Part of a series of articles about |
Quantum mechanics |
---|
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 William Wootters and Wojciech H. Zurek [2] as well as Dennis 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.
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).
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, Juan 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]
Suppose we have two quantum systems A and B with a common Hilbert space . Suppose we want to have a procedure to copy the state of quantum system A, over the state of quantum system B, for any original state (see bra–ket notation). That is, beginning with the state , we want to end up with the state . To make a "copy" of the state A, we combine it with system B in some unknown initial, or blank, state independent of , of which we have no prior knowledge.
The state of the initial composite system is then described by the following tensor product:
(in the following we will omit the symbol and keep it implicit).
There are only two permissible quantum operations with which we may manipulate the composite system:
The no-cloning theorem answers the following question in the negative: Is it possible to construct a unitary operator U, acting on , 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?
Theorem: There is no unitary operator U on such that for all normalised states and in
for some real number depending on and .
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.
To prove the theorem, we select an arbitrary pair of states and in the Hilbert space . Because U is supposed to be unitary, we would have
Since the quantum state is assumed to be normalized, we thus get
This implies that either or . Hence by the Cauchy–Schwarz inequality either or is orthogonal to . 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.
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.
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," i.e. treated as a pure state of a larger system. 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.[ clarification needed ] Thus the no-cloning theorem holds in full generality.
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]
Imperfect quantum cloning can be used as an eavesdropping attack on quantum cryptography protocols, among other uses in quantum information science.
Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread.
Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. The sender does not have to know the particular quantum state being transferred. Moreover, the location of the recipient can be unknown, but to complete the quantum teleportation, classical information needs to be sent from sender to receiver. Because classical information needs to be sent, quantum teleportation cannot occur faster than the speed of light.
Quantum decoherence is the loss of quantum coherence, the process in which a system's behaviour changes from that which can be explained by quantum mechanics to that which can be explained by classical mechanics. Beginning out of attempts to extend the understanding of quantum mechanics, the theory has developed in several directions and experimental studies have confirmed some of the key issues. Quantum computing relies on quantum coherence and is the primary practical applications of the concept.
In quantum information science, the Bell's states or EPR pairs are specific quantum states of two qubits that represent the simplest examples of quantum entanglement. The Bell's states are a form of entangled and normalized basis vectors. This normalization implies that the overall probability of the particle being in one of the mentioned states is 1: . Entanglement is a basis-independent result of superposition. Due to this superposition, measurement of the qubit will "collapse" it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will "collapse" the other qubit to a state whose measurement will yield one of two possible values, where the value depends on which Bell's state the two qubits are in initially. Bell's states can be generalized to certain quantum states of multi-qubit systems, such as the GHZ state for three or more subsystems.
In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information is a text document transmitted over the Internet.
LOCC, or local operations and classical communication, is a method in quantum information theory where a local (product) operation is performed on part of the system, and where the result of that operation is "communicated" classically to another part where usually another local operation is performed conditioned on the information received.
Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT transformations are represented on the Hilbert space of states.
In functional analysis and quantum information science, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalization of quantum measurement described by PVMs.
In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled quantum state that involves at least three subsystems. The four-particle version was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989, and the three-particle version was introduced by N. David Mermin in 1990. Extremely non-classical properties of the state have been observed. GHZ states for large numbers of qubits are theorized to give enhanced performance for metrology compared to other qubit superposition states.
The one-way or measurement-based quantum computer (MBQC) is a method of quantum computing that first prepares an entangled resource state, usually a cluster state or graph state, then performs single qubit measurements on it. It is "one-way" because the resource state is destroyed by the measurements.
In physics, the no-deleting theorem of quantum information theory is a no-go theorem which states that, in general, given two copies of some arbitrary quantum state, it is impossible to delete one of the copies. It is a time-reversed dual to the no-cloning theorem, which states that arbitrary states cannot be copied. This theorem seems remarkable, because, in many senses, quantum states are fragile; the theorem asserts that, in a particular case, they are also robust. Physicist Arun K. Pati along with Samuel L. Braunstein proved this theorem.
A decoherence-free subspace (DFS) is a subspace of a quantum system's Hilbert space that is invariant to non-unitary dynamics. Alternatively stated, they are a small section of the system Hilbert space where the system is decoupled from the environment and thus its evolution is completely unitary. DFSs can also be characterized as a special class of quantum error correcting codes. In this representation they are passive error-preventing codes since these subspaces are encoded with information that (possibly) won't require any active stabilization methods. These subspaces prevent destructive environmental interactions by isolating quantum information. As such, they are an important subject in quantum computing, where (coherent) control of quantum systems is the desired goal. Decoherence creates problems in this regard by causing loss of coherence between the quantum states of a system and therefore the decay of their interference terms, thus leading to loss of information from the (open) quantum system to the surrounding environment. Since quantum computers cannot be isolated from their environment and information can be lost, the study of DFSs is important for the implementation of quantum computers into the real world.
Entanglement distillation is the transformation of N copies of an arbitrary entangled state into some number of approximately pure Bell pairs, using only local operations and classical communication.
In quantum mechanics, and especially quantum information theory, the purity of a normalized quantum state is a scalar defined as
In quantum mechanics, weak measurements are a type of quantum measurement that results in an observer obtaining very little information about the system on average, but also disturbs the state very little. From Busch's theorem the system is necessarily disturbed by the measurement. In the literature weak measurements are also known as unsharp, fuzzy, dull, noisy, approximate, and gentle measurements. Additionally weak measurements are often confused with the distinct but related concept of the weak value.
The no-hiding theorem states that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot remain in the correlation between the system and the environment. This is a fundamental consequence of the linearity and unitarity of quantum mechanics. Thus, information is never lost. This has implications in black hole information paradox and in fact any process that tends to lose information completely. The no-hiding theorem is robust to imperfection in the physical process that seemingly destroys the original information.
In quantum information theory and operator theory, the Choi–Jamiołkowski isomorphism refers to the correspondence between quantum channels and quantum states, this is introduced by Man-Duen Choi and Andrzej Jamiołkowski. It is also called channel-state duality by some authors in the quantum information area, but mathematically, this is a more general correspondence between positive operators and the complete positive superoperators.
In quantum information theory and quantum optics, the Schrödinger–HJW theorem is a result about the realization of a mixed state of a quantum system as an ensemble of pure quantum states and the relation between the corresponding purifications of the density operators. The theorem is named after physicists and mathematicians Erwin Schrödinger, Lane P. Hughston, Richard Jozsa and William Wootters. The result was also found independently by Nicolas Gisin, and by Nicolas Hadjisavvas building upon work by Ed Jaynes, while a significant part of it was likewise independently discovered by N. David Mermin. Thanks to its complicated history, it is also known by various other names such as the GHJW theorem, the HJW theorem, and the purification theorem.
In quantum physics, the "monogamy" of quantum entanglement refers to the fundamental property that it cannot be freely shared between arbitrarily many parties.
Bell diagonal states are a class of bipartite qubit states that are frequently used in quantum information and quantum computation theory.