi，e))} langle phi | Psi rangle ^ 2.由于量子态 | e 被假定为正则化的，我们得到了

[math]\displaystyle{ |\langle \phi | \psi \rangle|^2 = |\langle \phi | \psi \rangle|. }[/math]

= langle phi | psi rangle | ^ 2 = langle phi | psi rangle | .

This implies that either [math]\displaystyle{ |\langle \phi | \psi \rangle| = 1 }[/math] or [math]\displaystyle{ |\langle \phi | \psi \rangle| = 0 }[/math]. Hence by the Cauchy–Schwarz inequality either [math]\displaystyle{ \phi = e^{i\beta}\psi }[/math] or [math]\displaystyle{ \phi }[/math] is orthogonal to [math]\displaystyle{ \psi }[/math]. However, this cannot be the case for two arbitrary states. Therefore, a single universal U cannot clone a general quantum state. This proves the no-cloning theorem.

This implies that either |\langle \phi | \psi \rangle| = 1 or |\langle \phi | \psi \rangle| = 0. Hence by the Cauchy–Schwarz inequality either \phi = e^{i\beta}\psi or \phi is orthogonal to \psi. However, this cannot be the case for two arbitrary states. Therefore, a single universal U cannot clone a general quantum state. This proves the no-cloning theorem.

这意味着 | langle phi | psi rangle | = 1或 | langle phi | psi rangle | = 0。因此通过 Cauchy-Schwarz 不等式，或者 φ = e ^ { i beta } ，或者 φ 与 psi 正交。但是，对于两个任意状态，情况就不一样了。因此，单个宇宙 u 不可能克隆一个普遍的量子态。这证明了不可克隆原理。

Take a qubit for example. It can be represented by two complex numbers, called probability amplitudes (normalised to 1), that is three real numbers (two polar angles and one radius). Copying three numbers on a classical computer using any copy and paste operation is trivial (up to a finite precision) but the problem manifests if the qubit is unitarily transformed (e.g. by the Hadamard quantum gate) to be polarised (which unitary transformation is a surjective isometry). In such a case the qubit can be represented by just two real numbers (one polar angle and one radius equal to 1), while the value of the third can be arbitrary in such a representation. Yet a realisation of a qubit (polarisation-encoded photon, for example) is capable of storing the whole qubit information support within its "structure". Thus no single universal unitary evolution U can clone an arbitrary quantum state according to the no-cloning theorem. It would have to depend on the transformed qubit (initial) state and thus would not have been universal.

Take a qubit for example. It can be represented by two complex numbers, called probability amplitudes (normalised to 1), that is three real numbers (two polar angles and one radius). Copying three numbers on a classical computer using any copy and paste operation is trivial (up to a finite precision) but the problem manifests if the qubit is unitarily transformed (e.g. by the Hadamard quantum gate) to be polarised (which unitary transformation is a surjective isometry). In such a case the qubit can be represented by just two real numbers (one polar angle and one radius equal to 1), while the value of the third can be arbitrary in such a representation. Yet a realisation of a qubit (polarisation-encoded photon, for example) is capable of storing the whole qubit information support within its "structure". Thus no single universal unitary evolution U can clone an arbitrary quantum state according to the no-cloning theorem. It would have to depend on the transformed qubit (initial) state and thus would not have been universal.

以量子位为例。它可以用两个复数表示，称为概率幅(规范化为1) ，即三个实数(两个极角和一个半径)。使用任何复制粘贴操作在经典计算机上复制三个数字都是很简单的(达到有限的精度) ，但是如果量子位进行单精度变换(例如。被 Hadamard 量子门极化(幺正是一个满射的等距术语)。在这种情况下，量子位只能由两个实数表示(一个极角和一个半径等于1) ，而第三个实数的值可以在这种表示中任意取得。然而，实现一个量子位(偏振编码光子，例如)是能够存储整个量子位的信息支持在其“结构”。因此，没有一个单一的宇宙幺正演化 u 可以克隆任意的量子态，根据不可克隆原理。它必须依赖于转换后的量子位(初始)状态，因此不具有通用性。

Generalization

= 归纳 =

In the statement of the theorem, two assumptions were made: the state to be copied is a pure state and the proposed copier acts via unitary time evolution. These assumptions cause no loss of generality. If the state to be copied is a mixed state, it can be purified.模板:Clarify Alternately, a different proof can be given that works directly with mixed states; in this case, the theorem is often known as the no-broadcast theorem.^[11]^[12] Similarly, an arbitrary quantum operation can be implemented via introducing an ancilla and performing a suitable unitary evolution.模板:Clarify Thus the no-cloning theorem holds in full generality.

