nLab unitary quantum channel

Idea

In quantum probability theory/open quantum systems, a unitary quantum channel is a quantum channel whose restriction to pure states acts by a unitary transformation just as a loss-less quantum gate does.

Concretely, in terms of Kraus decomposition, a quantum channel

$$\array{ \mathscr{H}_1 \otimes \mathscr{H}_1^\ast & \longrightarrow & \mathscr{H}_2 \otimes \mathscr{H}_2^\ast \\ \rho &\mapsto& ch(\rho) }$$

is unitary iff there exists a unitary operator $U \,\colon\, \mathscr{H}_1 \longrightarrow \mathscr{H}_2$ such that $ch$ is given by conjugation with this operator:

$$ch(\rho) \;=\; U \cdot \rho \cdot U^\dagger \,.$$

Properties

Quantum channels and decoherence

The crux of dynamical quantum decoherence is that fundamentally the (time-)evolution of any quantum system $\mathscr{H}$ may be assumed unitary (say via a Schrödinger equation) when taking the whole evolution of its environment $\mathscr{B}$ (the “bath”, ultimately the whole observable universe) into account, too, in that the evolution of the total system $\mathscr{H} \otimes \mathscr{B}$ is given by a unitary operator

$$\array{ \mathllap{ evolve \;\colon\; } \mathscr{H} \otimes \mathscr{B} &\longrightarrow& \mathscr{H} \otimes \mathscr{B} \\ \left\vert \psi, \beta \right\rangle &\mapsto& U_{tot} \left\vert \psi, \beta \right\rangle \mathrlap{\,,} }$$

after understanding the mixed states $\rho \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast$ (density matrices) of the given quantum system as coupled to any given mixed state $env \,\colon\, \mathscr{B} \otimes \mathscr{B}^\ast$ of the bath (via tensor product)

$$\array{ \mathllap{ couple \;\colon\; } \mathscr{H} \otimes \mathscr{H}^\ast & \longrightarrow & (\mathscr{H} \otimes \mathscr{B}) \otimes (\mathscr{H} \otimes \mathscr{B})^\ast \\ \rho &\mapsto& \rho \otimes env \mathrlap{\,;} }$$

…the only catch being that one cannot — and in any case does not (want or need to) — keep track of the precise quantum state of the environment/bath, instead only of its average effect on the given quantum system, which by the rule of quantum probability is the mixed state that remains after the partial trace over the environment:

(1)$$\array{ \mathllap{ average \;\colon\; } (\mathscr{H} \otimes \mathscr{B}) \otimes (\mathscr{H} \otimes \mathscr{B})^\ast &\longrightarrow& \mathscr{H} \otimes \mathscr{H}^\ast \\ \widehat{\rho} &\mapsto& Tr_{\mathscr{B}}\big(\widehat{\rho}\big) \mathrlap{\,.} }$$

In summary this means for practical purposes that the probabilistic evolution of quantum systems $\mathscr{H}$ is always of the composite form

$$\array{ \mathscr{H} \otimes \mathscr{H}^\ast & \xrightarrow{ \array{ \text{couple to} \\ \text{environment} } } & \left( \array{ \mathscr{H} \\ \otimes \\ \mathscr{B} } \right) \otimes \left( \array{ \mathscr{H} \\ \otimes \\ \mathscr{B} } \right)^\ast & \xrightarrow{ \array{ \text{total unitary} \\ \text{evolution} } } & \left( \array{ \mathscr{H} \\ \otimes \\ \mathscr{B} } \right) \otimes \left( \array{ \mathscr{H} \\ \otimes \\ \mathscr{B} } \right)^\ast & \xrightarrow{ \array{ \text{average over} \\ \text{environment} } } & \;\;\;\; \mathscr{H} \otimes \mathscr{H}^\ast \\ \rho &\mapsto& \rho \otimes env &\mapsto& \mathclap{ U_{tot} \cdot (\rho \otimes env) \cdot U_{tot}^\dagger } &\mapsto& \;\;\;\;\;\;\;\;\;\;\;\; \mathclap{ Tr_{\mathscr{B}} \big( U_{tot} \cdot (\rho \otimes env) \cdot U_{tot}^\dagger \big) } }$$

This composite turns out to be a “quantum channel” and in fact all quantum channels arise this way:

Proposition

(quantum channels and decoherence)

Every quantum channel

$$ch \;\;\colon\;\; \mathscr{H} \otimes \mathscr{H}^\ast \longrightarrow \mathscr{H} \otimes \mathscr{H}^\ast$$

may be written as

1. a unitary quantum channel, induced by a unitary operator $U_{tot} \,\colon\, \mathscr{H} \otimes \mathscr{B} \to \mathscr{H} \otimes \mathscr{B}$

2. on a compound system with some $\mathscr{B}$ (the “bath”), yielding a total system Hilbert space $\mathscr{H} \otimes \mathscr{B}$ (tensor product),

3. and acting on the given mixed state $\rho$ coupled (tensored) with a fixed mixed state $env \,\colon\, \mathscr{B} \otimes \mathscr{B}^\ast$ of the bath system,

4. followed by partial trace (averaging) over $\mathscr{B}$ (leading to decoherence in the remaining state)

in that

(2)$$ch(\rho) \;\;=\;\; Tr_{\mathscr{B}} \big( U_{tot} \cdot (\rho \otimes env) \cdot U_{tot}^\dagger \big) \,.$$

Conversely, every operation of the form (2) is a quantum channel.

For exposition: Nielsen & Chuang 2000 §8.2.2-8.2.3

Detailed proof, including the infinite-dimensional case: Attal, Thm. 6.5 & 6.7.

References

See the references at quantum channel, for instance:

• Stéphane Attal, Quantum Channels, Lecture 6 in: Lectures on Quantum Noises [pdf, pdf, webpage]