# No Communication Theorem

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## Introduction

The no-communication theorem or no-signal principle in physics is a forbidden theorem in quantum information theory that during the measurement of an entangled quantum state it is impossible for a single observer to measure it by making a measurement of the subsystem. said. Communicate overall state information to another observer. This theorem is important because in quantum mechanics, quantum entanglement is an effect that allows certain events that are widely separated to be correlated in ways that seemingly suggest the possibility of faster-than-light communication. The no-communication theorem gives the condition that such transfer of information between two observers is impossible. These results can be applied to understand 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 the failure of local realism does not lead to so-called ``eerie communication at a distance'' (Einstein called quantum entanglement ``eerie at a distance''). (assuming QM is perfect).

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## Informal overview

The no-communication theorem states that in the context of quantum mechanics, it is impossible to transmit classical information bits using carefully prepared mixed or pure states, with or without entanglement. . This theorem is just a sufficient condition that if the Claus matrices are commutative, there can be no communication via entangled states, which is true for all communication. The theory of relativity and quantum fields also do not allow faster-than-light or "instantaneous" communication. There may be additional cases where communication is not permitted because it is only a sufficient condition, and there may be cases where communication is still possible via quantum channel coding over classical information. As for communication, quantum channels can always be used to transfer classical information by means of shared quantum states. In 2008 Matthew Hastings proved a counterexample that the minimum output entropy does not add up over all quantum channels. Thus, according to Peter Shor's equivalence results, the Holevo capacity is not merely additive, but hyperadditive, as is the entropy, resulting in a quantum channel capable of transferring quantities exceeding the classical capacity. There may be several The entire communication usually happens simultaneously through quantum and non-quantum channels, and in general temporal order and causality cannot be violated. The basic assumption going into the theorem is that a quantum mechanical system is prepared with an initial state containing some entangled states, and this initial state can be described as a mixed or pure state in the Hilbert space H. The system is split into two parts, each containing some unentangled and half-entangled states, the two parts being spatially distinct A and B, free from quantum mechanics. are sent to Alice and Bob, two separate observers who can run A measurement of that part of the whole system (that is, A and B). The question is, are there any actions that Alice can perform on A that can be detected by Bob observing B? The theorem answer is "no". A key assumption going into the theorem is that neither Alice nor Bob are allowed to influence the initial state preparation in any way. If Alice can participate in the preparation of the initial state, it is trivial for her to encode the message into the initial state. Therefore neither Alice nor Bob participates in the preparation of the initial state. This theorem does not require that the initial state is somehow "random", "balanced" or "uniform". In fact, a third party that prepares the initial state can easily encode into it the messages that Alice and Bob receive. Simply put, this theorem states that given an initial state prepared in some way, there is no action that Alice can take that Bob can detect. The proof proceeds by defining how the total Hilbert space H is split into two parts HA and HB, describing the subspaces accessible to Alice and Bob. The overall state of the system is assumed to be described by the density matrix σ. This seems to be a reasonable assumption, since density matrices are sufficient to describe both pure and mixed states in quantum mechanics. Another important part of the theorem is that the measurement is performed by applying the generalized projection operator P to the states σ. This is also reasonable, since the projection operator gives a good mathematical description of quantum measurements. After measurements by Alice, the state of the whole system is said to have collapsed to the state P(σ). The purpose of this theorem is to prove that Bob can never distinguish between the pre-measurement state σ and the post-measurement state P(σ). This is achieved mathematically by comparing the traces of σ and P(σ) using the traces obtained over the subspace HA. Technically it is called a partial trace because the trace is only on a subspace. The key to this step is the assumption that the (partial) trace adequately summarizes the system from Bob's point of view. In other words, everything Bob can access, or can access, measure, or detect is completely described by the partial traces on the HA of the system σ. Again, this is a reasonable assumption as it is part of standard quantum mechanics. The fact that this trace never changes when Alice performs measurements concludes the proof of the no-communication theorem.