In the statement of the theorem, two assumptions were made: the state to be copied is a pure state and the proposed copier acts via unitary time evolution. These assumptions cause no loss of generality. If the state to be copied is a mixed state, it can be purified. Alternately, a different proof can be given that works directly with mixed states; in this case, the theorem is often known as the no-broadcast theorem. Similarly, an arbitrary quantum operation can be implemented via introducing an ancilla and performing a suitable unitary evolution. Thus the no-cloning theorem holds in full generality.

在定理的陈述中，提出了两个假设: 被复制的状态是一个纯态，复印机通过幺正时间演化作用。这些假设不会导致一般性的损失。如果要复制的状态是混合状态，则可以对其进行净化。另外，可以给出一个不同的直接与混合状态相关的证明; 在这种情况下，这个定理通常被称为不广播定理。类似地，通过引入一个 ancilla 并执行一个合适的幺正演化，可以实现任意的量子操作。因此，不可克隆原理的观点是完全概括的。

Consequences

The no-cloning theorem prevents the use of certain classical error correction techniques on quantum states. For example, backup copies of a state in the middle of a quantum computation cannot be created and used for correcting subsequent errors. Error correction is vital for practical quantum computing, and for some time it was unclear whether or not it was possible. In 1995, Shor and Steane showed that it is by independently devising the first quantum error correcting codes, which circumvent the no-cloning theorem.
Similarly, cloning would violate the no-teleportation theorem, which says that it is impossible to convert a quantum state into a sequence of classical bits (even an infinite sequence of bits), copy those bits to some new location, and recreate a copy of the original quantum state in the new location. This should not be confused with entanglement-assisted teleportation, which does allow a quantum state to be destroyed in one location, and an exact copy to be recreated in another location.
The no-cloning theorem is implied by the no-communication theorem, which states that quantum entanglement cannot be used to transmit classical information (whether superluminally, or slower). That is, cloning, together with entanglement, would allow such communication to occur. To see this, consider the EPR thought experiment, and suppose quantum states could be cloned. Assume parts of a maximally entangled Bell state are distributed to Alice and Bob. Alice could send bits to Bob in the following way: If Alice wishes to transmit a "0", she measures the spin of her electron in the z direction, collapsing Bob's state to either [math]\displaystyle{ |z+\rangle_B }[/math] or [math]\displaystyle{ |z-\rangle_B }[/math]. To transmit "1", Alice does nothing to her qubit. Bob creates many copies of his electron's state, and measures the spin of each copy in the z direction. Bob will know that Alice has transmitted a "0" if all his measurements will produce the same result; otherwise, his measurements will have outcomes [math]\displaystyle{ |z+\rangle_B }[/math] or [math]\displaystyle{ |z-\rangle_B }[/math] with equal probability. This would allow Alice and Bob to communicate classical bits between each other (possibly across space-like separations, violating causality).
Quantum states cannot be discriminated perfectly.^[13]
The no cloning theorem prevents an interpretation of the holographic principle for black holes as meaning that there are two copies of information, one lying at the event horizon and the other in the black hole interior. This leads to more radical interpretations, such as black hole complementarity.
The no-cloning theorem applies to all dagger compact categories: there is no universal cloning morphism for any non-trivial category of this kind.^[14] Although the theorem is inherent in the definition of this category, it is not trivial to see that this is so; the insight is important, as this category includes things that are not finite-dimensional Hilbert spaces, including the category of sets and relations and the category of cobordisms.

