No-communication theorem

In quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous 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 "spooky action at a distance").

Formulation

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]

Alice and Bob perform measurements on system S whose underlying Hilbert space is

$H = H_A \otimes H_B.$

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:

$\sigma = \sum_i T_i \otimes S_i$

where T_i and S_i are operators on H_A and H_B which however need not be states on the subsystems (that is non-negative of trace 1). In fact, the claim 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

$P(\sigma) = \sum_k (V_k \otimes I_{H_B})^* \ \sigma \ (V_k \otimes I_{H_B}),$

where V_k are called Kraus matrices which satisfy

$\sum_k V_k V_k^* = I_{H_A}.$

The term

$I_{H_B}$

from the expression

$(V_k \otimes I_{H_B})$

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

$\operatorname{tr}_{H_A}(P(\sigma))$

where $\operatorname{tr}_{H_A}$ is the partial trace mapping with respect to Alice's system.

One can directly calculate this state:

$\operatorname{tr}_{H_A}(P(\sigma)) = \operatorname{tr}_{H_A} \left(\sum_k (V_k \otimes I_{H_B})^* \sigma (V_k \otimes I_{H_B} )\right)$

$= \operatorname{tr}_{H_A} \left(\sum_k \sum_i V_k^* T_i V_k \otimes S_i \right)$

$= \sum_i \sum_k \operatorname{tr}(V_k^* T_i V_k) S_i$

$= \sum_i \sum_k \operatorname{tr}(T_i V_k V_k^*) S_i$

$= \sum_i \operatorname{tr}\left(T_i \sum_k V_k V_k^*\right) S_i$

$= \sum_i \operatorname{tr}(T_i) S_i$

$= \operatorname{tr}_{H_A}(\sigma).$

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).

Some comments

• Notice that once time evolution operates on the density state, then the calculation in the proof fails. In the case of the (non-relativistic) Schrödinger equation which has infinite propagation speed, then of course the above analysis will fail for positive times. Clearly, the importance of the no-communication theorem for positive times is for relativistic systems.

• The no-communication theorem thus says shared entanglement alone can not be used to transmit quantum information. Compare this with the no teleportation theorem, which states a classical information channel can not transmit quantum information. (By transmit, we mean transmission with full fidelity.) However, quantum teleportation schemes utilize both resources to achieve what is impossible for either alone.

Opposing viewpoint

Some authors have argued that most of the proofs of the no-communication theorem are actual circular.^[2][3] In their view, a no-signalling condition is built into the assumptions of the bipartite Hilbert space (the tensor product of the two individual Hilbert spaces) and the locally restricted operators. Therefore, proofs like the one above do not forbid superluminal communication, but show that the formalism of quantum mechanics is consistent in that no superluminal causal interactions appear when the base assumptions do not include them.

Others have questioned if the no-communication theorem holds for signalling methods using ensembles of entangled particle pairs. As the no-communication theorem is a mathematical derivation on the Hilbert space of a single particle, its implications are not as clear for an ensemble of particles; where one is not attempting to transmit a single bit through a single particle, but instead a single bit through many particles (partial information through each particle).^[4][5][6] In this case the binary basis state would be over the state of the ensemble, not a property of the Hilbert state of any particular particle. Thus only a measurement on the ensemble as a whole would resolve a bit. However, typically these kind of quantum eraser experiments also require a subluminal classical channel for coincidence detection. Physicist John G. Cramer at the University of Washington is attempting to replicate one of these experiments and demonstrate whether or not it can produce superluminal communication.^[7][8]

References

1. ^ Peres, A. and Terno, D. (2004). quant-ph/0212023 "Quantum Information and Relativity Theory". Rev. Mod. Phys. 76: 93–123. http://arxiv.org/abs/quant-ph/0212023 quant-ph/0212023. 
2. ^ J.B. Kennedy (1995). "On the empirical foundations of the quantum no-signalling proofs". Philosophy of Science 62: 543–560. 
3. ^ Peacock, K.A.; Hepburn, B. (1999). "Begging the Signaling Question: Quantum Signaling and the Dynamics of Multiparticle Systems". Proceedings of the Meeting of the Society of Exact Philosophy. 
4. ^ Millis, M.G.; Davis, E.W., eds (2009). Frontiers of Propulsion Science. Progress in astronautics and aeronautics. American Institute of Aeronautics and Astronautics. pp. 509–530. 
5. ^ Dopfer, Birgit (1998). PhD Thesis. Univ. Innsbruck. 
6. ^ Zeilinger, Anton (1999). "Experiment and the foundations of quantum physics". Rev. Mod. Physics 71: 288–297. 
7. ^ Paulson, Tom (14 November 2006). "Going for a blast into the real past". Seattle Post-Intelligencer. http://www.seattlepi.com/default/article/Going-for-a-blast-into-the-real-past-1219821.php. Retrieved 11 July 2011. 
8. ^ Barry, Patrick (2006). "What's done is done… or is it?". New Scientist 191 (2571): 36–39. 

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