In physics, the no-communication theorem (also referred to as the no-signaling principle) is a no-go theorem in quantum information theory. It asserts that during the measurement of an entangled quantum state, it is impossible for one observer to transmit information to another observer, regardless of their spatial separation. This conclusion preserves the principle of causality in quantum mechanics and ensures that information transfer does not violate special relativity by exceeding the speed of light.

The theorem is significant because quantum entanglement creates correlations between distant events that might initially appear to enable faster-than-light communication. The no-communication theorem demonstrates that the failure of local causality does not imply that "spooky action at a distance," a phrase originally coined by Einstein, can be used to communicate faster than light.

The no-communication theorem states that, within the context of quantum mechanics, it is not possible to transmit classical bits of information by means of carefully prepared mixed or pure states, whether entangled or not. The theorem is only a sufficient condition that states that if the Kraus matrices commute then there can be no communication through the quantum entangled states and that this is applicable to all communication. From a relativity and quantum field perspective, also faster than light or "instantaneous" communication is disallowed. Being only a sufficient condition, there can be other reasons communication is not allowed.

The basic premise entering into the theorem is that a quantum-mechanical system is prepared in an initial state with some entangled states, and that this initial state is describable as a mixed or pure state in a Hilbert space H. After a certain amount of time, the system is divided in two parts each of which contains some non-entangled states and half of the quantum entangled states, and the two parts become spatially distinct, A and B, sent to two distinct observers, Alice and Bob, who are free to perform quantum mechanical measurements on their portion of the total system (viz, A and B). The question is: is there any action that Alice can perform on A that would be detectable by Bob making an observation of B? The theorem replies 'no'.

An important assumption going into the theorem is that neither Alice nor Bob is allowed, in any way, to affect the preparation of the initial state. If Alice were allowed to take part in the preparation of the initial state, it would be trivially easy for her to encode a message into it; thus neither Alice nor Bob participates in the preparation of the initial state. The theorem does not require that the initial state be somehow 'random' or 'balanced' or 'uniform': indeed, a third party preparing the initial state could easily encode messages in it, received by Alice and Bob. Simply, the theorem states that, given some initial state, prepared in some way, there is no action that Alice can take that would be detectable by Bob.

The proof proceeds by defining how the total Hilbert space H can be split into two parts, H_A and H_B, describing the subspaces accessible to Alice and Bob. The total state of the system is described by a density matrix σ. The goal of the theorem is to prove that Bob cannot in any way distinguish the pre-measurement state σ from the post-measurement state P(σ). This is accomplished mathematically by comparing the trace of σ and the trace of P(σ), with the trace being taken over the subspace H_A. Since the trace is only over a subspace, it is technically called a partial trace. Key to this step is that the (partial) trace adequately summarizes the system from Bob's point of view. That is, everything that Bob has access to, or could ever have access to, measure, or detect, is completely described by a partial trace over H_A of the system σ. The fact that this trace never changes as Alice performs her measurements is the conclusion of the proof of the no-communication theorem.

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.

Alice and Bob perform measurements on system S whose underlying Hilbert space is $${\displaystyle H=H_{A}\otimes H_{B}.}$$