The no-cloning theorem prevents the use of certain classical error correction techniques on quantum states. For example, backup copies of a state in the middle of a quantum computation cannot be created and used for correcting subsequent errors. Error correction is vital for practical quantum computing, and for some time it was unclear whether or not it was possible. In 1995, Shor and Steane showed that it is by independently devising the first quantum error correcting codes, which circumvent the no-cloning theorem.
Similarly, cloning would violate the no-teleportation theorem, which says that it is impossible to convert a quantum state into a sequence of classical bits (even an infinite sequence of bits), copy those bits to some new location, and recreate a copy of the original quantum state in the new location. This should not be confused with entanglement-assisted teleportation, which does allow a quantum state to be destroyed in one location, and an exact copy to be recreated in another location.
The no-cloning theorem is implied by the no-communication theorem, which states that quantum entanglement cannot be used to transmit classical information (whether superluminally, or slower). That is, cloning, together with entanglement, would allow such communication to occur. To see this, consider the EPR thought experiment, and suppose quantum states could be cloned. Assume parts of a maximally entangled Bell state are distributed to Alice and Bob. Alice could send bits to Bob in the following way: If Alice wishes to transmit a "0", she measures the spin of her electron in the z direction, collapsing Bob's state to either |z+\rangle_B or |z-\rangle_B. To transmit "1", Alice does nothing to her qubit. Bob creates many copies of his electron's state, and measures the spin of each copy in the z direction. Bob will know that Alice has transmitted a "0" if all his measurements will produce the same result; otherwise, his measurements will have outcomes |z+\rangle_B or |z-\rangle_B with equal probability. This would allow Alice and Bob to communicate classical bits between each other (possibly across space-like separations, violating causality).
Quantum states cannot be discriminated perfectly.
The no cloning theorem prevents an interpretation of the holographic principle for black holes as meaning that there are two copies of information, one lying at the event horizon and the other in the black hole interior. This leads to more radical interpretations, such as black hole complementarity.
The no-cloning theorem applies to all dagger compact categories: there is no universal cloning morphism for any non-trivial category of this kind.S. Abramsky, "No-Cloning in categorical quantum mechanics", (2008) Semantic Techniques for Quantum Computation, I. Mackie and S. Gay (eds), Cambridge University Press. Although the theorem is inherent in the definition of this category, it is not trivial to see that this is so; the insight is important, as this category includes things that are not finite-dimensional Hilbert spaces, including the category of sets and relations and the category of cobordisms.

= = =

不可克隆原理阻止了对量子态使用某些经典的纠错技术。例如，无法创建量子计算中某个状态的备份副本并用于纠正后续错误。纠错对于实际的量子计算是至关重要的，有一段时间我们不清楚它是否可行。1995年，Shor 和 Steane 展示了它是通过独立设计第一个量子纠错码来绕过不可克隆原理。
同样，克隆将违反禁止量子传输定理，即不可能将一个量子态转化为一系列经典比特(甚至是无限的比特序列) ，将这些比特复制到某个新的位置，并在新的位置复制原来的量子态。这不应该与纠缠辅助的隐形传态混淆，后者允许量子态在一个位置被摧毁，在另一个位置被重新创建。
无通信定理暗示了不可克隆原理，即量子纠缠不能用于传输经典信息(无论是超光速传输还是慢速传输)。也就是说，克隆技术和纠缠技术一起，将允许这样的通信发生。要看到这一点，考虑 EPR 思想实验，并假设量子态可以被克隆。假设部分最大纠缠的贝尔态被分配给爱丽丝和鲍勃。爱丽丝可以通过以下方式向鲍勃发送比特: 如果爱丽丝希望传送一个“0”，她测量她的电子在 z 方向的自旋，将鲍勃的状态压缩到 | z + rangle _ b 或 | z-rangle _ b。为了传输“1”，爱丽丝对她的量子位什么也不做。Bob 创建了许多电子状态的副本，并测量每个副本在 z 方向上的自旋。如果 Bob 的所有测量结果都相同，那么他就知道 Alice 传递了一个“0”; 否则，他的测量结果将以相同的概率 | z + rangle _ b 或 | z-rangle _ b。这将允许 Alice 和 Bob 在彼此之间传递经典的比特信息(可能跨越类似空间的分离，违反了因果关系)。
量子态不能完全区分。
不可克隆定理阻止了对黑洞全息原理的解释，即信息有两个副本，一个在事件视界，另一个在黑洞内部。这导致了更激进的解释，比如黑洞互补性。
不可克隆原理适用于所有匕首紧凑类别: 这个类别的任何非平凡类别都没有通用的克隆形态。量子计算的语义技术》，量子力学出版社，2008年。虽然这个定理是这个范畴的定义中固有的，但是看到这一点并不是微不足道的; 洞察力是重要的，因为这个范畴包括不是有限维希尔伯特空间的事物，包括集合和关系的范畴以及余子集的范畴。

