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Quantum Channels

Abstract

The details of how the dynamics of quantum states are modelled.

Keywords:DynamicsChannelsEvolutionPositiveTrace Preserving.

Definition

A quantum channel is a linear, completely positive, trace-preserving map (from the set of density operators to itself).

A channel mapping from states in an input Hilbert space Hin\mathcal{H}_{in} to an output Hilbert space Hout\mathcal{H}_{out} is denoted as

E:Hin→Hout\mathcal{E}: \mathcal{H}_{in} \rightarrow \mathcal{H}_{out}
(1)

A quantum channel is a more general notion of evolution then that captured via the Schrödinger equation and unitary dynamics as it models interactions with the environment. Unitary dynamics are subset of quantum channels.

Linear

A map, E\mathcal{E}, is linear if

E(ρ+σ)=E(ρ)+E(σ)\mathcal{E}(\rho + \sigma) = \mathcal{E}(\rho) + \mathcal{E}(\sigma)
(2)

Positive

A linear map, E\mathcal{E}, is positive if it maps positive elements to positive elements,

ρ≥0  ⟹  E(ρ)≥0.\rho \geq 0 \implies \mathcal{E}(\rho) \geq 0.
(3)

Completely Positive

A linear map, E\mathcal{E}, is completely positive if it remains a positive map when embedded into a high dimensional space,

ρ≥0  ⟹  (E⊗I)(ρ)≥0,\rho \geq 0 \implies (\mathcal{E} \otimes \mathcal{I}) (\rho) \geq 0,
(4)

where I\mathcal{I} is the identity channel on an arbitrarily large ancilla.

This additional restriction is needed to ensure that acting that map on part of a larger, potentially entangled state, still outputs a physical quantum state.

This ensure that all operators output from a channel remain positive semi-definite.

Trace-preserving

A map, E\mathcal{E}, is trace preserving if

tr[E(ρ)]=1,\textrm{tr} \big[ \mathcal{E}(\rho) \big] = 1,
(5)

for all ρ∈Hin\rho \in \mathcal{H}_{in}, where ρ\rho is a valid density operator, tr[ρ]=1,ρ≥0 \textrm{tr} \big[ \rho \big] = 1, \rho \geq 0.

This ensure that all operators output from a channel still have trace one.

Channel Descriptions

There are multiple different ways to represent a quantum channel. Each different channel description can be used whenever useful.

Stinespring Dilation
Kraus Decomposition
Choi-Jamiolkowski isomorphism

Let E:HS→HS′\mathcal{E}: \mathcal{H}_{S} \rightarrow \mathcal{H}_{S'} be a quantum channel and ρ ∈ HS\rho~\in~\mathcal{H}_{S}.

There exists a state, τE ∈ HE\tau_{E}~\in~\mathcal{H}_{E}, and isometry, USEU_{SE}, such that

E(ρ)=trE[USE(ρS⊗τE)USE†],\mathcal{E}(\rho) = \textrm{tr}_{E} \big[ U_{SE}(\rho_{S} \otimes \tau_{E}) U_{SE}^{\dagger} \big],
(6)

where the subscript SS and EE mean system and environment respectively.

The Stinespring dilation says that all channels can be consider unitary with respect to a higher dimensional space. That is, all channels are the interaction of ρ\rho and some environment state, τ\tau under a global unitary.

Properties

  • dim τE≤d2\textrm{dim} ~ \tau_{E} \leq d^{2} where dim ρ=d\textrm{dim} ~ \rho = d
  • Tracing out the system instead of the enviroment gives the complementary, Ec\mathcal{E}^{c}, which can be thought as the channel from the perspective of the environment,
Ec(ρ)=trS[USE(ρS⊗τE)USE†].\mathcal{E}^{c}(\rho) = \textrm{tr}_{S} \big[ U_{SE}(\rho_{S} \otimes \tau_{E}) U_{SE}^{\dagger} \big].
(7)

Useful Channels

Dephasing Channel
Depolarising Channel

Qubits

Let ρ∈H2 \rho \in \mathcal{H}^2. The dephasing channel acting on ρ \rho is given by

Dpph(ρ)=pρ+(1−p)∑n∣n⟩⟨n∣ρ∣n⟩⟨n∣.\mathcal{D}^{\textrm{ph}}_p(\rho) = p \rho + (1-p) \sum_{n} \ket{n}\bra{n} \rho \ket{n}\bra{n}.
(13)

With respect to the standard basis this becomes

Dpph(ρ)=pρ+(1−p)(∣0⟩⟨0∣ρ∣0⟩⟨0∣+∣1⟩⟨1∣ρ∣1⟩⟨1∣)\mathcal{D}^{\textrm{ph}}_p(\rho) = p \rho + (1-p) \bigg( \ket{0}\bra{0} \rho \ket{0}\bra{0} + \ket{1}\bra{1} \rho \ket{1}\bra{1} \bigg)
(14)
  1. The Kraus operators are
K0=1−pI,  K1=p∣0⟩⟨0∣,  K2=p∣1⟩⟨1∣.\begin{align*} K_0 &= \sqrt{ 1 - p} \mathbb{I}, ~ ~ K_1 = \sqrt{ p } \ket{0}\bra{0}, ~ ~ K_2 = \sqrt{ p } \ket{1}\bra{1}. \end{align*}
(15)
  1. The Choi-state is
Jph=12(1+p)∣Φ00+⟩⟨Φ00+∣+12(1−p)∣Φ01+⟩⟨Φ01+∣,\mathcal{J}^{\textrm{ph}} = \frac{1}{2}(1+p) \ket{\Phi^+_{00}} \bra{\Phi^+_{00}} + \frac{1}{2}(1-p) \ket{\Phi^+_{01}} \bra{\Phi^+_{01}},
(16)

where

∣Φ00+⟩=∣0⟩∣0⟩+∣1⟩∣1⟩2,∣Φ01+⟩=∣0⟩∣0⟩−∣1⟩∣1⟩2.\begin{align*} \ket{\Phi_{00}^{+}} &= \frac{\ket{0}\ket{0}+\ket{1} \ket{1}}{\sqrt{2}}, \\ \ket{\Phi_{01}^{+}} &= \frac{\ket{0}\ket{0}-\ket{1} \ket{1}}{\sqrt{2}}. \end{align*}
(17)

d-dimensional

See the appendix of Stratton et al. (2024) for the Choi-state of the d-dimensional dephasing channel.

References
  1. Jamiołkowski, A. (1972). Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3(4), 275–278. 10.1016/0034-4877(72)90011-0
  2. Choi, M.-D. (1975). Completely positive linear maps on complex matrices. Linear Algebra and Its Applications, 10(3), 285–290. 10.1016/0024-3795(75)90075-0
  3. Stratton, B., Hsieh, C.-Y., & Skrzypczyk, P. (2024). Operational Interpretation of the Choi Rank Through k-State Exclusion. 10.48550/ARXIV.2406.08360
Quantum Formalism
Density Operators
Quantum Formalism
Generalised Measurements