This article needs additional citations for verification .(February 2018) |
In physics, the no-communication theorem or no-signaling principle is a no-go theorem from quantum information theory which states that, during measurement of an entangled quantum state, it is not possible for one observer, by making a measurement of a subsystem of the total state, to communicate information to another observer. The theorem is important because, in quantum mechanics, quantum entanglement is an effect by which certain widely separated events can be correlated in ways that, at first glance, suggest the possibility of communication faster-than-light. The no-communication theorem gives conditions under which such transfer of information between two observers is impossible. These results can be applied to understand the so-called paradoxes in quantum mechanics, such as the EPR paradox, or violations of local realism obtained in tests of Bell's theorem. In these experiments, the no-communication theorem shows that failure of local realism does not lead to what could be referred to as "spooky communication at a distance" (in analogy with Einstein's labeling of quantum entanglement as requiring "spooky action at a distance" on the assumption of QM's completeness).
The no-communication theorem states that, within the context of quantum mechanics, it is not possible to transmit classical bits of information by means of carefully prepared mixed or pure states, whether entangled or not. The theorem is only a sufficient condition that states that if the Kraus matrices commute then there can be no communication through the quantum entangled states and this is applicable to all communication. From a relativity and quantum field perspective also faster than light or "instantaneous" communication is disallowed. [1] : 100 Being only a sufficient condition there can be extra cases where communication is not allowed and there can be also cases where is still possible to communicate through the quantum channel encoding more than the classical information.
In regards to communication, a quantum channel can always be used to transfer classical information by means of shared quantum states. [2] [3] In 2008 Matthew Hastings proved a counterexample where the minimum output entropy is not additive for all quantum channels. Therefore, by an equivalence result due to Peter Shor, [4] the Holevo capacity is not just additive, but super-additive like the entropy, and by consequence there may be some quantum channels where you can transfer more than the classical capacity. [5] [6] Typically overall communication happens at the same time via quantum and non quantum channels, and in general time ordering and causality cannot be violated.
The basic assumption entering into the theorem is that a quantum-mechanical system is prepared in an initial state with some entangled states, and that this initial state is describable as a mixed or pure state in a Hilbert space H. After a certain amount of time the system is divided in two parts each of which contains some non entangled states and half of quantum entangled states and the two parts becomes spatially distinct, A and B, sent to two distinct observers, Alice and Bob, who are free to perform quantum mechanical measurements on their portion of the total system (viz, A and B). The question is: is there any action that Alice can perform on A that would be detectable by Bob making an observation of B? The theorem replies 'no'.
An important assumption going into the theorem is that neither Alice nor Bob is allowed, in any way, to affect the preparation of the initial state. If Alice were allowed to take part in the preparation of the initial state, it would be trivially easy for her to encode a message into it; thus neither Alice nor Bob participates in the preparation of the initial state. The theorem does not require that the initial state be somehow 'random' or 'balanced' or 'uniform': indeed, a third party preparing the initial state could easily encode messages in it, received by Alice and Bob. Simply, the theorem states that, given some initial state, prepared in some way, there is no action that Alice can take that would be detectable by Bob.
The proof proceeds by defining how the total Hilbert space H can be split into two parts, HA and HB, describing the subspaces accessible to Alice and Bob. The total state of the system is assumed to be described by a density matrix σ. This appears to be a reasonable assumption, as a density matrix is sufficient to describe both pure and mixed states in quantum mechanics. Another important part of the theorem is that measurement is performed by applying a generalized projection operator P to the state σ. This again is reasonable, as projection operators give the appropriate mathematical description of quantum measurements. After a measurement by Alice, the state of the total system is said to have collapsed to a state P(σ).
The goal of the theorem is to prove that Bob cannot in any way distinguish the pre-measurement state σ from the post-measurement state P(σ). This is accomplished mathematically by comparing the trace of σ and the trace of P(σ), with the trace being taken over the subspace HA. Since the trace is only over a subspace, it is technically called a partial trace. Key to this step is the assumption that the (partial) trace adequately summarizes the system from Bob's point of view. That is, everything that Bob has access to, or could ever have access to, measure, or detect, is completely described by a partial trace over HA of the system σ. Again, this is a reasonable assumption, as it is a part of standard quantum mechanics. The fact that this trace never changes as Alice performs her measurements is the conclusion of the proof of the no-communication theorem.
The proof of the theorem is commonly illustrated for the setup of Bell tests in which two observers Alice and Bob perform local observations on a common bipartite system, and uses the statistical machinery of quantum mechanics, namely density states and quantum operations. [1] [7] [8]
Alice and Bob perform measurements on system S whose underlying Hilbert space is
It is also assumed that everything is finite-dimensional to avoid convergence issues. The state of the composite system is given by a density operator on H. Any density operator σ on H is a sum of the form: where Ti and Si are operators on HA and HB respectively. For the following, it is not required to assume that Ti and Si are state projection operators: i.e. they need not necessarily be non-negative, nor have a trace of one. That is, σ can have a definition somewhat broader than that of a density matrix; the theorem still holds. Note that the theorem holds trivially for separable states. If the shared state σ is separable, it is clear that any local operation by Alice will leave Bob's system intact. Thus the point of the theorem is no communication can be achieved via a shared entangled state.
Alice performs a local measurement on her subsystem. In general, this is described by a quantum operation, on the system state, of the following kind where Vk are called Kraus matrices which satisfy
The term from the expression means that Alice's measurement apparatus does not interact with Bob's subsystem.