Imperfect cloning

Even though it is impossible to make perfect copies of an unknown quantum state, it is possible to produce imperfect copies. This can be done by coupling a larger auxiliary system to the system that is to be cloned, and applying a unitary transformation to the combined system. If the unitary transformation is chosen correctly, several components of the combined system will evolve into approximate copies of the original system. In 1996, V. Buzek and M. Hillery showed that a universal cloning machine can make a clone of an unknown state with the surprisingly high fidelity of 5/6.^[15]

Even though it is impossible to make perfect copies of an unknown quantum state, it is possible to produce imperfect copies. This can be done by coupling a larger auxiliary system to the system that is to be cloned, and applying a unitary transformation to the combined system. If the unitary transformation is chosen correctly, several components of the combined system will evolve into approximate copies of the original system. In 1996, V. Buzek and M. Hillery showed that a universal cloning machine can make a clone of an unknown state with the surprisingly high fidelity of 5/6.

= = 不完美的克隆即使不可能完美地复制一个未知的量子态，也可能产生不完美的复制品。这可以通过将一个更大的辅助系统耦合到将要克隆的系统上，并在组合系统上应用一个幺正控制器来实现。如果正确地选择幺正，组合系统的几个组成部分将演变成原系统的近似副本。1996年，v · 布泽克和 m · 希勒里证明了一台通用克隆机器可以制造出一个未知状态的克隆，其保真度达到惊人的5/6。

Imperfect quantum cloning can be used as an eavesdropping attack on quantum cryptography protocols, among other uses in quantum information science.

