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quantum channels and decoherence -- section
Quantum channels and decoherence
Quantum channels and decoherence
The crux of dynamical quantum decoherence is that fundamentally the (time-)evolution of any quantum system ℋ \mathscr{H} may be assumed unitary (say via a Schrödinger equation ) when taking the whole evolution of its environment ℬ \mathscr{B} (the “bath”, ultimately the whole observable universe ) into account, too, in that the evolution of the total system ℋ ⊗ ℬ \mathscr{H} \otimes \mathscr{B} is given by a unitary operator
evolve : ℋ ⊗ ℬ ⟶ ℋ ⊗ ℬ | ψ , β ⟩ ↦ U tot | ψ , β ⟩ ,
\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 ρ : ℋ ⊗ ℋ * \rho \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast (density matrices ) of the given quantum system as coupled to any given mixed state env : ℬ ⊗ ℬ * env \,\colon\, \mathscr{B} \otimes \mathscr{B}^\ast of the bath (via tensor product )
couple : ℋ ⊗ ℋ * ⟶ ( ℋ ⊗ ℬ ) ⊗ ( ℋ ⊗ ℬ ) * ρ ↦ ρ ⊗ env ;
\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) average : ( ℋ ⊗ ℬ ) ⊗ ( ℋ ⊗ ℬ ) * ⟶ ℋ ⊗ ℋ * ρ ^ ↦ Tr ℬ ( ρ ^ ) .
\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 ℋ \mathscr{H} is always of the composite form
ℋ ⊗ ℋ * → couple to environment ( ℋ ⊗ ℬ ) ⊗ ( ℋ ⊗ ℬ ) * → total unitary evolution ( ℋ ⊗ ℬ ) ⊗ ( ℋ ⊗ ℬ ) * → average over environment ℋ ⊗ ℋ * ρ ↦ ρ ⊗ env ↦ U tot ⋅ ( ρ ⊗ env ) ⋅ U tot † ↦ Tr ℬ ( U tot ⋅ ( ρ ⊗ env ) ⋅ U tot † )
\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 : ℋ ⊗ ℋ * ⟶ ℋ ⊗ ℋ *
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 : ℋ ⊗ ℬ → ℋ ⊗ ℬ U_{tot} \,\colon\, \mathscr{H} \otimes \mathscr{B} \to \mathscr{H} \otimes \mathscr{B}
on a compound system with some ℬ \mathscr{B} (the “bath”), yielding a total system Hilbert space ℋ ⊗ ℬ \mathscr{H} \otimes \mathscr{B} (tensor product ),
and acting on the given mixed state ρ \rho coupled (tensored) with a fixed mixed state env : ℬ ⊗ ℬ * env \,\colon\, \mathscr{B} \otimes \mathscr{B}^\ast of the bath system,
followed by partial trace (averaging) over ℬ \mathscr{B} (leading to decoherence in the remaining state)
in that
(2) ch ( ρ ) = Tr ℬ ( U tot ⋅ ( ρ ⊗ env ) ⋅ U tot † ) .
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 .
Last revised on September 18, 2023 at 14:40:24.
See the history of this page for a list of all contributions to it.