This conclusion preserves the principle of causality in quantum mechanics and ensures that information transfer does not violate special relativity by exceeding the speed of light.
The theorem is significant because quantum entanglement creates correlations between distant events that might initially appear to enable faster-than-light communication.
The no-communication theorem establishes conditions under which such transmission is impossible, thus resolving paradoxes like the Einstein-Podolsky-Rosen (EPR) paradox and addressing the violations of local realism observed in Bell's theorem.
Specifically, it demonstrates that the failure of local realism does not imply the existence of "spooky action at a distance," a phrase originally coined by Einstein.
From a relativity and quantum field perspective, also faster than light or "instantaneous" communication is disallowed.
The question is: is there any action that Alice can perform on A that would be detectable by Bob making an observation of B?
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 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 described by a density matrix σ.
Key to this step is that the (partial) trace adequately summarizes the system from Bob's point of view.
The fact that this trace never changes as Alice performs her measurements is the conclusion of the proof of the no-communication theorem.
[1]: 100 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]: 100 [2][3]: 96 Alice and Bob perform measurements on system S whose underlying Hilbert space is
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.
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.
In general, this is described by a quantum operation, on the system state, of the following kind
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.
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).
In 1978, Phillippe H. Eberhard's paper, Bell's Theorem and the Different Concepts of Locality, rigorously demonstrated the impossibility of faster-than-light communication through quantum systems.
[5] Eberhard introduced several mathematical concepts of locality and showed how quantum mechanics contradicts most of them while preserving causality.
Further, in 1988, the paper Quantum Field Theory Cannot Provide Faster-Than-Light Communication by Eberhard and Ronald R. Ross analyzed how relativistic quantum field theory inherently forbids faster-than-light communication.
[6] This work elaborates on how misinterpretations of quantum field properties had led to claims of superluminal communication and pinpoints the mathematical principles that prevent it.
[7][8] In 2008 Matthew Hastings proved a counterexample where the minimum output entropy is not additive for all quantum channels.
In August 24th 2015, a team led by physicist Ronald Hanson from Delft University of Technology in the Netherlands uploaded their latest paper to the preprint website arXiv, reporting the first Bell experiment that simultaneously addressed both the detection loophole and the communication loophole.
The research team used a clever technique known as "entanglement swapping," which combines the benefits of photons and matter particles.
The final measurements showed coherence between the two electrons that exceeded the Bell limit, once again supporting the standard view of quantum mechanics and rejecting Einstein's hidden variable theory.