不完美的量子克隆技术可以用来窃听量子密码学协议，以及其他在量子信息科学中的应用。

References

↑ ^1.0 ^1.1 Park, James (1970). "The concept of transition in quantum mechanics". Foundations of Physics. 1 (1): 23–33. Bibcode:1970FoPh....1...23P. CiteSeerX 10.1.1.623.5267. doi:10.1007/BF00708652. S2CID 55890485.
↑ ^2.0 ^2.1 Wootters, William; Zurek, Wojciech (1982). "A Single Quantum Cannot be Cloned". Nature. 299 (5886): 802–803. Bibcode:1982Natur.299..802W. doi:10.1038/299802a0. S2CID 4339227.
↑ ^3.0 ^3.1 Dieks, Dennis (1982). "Communication by EPR devices". Physics Letters A. 92 (6): 271–272. Bibcode:1982PhLA...92..271D. CiteSeerX 10.1.1.654.7183. doi:10.1016/0375-9601(82)90084-6. hdl:1874/16932.
↑ Baez, John; Stay, Mike (2010). "Physics, Topology, Logic and Computation: A Rosetta Stone". New Structures for Physics. Berlin: Springer. pp. 95–172. ISBN 978-3-642-12821-9. http://math.ucr.edu/home/baez/rosetta.pdf.
↑ Coecke, Bob (2009). "Quantum Picturalism". Contemporary Physics. 51: 59–83. arXiv:0908.1787. doi:10.1080/00107510903257624. S2CID 752173.
↑ Peres, Asher (2003). "How the No-Cloning Theorem Got its Name". Fortschritte der Physik. 51 (45): 458–461. arXiv:quant-ph/0205076. Bibcode:2003ForPh..51..458P. doi:10.1002/prop.200310062. S2CID 16588882.
↑ Kaiser, David (2011). How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival. W. W. Norton. ISBN 978-0-393-07636-3.
↑ Herbert, Nick (1982). "FLASH—A superluminal communicator based upon a new kind of quantum measurement". Foundations of Physics. 12 (12): 1171–1179. Bibcode:1982FoPh...12.1171H. doi:10.1007/BF00729622. S2CID 123118337.
↑ ^9.0 ^9.1 Ghirardi, GianCarlo (2013), "Entanglement, Nonlocality, Superluminal Signaling and Cloning", in Bracken, Paul (ed.), Advances in Quantum Mechanics, IntechOpen (published April 3, 2013), arXiv:1305.2305, doi:10.5772/56429, ISBN 978-953-51-1089-7, S2CID 118778014
↑ Ortigoso, Juan (2018). "Twelve years before the quantum no-cloning theorem". American Journal of Physics. 86 (3): 201–205. arXiv:1707.06910. Bibcode:2018AmJPh..86..201O. doi:10.1119/1.5021356. S2CID 119192142.
↑ Barnum, Howard; Caves, Carlton M.; Fuchs, Christopher A.; Jozsa, Richard; Schumacher, Benjamin (1996-04-08). "Noncommuting Mixed States Cannot Be Broadcast". Physical Review Letters. 76 (15): 2818–2821. arXiv:quant-ph/9511010. Bibcode:1996PhRvL..76.2818B. doi:10.1103/PhysRevLett.76.2818. PMID 10060796. S2CID 11724387.
↑ Kalev, Amir; Hen, Itay (2008-05-29). "No-Broadcasting Theorem and Its Classical Counterpart". Physical Review Letters. 100 (21): 210502. arXiv:0704.1754. Bibcode:2008PhRvL.100u0502K. doi:10.1103/PhysRevLett.100.210502. PMID 18518590. S2CID 40349990.
↑ Bae, Joonwoo; Kwek, Leong-Chuan (2015-02-27). "Quantum state discrimination and its applications". Journal of Physics A: Mathematical and Theoretical. 48 (8): 083001. arXiv:1707.02571. Bibcode:2015JPhA...48h3001B. doi:10.1088/1751-8113/48/8/083001. ISSN 1751-8113. S2CID 119199057.
↑ S. Abramsky, "No-Cloning in categorical quantum mechanics", (2008) Semantic Techniques for Quantum Computation, I. Mackie and S. Gay (eds), Cambridge University Press. 模板:Arxiv
↑ Bužek, V.; Hillery, M. (1996). "Quantum Copying: Beyond the No-Cloning Theorem". Phys. Rev. A. 54 (3): 1844–1852. arXiv:quant-ph/9607018. Bibcode:1996PhRvA..54.1844B. doi:10.1103/PhysRevA.54.1844. PMID 9913670. S2CID 1446565.

Other sources

V. Buzek and M. Hillery, Quantum cloning, Physics World 14 (11) (2001), pp. 25–29.

模板:Quantum computing

V. Buzek and M. Hillery, Quantum cloning, Physics World 14 (11) (2001), pp. 25–29.

Category:Quantum information science Category:Theorems in quantum mechanics Category:Articles containing proofs

= = = 其他来源 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =。25–29.范畴: 量子信息科学范畴: 量子力学范畴中的定理: 包含证明的文章

This page was moved from wikipedia:en:No-cloning theorem. Its edit history can be viewed at 量子不可克隆定理/edithistory

[park-1] 1.0 ^1.1 Park, James (1970). "The concept of transition in quantum mechanics". Foundations of Physics. 1 (1): 23–33. Bibcode:1970FoPh....1...23P. CiteSeerX 10.1.1.623.5267. doi:10.1007/BF00708652. S2CID 55890485.

[wootterszurek-2] 2.0 ^2.1 Wootters, William; Zurek, Wojciech (1982). "A Single Quantum Cannot be Cloned". Nature. 299 (5886): 802–803. Bibcode:1982Natur.299..802W. doi:10.1038/299802a0. S2CID 4339227.

