No-teleportation theorem

In quantum information theory, the no-teleportation theorem states that an arbitrary quantum state cannot be measured with complete accuracy. In essence, it states that the unit of quantum information, the qubit, cannot be exactly, precisely converted into classical information bits. The somewhat misleading name of this theorem does not imply that quantum teleportation is impossible; rather that teleportation is impossible by first converting quantum state into classical bits, and then moving the bits, and constructing a specific quantum state elsewhere.

In crude terms, the no-teleportation theorem stems from the Heisenberg uncertainty principle and the EPR paradox: although a qubit $|\psi\rangle$ can be imagined to be a specific direction on the Bloch sphere, that direction cannot be measured precisely, for the general case $|\psi\rangle$ ; for if it could, the results of that measurement would be describable with words, i.e. classical information.

The no-teleportation theorem is implied by the no-cloning theorem: if it were possible to convert a qubit into classical bits, then a qubit would be easy to copy (since classical bits are trivially copyable).

Formulation

The term quantum information refers to information stored in the state of a quantum system. Two quantum states ρ₁ and ρ₂ are identical if the measurement results of any physical observable have the same expectation value for ρ₁ and ρ₂. Thus measurement can be viewed as an information channel with quantum input and classical output, that is, performing measurement on a quantum system transforms quantum information into classical information. On the other hand, preparing a quantum state takes classical information to quantum information.

In general, a quantum state is described by a density matrix. Suppose one has a quantum system in some mixed state ρ. Prepare an ensemble, of the same system, as follows:

Perform a measurement on ρ.
According to the measurement outcome, prepare a system in some pre-specified state.

The no-teleportation theorem states that the result will be different from ρ, irrespective of how the preparation procedure is related to measurement outcome. A quantum state cannot be determined via a single measurement. In other words, if a quantum channel measurement is followed by preparation, it cannot be the identity channel. Once converted to classical information, quantum information cannot be recovered.

In contrast, perfect transmission is possible if one wishes to convert classical information to quantum information then back to classical information. For classical bits, this can be done by encoding them in orthogonal quantum states, which can always be distinguished.

No-teleportation theorem

Formulation

See also