Definitions

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

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 Ti and Si are operators on HA and HB 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

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

where Vk are called Kraus matrices which satisfy

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

The term

$I_\left\{H_B\right\}$

from the expression

$\left(V_k otimes I_\left\{H_B\right\}\right)$

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

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

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

One can directly calculate this state:

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

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

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

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

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

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

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

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