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-comunication theorem shows that failure of local realism does not lead to what could be referred to as "spooky communication at a distance".

Formulation

We will first show this result for the setup of Bell tests in which two observers Alice and Bob perform sequences of observations on a common system consisting of a pair of (possibly entangled) electrons.

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

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. The density operator σ is a sum of separable density states:

\sigma = \sum_i T_i \otimes S_i

Alice makes measurements which are given 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 Vk are so-called Kraus matrices which satisfy

\sum_k V_k^* V_k = I_{H_A}

Bob observes the state σ. Let us assume for purposes of argument a non-relativistic situation, if Bob looks at the state immediately (with no time delay) after Alice performs her measurement, he will see the relative state given by the partial trace:

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

In the above sequence trH denotes the relative trace mapping. In conclusion, statistically, Bob cannot tell the difference what Alice did (or whether she did anything at all).

Notice that once time evolution operates on the density state, then the above calculation 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.

References

See also: No-communication theorem, Alice and Bob, Bell's theorem, Density state, EPR paradox, Local realism, Quantum information theory, Quantum mechanics, Quantum operation, Schrödinger equation