No-communication theorem

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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').

[edit] Formulation

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We will illustrate this result for the setup of Bell tests in which two observers Alice and Bob perform local observations on a common bipartite system.

Theorem. In a Bell test, the statistics of Bob's measurements are unaffected by anything Alice does locally.

To prove this, we use the statistical machinery of quantum mechanics, namely density states and quantum operations. Alice and Bob perform measurements on system S whose underlying Hilbert space is

$H = H_A \otimes H_B.$

We also assume 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.

Suppose the combined system is prepared in state σ. Assume 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:

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

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

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

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

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

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

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

In conclusion, statistically, Bob cannot tell the difference between what Alice did and a random measurement (or whether she did anything at all).

[edit] 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.

[edit] References

Florig, M. and Summers, S. J. On the statistical independence of algebras of observables, J. Math. Phys. 38 (1997) 1318- 1328
Peres, A. and Terno, D. Quantum Information and Relativity Theory, Rev. Mod. Phys. 76, 93 (2004), arXiv quant-ph/0212023