Channel kets, entangled states, and the location of quantum information

The well-known duality relating entangled states and noisy quantum channels is expressed in terms of a channel ket, a pure state on a suitable tripartite system, which functions as a pre-probability allowing the calculation of statistical correlations between, for example, the entrance and exit of a channel, once a framework has been chosen so as to allow a consistent set of probabilities. In each framework the standard notions of ordinary (classical) information theory apply, and it makes sense to ask whether information of a particular sort about one system is or is not present in another system. Quantum effects arise when a single pre-probability is used to compute statistical correlations in different incompatible frameworks, and various constraints on the presence and absence of different kinds of information are expressed in a set of all-or-nothing theorems which generalize or give a precise meaning to the concept of “no-cloning.” These theorems are used to discuss the location of information in quantum channels modeled using a mixed-state environment, the classical-quantum channels introduced by Holevo, and the location of information in the physical carriers of a quantum code. It is proposed that both channel and entanglement problems be classified in terms of pure states (functioning as pre-probabilities) on systems of $p⩾2$ parts, with mixed bipartite entanglement and simple noisy channels belonging to the category $p=3$, a five-qubit code to the category $p=6$, etc., then by the dimensions of the Hilbert spaces of the component parts, along with other criteria yet to be determined.

Figure 1

Circuit illustrating map $M$ - ket $\mid \psi ⟩$ duality.

Figure 2

Quantum channel using a unitary transformation $T$ or isometry $V$.

Figure 3

Circuit for visualizing the channel ket $\mid \Psi ⟩$.

Figure 4

Channel and channel ket $\mid \Psi ⟩$ for a mixed-state environment.