[dieks-3] 3.0 ^3.1 Dieks, Dennis (1982). "Communication by EPR devices". Physics Letters A. 92 (6): 271–272. Bibcode:1982PhLA...92..271D. CiteSeerX 10.1.1.654.7183. doi:10.1016/0375-9601(82)90084-6. hdl:1874/16932.

[4] Baez, John; Stay, Mike (2010). "Physics, Topology, Logic and Computation: A Rosetta Stone". New Structures for Physics. Berlin: Springer. pp. 95–172. ISBN 978-3-642-12821-9. http://math.ucr.edu/home/baez/rosetta.pdf.

[5] Coecke, Bob (2009). "Quantum Picturalism". Contemporary Physics. 51: 59–83. arXiv:0908.1787. doi:10.1080/00107510903257624. S2CID 752173.

[6] Peres, Asher (2003). "How the No-Cloning Theorem Got its Name". Fortschritte der Physik. 51 (45): 458–461. arXiv:quant-ph/0205076. Bibcode:2003ForPh..51..458P. doi:10.1002/prop.200310062. S2CID 16588882.

[7] Kaiser, David (2011). How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival. W. W. Norton. ISBN 978-0-393-07636-3.

[8] Herbert, Nick (1982). "FLASH—A superluminal communicator based upon a new kind of quantum measurement". Foundations of Physics. 12 (12): 1171–1179. Bibcode:1982FoPh...12.1171H. doi:10.1007/BF00729622. S2CID 123118337.

[Ghirardi2013-9] 9.0 ^9.1 Ghirardi, GianCarlo (2013), "Entanglement, Nonlocality, Superluminal Signaling and Cloning", in Bracken, Paul (ed.), Advances in Quantum Mechanics, IntechOpen (published April 3, 2013), arXiv:1305.2305, doi:10.5772/56429, ISBN 978-953-51-1089-7, S2CID 118778014

[10] Ortigoso, Juan (2018). "Twelve years before the quantum no-cloning theorem". American Journal of Physics. 86 (3): 201–205. arXiv:1707.06910. Bibcode:2018AmJPh..86..201O. doi:10.1119/1.5021356. S2CID 119192142.

[11] Barnum, Howard; Caves, Carlton M.; Fuchs, Christopher A.; Jozsa, Richard; Schumacher, Benjamin (1996-04-08). "Noncommuting Mixed States Cannot Be Broadcast". Physical Review Letters. 76 (15): 2818–2821. arXiv:quant-ph/9511010. Bibcode:1996PhRvL..76.2818B. doi:10.1103/PhysRevLett.76.2818. PMID 10060796. S2CID 11724387.

[12] Kalev, Amir; Hen, Itay (2008-05-29). "No-Broadcasting Theorem and Its Classical Counterpart". Physical Review Letters. 100 (21): 210502. arXiv:0704.1754. Bibcode:2008PhRvL.100u0502K. doi:10.1103/PhysRevLett.100.210502. PMID 18518590. S2CID 40349990.

[13] Bae, Joonwoo; Kwek, Leong-Chuan (2015-02-27). "Quantum state discrimination and its applications". Journal of Physics A: Mathematical and Theoretical. 48 (8): 083001. arXiv:1707.02571. Bibcode:2015JPhA...48h3001B. doi:10.1088/1751-8113/48/8/083001. ISSN 1751-8113. S2CID 119199057.

[14] S. Abramsky, "No-Cloning in categorical quantum mechanics", (2008) Semantic Techniques for Quantum Computation, I. Mackie and S. Gay (eds), Cambridge University Press. 模板:Arxiv

[15] Bužek, V.; Hillery, M. (1996). "Quantum Copying: Beyond the No-Cloning Theorem". Phys. Rev. A. 54 (3): 1844–1852. arXiv:quant-ph/9607018. Bibcode:1996PhRvA..54.1844B. doi:10.1103/PhysRevA.54.1844. PMID 9913670. S2CID 1446565.

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量子不可克隆定理

目录

History

History

= 历史 =

Theorem and proof

Generalization

Generalization

= 归纳 =

Consequences

= = =

Imperfect cloning

See also

References

Other sources

导航菜单

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