nLab mixed unitary quantum channel

Idea

In quantum information theory, a mixed unitary quantum channel is a quantum channel

$$chan \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast \longrightarrow \mathscr{H} \otimes \mathscr{H}^\ast$$

which admits an operator-sum decomosition all whose Kraus operators are multiples of unitary operators

$$s \,\colon\, S \;\;\;\; \vdash \;\;\;\; U \,\colon\, \mathscr{H} \to \mathscr{H}$$

as

$$chan \;\colon\; \rho \;\mapsto\; \underset{s}{\sum} \, p_s \, U_s \cdot \rho \cdot U^\dagger \,,$$

where the coefficients

$$p_{(-)} \,\colon\, S \to [0,1]$$

necessarily form a probability distribution over the finite index set $S$.

Examples

Example

Of course, all unitary quantum channels are examples of mixed unitary quantum channels, this being the case (in particular) where the index set $S = \ast$ is the singleton set.

Example

A bit-flip quantum channel

$$\array{ QBit \otimes QBit^\ast &\xrightarrow{\phantom{-------}}& QBit \otimes QBit^\ast \\ \rho &\mapsto& (1-p)\,\rho + p \, X \cdot \rho \cdot X }$$

is a mixed unitary quantum channel, where the two unitary operators are the identity matrix on qbits and the $X$-Pauli gate-operator, respectively.

In fact:

Example

Every unital quantum channel on a single qbit is a mixed unitary quantum channel.

This is attributed by Müller-Hermes & Perry 2019 (inside the proof of Cor. 1.4) to Ruskai, Szarek & Werner 2002, where it may be gleaned from combining Thm. 14 (every channel on qbits is the convex combination of two channels in the closure of extremal channels) there and the remark on p. 163 (that the extremal unital channels are the unitary channels).

References

See most references on quantum channels.

Discussion of mixed unitary quantum channels specifically on single qbits (cf. DQC1):

• Mary Beth Ruskai, Stanislaw Szarek, Elizabeth Werner, An analysis of completely-positive trace-preserving maps on $M_2$, Linear Algebra and its Applications 347 1–3 (2002) 159-187 [doi:10.1016/S0024-3795(01)00547-X]

• Alexander Müller-Hermes, Christopher Perry, All unital qubit channels are 4-noisy operations, Letters in Mathematical Physics 109 (2019) 1–9 [doi:10.1007/s11005-018-1104-x, arXiv:1802.01337]