nLab quantum decoherence

Contents

Context

Quantum systems

Idea
Properties

Quantum channels and decoherence

Related concepts
References

Idea

In quantum physics decoherence refers to the disappearance of quantum entanglement and superposition in the limit where small quantum mechanical systems are coupled to large thermal baths?.

This has been argued to resolve (and has been argued not to resolve) the problem with the interpretation of quantum measurement.

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

a unitary quantum channel, induced by a unitary operator $U_{tot} \,\colon\, \mathscr{H} \otimes \mathscr{B} \to \mathscr{H} \otimes \mathscr{B}$
on a compound system with some $\mathscr{B}$ (the “bath”), yielding a total system Hilbert space $\mathscr{H} \otimes \mathscr{B}$ (tensor product),
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,
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.

Related concepts

References

Original discusssion identifying quantum decoherence as interaction with an averaged environment (“bath”):

Erich Joos, H. Dieter Zeh, The emergence of classical properties through interaction with the environment, Z. Physik B – Condensed Matter 59 (1985) 223–243 [doi:10.1007/BF01725541]
Erich Joos, www.decoherence.de
Wojciech Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75 (2003) 715-775 [quant-ph/0105127, doi:10.1103/RevModPhys.75.715]

With regards to the measurement problem and interpretations of quantum mechanics:

Roland Omnès, §7 of: The Interpretation of Quantum Mechanics, Princeton University Press (1994) [ISBN:9780691036694]
Maximilian Schlosshauer, Decoherence, the measurement problem, and interpretations of quantum mechanics, Rev. Mod. Phys. 76 (2004) 1267-1305 [arXiv;quant-ph/0312059, doi:10.1103/RevModPhys.76.1267]
Guido Bacciagaluppi, The Role of Decoherence in Quantum Mechanics, Stanf. Enc. of Phil. (2020) [web]

Textbook accounts:

Heinz-Peter Breuer, Francesco Petruccione, Decoherence, Chapter 4 in: The Theory of Open Quantum Systems, Oxford University Press (2007) [book:doi:10.1093/acprof:oso/9780199213900.001.0001, chapter:.003.04]
Maximilian Schlosshauer, Decoherence and the Quantum-To-Classical Transition, The Frontiers Collection, Springer (2007) [doi:10.1007/978-3-540-35775-9]

Further review:

Maximilian Schlosshauer, Quantum Decoherence, Phys. Rep. 831 (2019) 1-57 [arXiv:1911.06282, doi:10.1016/j.physrep.2019.10.001]