Supposing the combined system is prepared in state σ and assuming, for purposes of argument, a non-relativistic situation, immediately (with no time delay) after Alice performs her measurement, the relative state of Bob's system is given by the partial trace of the overall state with respect to Alice's system. In symbols, the relative state of Bob's system after Alice's operation is where is the partial trace mapping with respect to Alice's system.
One can directly calculate this state:
From this it is argued that, statistically, Bob cannot tell the difference between what Alice did and a random measurement (or whether she did anything at all).
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.
Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measurement. "Local" here refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields cannot propagate faster than the speed of light. "Hidden variables" are supposed properties of quantum particles that are not included in quantum theory but nevertheless affect the outcome of experiments. In the words of physicist John Stewart Bell, for whom this family of results is named, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."
In quantum mechanics, a density matrix is a matrix that describes an ensemble of physical systems as quantum states. It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed ensembles. Mixed ensembles arise in quantum mechanics in two different situations:
In physics, the CHSH inequality can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics cannot be reproduced by local hidden-variable theories. Experimental verification of the inequality being violated is seen as confirmation that nature cannot be described by such theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony, and Richard Holt, who described it in a much-cited paper published in 1969. They derived the CHSH inequality, which, as with John Stewart Bell's original inequality, is a constraint—on the statistical occurrence of "coincidences" in a Bell test—which is necessarily true if an underlying local hidden-variable theory exists. In practice, the inequality is routinely violated by modern experiments in quantum mechanics.
In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule. For example, a quantum particle like an electron can be described by a quantum state that associates to each point in space a complex number called a probability amplitude. Applying the Born rule to these amplitudes gives the probabilities that the electron will be found in one region or another when an experiment is performed to locate it. This is the best the theory can do; it cannot say for certain where the electron will be found. The same quantum state can also be used to make a prediction of how the electron will be moving, if an experiment is performed to measure its momentum instead of its position. The uncertainty principle implies that, whatever the quantum state, the range of predictions for the electron's position and the range of predictions for its momentum cannot both be narrow. Some quantum states imply a near-certain prediction of the result of a position measurement, but the result of a momentum measurement will be highly unpredictable, and vice versa. Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about "local hidden variables" within quantum systems.
In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan. The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation, a quantum operation is called a quantum channel.
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in quantum information and decoherence which is relevant for quantum measurement and thereby to the decoherent approaches to interpretations of quantum mechanics, including consistent histories and the relative state interpretation.
In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is
In quantum information theory, superdense coding is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the assumption of sender and receiver pre-sharing an entangled resource. In its simplest form, the protocol involves two parties, often referred to as Alice and Bob in this context, which share a pair of maximally entangled qubits, and allows Alice to transmit two bits to Bob by sending only one qubit. This protocol was first proposed by Charles H. Bennett and Stephen Wiesner in 1970 and experimentally actualized in 1996 by Klaus Mattle, Harald Weinfurter, Paul G. Kwiat and Anton Zeilinger using entangled photon pairs. Superdense coding can be thought of as the opposite of quantum teleportation, in which one transfers one qubit from Alice to Bob by communicating two classical bits, as long as Alice and Bob have a pre-shared Bell pair.
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.
In quantum information theory, an entanglement witness is a functional which distinguishes a specific entangled state from separable ones. Entanglement witnesses can be linear or nonlinear functionals of the density matrix. If linear, then they can also be viewed as observables for which the expectation value of the entangled state is strictly outside the range of possible expectation values of any separable state.
Holevo's theorem is an important limitative theorem in quantum computing, an interdisciplinary field of physics and computer science. It is sometimes called Holevo's bound, since it establishes an upper bound to the amount of information that can be known about a quantum state. It was published by Alexander Holevo in 1973.
In quantum mechanics, notably in quantum information theory, fidelity quantifies the "closeness" between two density matrices. It expresses the probability that one state will pass a test to identify as the other. It is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.
In quantum information theory, quantum relative entropy is a measure of distinguishability between two quantum states. It is the quantum mechanical analog of relative entropy.
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 where is the density matrix of the state and is the trace operation. The purity defines a measure on quantum states, giving information on how much a state is mixed.
In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distanceT is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions.
In quantum information theory, strong subadditivity of quantum entropy (SSA) is the relation among the von Neumann entropies of various quantum subsystems of a larger quantum system consisting of three subsystems. It is a basic theorem in modern quantum information theory. It was conjectured by D. W. Robinson and D. Ruelle in 1966 and O. E. Lanford III and D. W. Robinson in 1968 and proved in 1973 by E.H. Lieb and M.B. Ruskai, building on results obtained by Lieb in his proof of the Wigner-Yanase-Dyson conjecture.
Quantum refereed game in quantum information processing is a class of games in the general theory of quantum games. It is played between two players, Alice and Bob, and arbitrated by a referee. The referee outputs the pay-off for the players after interacting with them for a fixed number of rounds, while exchanging quantum information.
The min-entropy, in information theory, is the smallest of the Rényi family of entropies, corresponding to the most conservative way of measuring the unpredictability of a set of outcomes, as the negative logarithm of the probability of the most likely outcome. The various Rényi entropies are all equal for a uniform distribution, but measure the unpredictability of a nonuniform distribution in different ways. The min-entropy is never greater than the ordinary or Shannon entropy and that in turn is never greater than the Hartley or max-entropy, defined as the logarithm of the number of outcomes with nonzero probability.
{{cite journal}}
: CS1 maint: date and year (link)