## Formulation

The proof of this theorem is usually done in a Bell test setup where two observers, Alice and Bob, perform local observations in a common bipartite system and use the statistical machinery of quantum mechanics, i.e. density states and quantum operations. is indicated. Alice and Bob run. Measurements in a system S whose underlying Hilbert space is Also, everything is assumed to be finite-dimensional to avoid convergence problems. The state of the composite system is given by the density operator on H. The density operator σ on H is a sum of the form where Ti and Si are the operators of HA and HB respectively. There is no need to assume that Ti and Si are state projection operators if: That is, they need not necessarily be non-negative or have a negative signature. That is, σ can have a slightly broader definition than that of the density matrix. The theorem still holds. Note that this theorem holds trivially for separable states. If the shared state σ is separable, it is clear that Alice's local manipulation leaves Bob's system intact. So the gist of the theorem is that communication cannot be achieved via a shared entangled state. Alice performs local measurements on the subsystem. In general, this is described by the following quantum operations on the system state: where Vk is called a Claus matrix that satisfies the term from the expression This means that Alice's measurement device does not interact with Bob's subsystem. Assuming the coupled system is prepared in state σ, and for discussion purposes, assuming a non-relativistic situation immediately after Alice performs the measurement (with no time delay), the relative state of Bob's system is The overall state of Alice's system. In symbols, the relative state of Bob's system after Alice's surgery is: where tr H. a {\displaystyle \operatorname {tr} _{H_{A}}} is a partial trace mapping for Alice's system. This state can be calculated directly. From this it is argued that statistically Bob cannot distinguish between what Alice did and the difference between random measurements (or whether Alice did something).

## Some comments

For Density Operator P. ( σ ) {\displaystyle P(\sigma )} is allowed to evolve under the influence of non-local interactions between A and B, in general the computations in the proof no longer hold unless a suitable commutation relation is assumed. Therefore, the non-communication theorem states that information cannot be transmitted using shared entanglement alone. Compare this with the non-teleportation theorem, which states that quantum information cannot be transmitted in classical information channels. (By transmission, we mean transmission with perfect fidelity.) However, quantum teleportation schemes take advantage of both resources to achieve things that neither can do alone. The no-communication theorem implies the no-cloning theorem that quantum states cannot be (perfectly) copied. Thus, cloning is a sufficient condition for classical information transfer to occur. To see this, suppose we can clone the quantum state. Suppose a portion of the maximally entangled Bell state is distributed to Alice and Bob. Alice can send bits to Bob by: If Alice wants to send a '0', she measures her electron's spin in the z direction and causes Bob's state to collapse to one of the following: | z + ⟩ B. {\displaystyle |z+\rangle _{B}} again | z − ⟩ B. {\displaystyle |z-\rangle _{B}} . To send a '1', Alice does nothing to the qubit. Bob makes many copies of the electronic state and measures the spin in the z direction of each copy. Bob knows that Alice sent a '0' if all the measurements produce the same result. otherwise his measurements will have consequences | z + ⟩ B. {\displaystyle |z+\rangle _{B}} again | z − ⟩ B. {\displaystyle |z-\rangle _{B}} with equal probability. This allows Alice and Bob to communicate classical bits to each other (perhaps in violation of causality, across space-like separations). The version of the non-communication theorem described in this article states that the quantum system shared by Alice and Bob is a composite system, i.e. its underlying Hilbert space is the tensor product whose first factor is Alice's interaction It assumes that you describe the parts of the system that can. and its second element represent the parts of the system that Bob can interact with. In quantum field theory, this assumption can be replaced by the assumption that Alice and Bob are spatially separated. Another version of this no-communication theorem states that faster-than-light communication cannot be achieved using processes that obey the laws of quantum field theory. The non-communication theorem proof assumes that all measurable properties of Bob's system can be computed from its reduced density matrix. This is true considering the Bourne rule for calculating probabilities of taking various measurements. However, this equivalence with the Bourne rule can also be essentially deduced in the opposite direction. In other words, it can be shown that the Bourne Rule is based on the assumption that spatially separated events cannot violate causality by influencing each other.

## See also

broadcast prohibition theorem irreproducible theorem non-deletion theorem non-hiding theorem Non-teleportation theorem

## References

Hall, Michael J.W. (1987). "Inaccurate measurements and nonlocality in quantum mechanics". Physics Letters A. Elsevier BV. 125 (2-3): 89-91. Bibcode: 1987PhLA..125...89H. Doi: 10.1016/0375-9601(87)90127-7. ISSN 0375-9601. Ghirardi, GC; Grassi, R; Rimini, A; Weber, T (1988-05-15). "In EPR-type experiments with CP violations, faster-than-light communication between distant observers is not possible." Europhysics Letters (EPL). IOP Publishing. 6(2): 95-100. Bibcode: 1988EL....6...95G. Doi: 10.1209/0295-5075/6/2/001. ISSN 0295-5075. S2CID 250762344. Florigg, Martin. Summers, Stephen J. (1997). "On the statistical independence of the algebra of observables". Journal of Mathematical Physics. AIP Publishing. 38(3): 1318–1328. Bibcode: 1997JMP....38.1318F. Doi: 10.1063/1.531812. ISSN 0022-2488.

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### Definition & Meaning

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Noun

- a proposition deducible from basic postulates an idea accepted as a demonstrable truth