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 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).

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

The no-communication theorem says that whether or not Alice performs a measurement on a quantum system which Alice and Bob interact with, Bob will see no difference in the probability of outcome i of whatever observable operation he is performing (i = 1,2,3...). Thus if Alice has the choice of performing observables X or Y, then regardless, the conditional probabilities p(i|X) and p(i|Y) will be the same. This is the conclusion above: that instantaneous communication is impossible.

The trouble with this conclusion applies not only to attempts at communication, but to any experiment. So even if Alice and Bob collect data in coincidence (and hence not for communication purposes), each individual's data should show no indication of the other's actions. Several experiments may suggest otherwise, but it is impossible to experimentally verify that any transfer of information was actually instantaneous.

For example B. Dopfer, a graduate student of Anton Zeilinger, has indicated via experiment that it is possible to cause or prohibit an ensemble of photons into making an interference pattern on a screen, by remotely manipulating their entangled twins. Physicist John Cramer is currently attempting to replicate Zeilinger's experiment for the purpose of communication. (The first experiment, attributed to A. Zeilinger, was actually done by Zeilinger's graduate student B. Dopfer).

Of course Zeilinger and Dopfer's experiment does not prove superluminal communication, but neither does the no-communication prohibit all forms of communication. If superluminal communication is prohibited, it is not because of the no-communication theorem. Thus, the question of superluminal communication remains open.

--- ---

A communication system using quantum entanglement could be something like this:

Alice and Bob are measuring corresponding particles from pairs of quantum mechanically entangled photons - so-called Bell-couples. The distance between Alice and the source are shorter than between Bob and the source. Alice can change her measuring set-up by inserting a mirror or not. She keeps her choice for an agreed period – for instance 1/100.000 sec. By measuring his part of the pairs Bob should, with at least 99% probability, guess what Alice has chosen? Can he do this there is a practical basis for communication? This will for a growing distance between Alice and Bob create superluminal communication.

How the transmitter and the receiver should be build up is still a discussion, but of course Alice's measurement apparatus will interact with Bob's while they communicate, and the theorem has therefore a serious problem. UChr (talk) 17:37, 27 March 2011 (UTC)

References

• Hall, M.J.W. Imprecise measurements and non-locality in quantum mechanics, Phys. Lett. A (1987) 89-91
• Ghirardi, G.C. et al. Experiments of the EPR Type Involving CP-Violation Do not Allow Faster-than-Light Communication between Distant Observers, Europhys. Lett. 6 (1988) 95-100
• 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