Abstract¶ The details of how the dynamics of quantum states are modelled.
Keywords: Dynamics Channels Evolution Positive Trace 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 H i n \mathcal{H}_{in} H in H o u t \mathcal{H}_{out} H o u t
E : H i n → H o u t \mathcal{E}: \mathcal{H}_{in} \rightarrow \mathcal{H}_{out} E : H in → H o u t 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} E
E ( ρ + σ ) = E ( ρ ) + E ( σ ) \mathcal{E}(\rho + \sigma) = \mathcal{E}(\rho) + \mathcal{E}(\sigma) E ( ρ + σ ) = E ( ρ ) + E ( σ ) Positive ¶ A linear map, E \mathcal{E} E
ρ ≥ 0 ⟹ E ( ρ ) ≥ 0. \rho \geq 0 \implies \mathcal{E}(\rho) \geq 0. ρ ≥ 0 ⟹ E ( ρ ) ≥ 0. Completely Positive ¶ A linear map, E \mathcal{E} E
ρ ≥ 0 ⟹ ( E ⊗ I ) ( ρ ) ≥ 0 , \rho \geq 0 \implies (\mathcal{E} \otimes \mathcal{I}) (\rho) \geq 0, ρ ≥ 0 ⟹ ( E ⊗ I ) ( ρ ) ≥ 0 , where I \mathcal{I} I
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} E
tr [ E ( ρ ) ] = 1 , \textrm{tr} \big[ \mathcal{E}(\rho) \big] = 1, tr [ E ( ρ ) ] = 1 , for all ρ ∈ H i n \rho \in \mathcal{H}_{in} ρ ∈ H in ρ \rho ρ tr [ ρ ] = 1 , ρ ≥ 0 \textrm{tr} \big[ \rho \big] = 1, \rho \geq 0 tr [ ρ ] = 1 , ρ ≥ 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.
Let E : H S → H S ′ \mathcal{E}: \mathcal{H}_{S} \rightarrow \mathcal{H}_{S'} E : H S → H S ′ ρ ∈ H S \rho~\in~\mathcal{H}_{S} ρ ∈ H S
There exists a state, τ E ∈ H E \tau_{E}~\in~\mathcal{H}_{E} τ E ∈ H E U S E U_{SE} U SE
E ( ρ ) = tr E [ U S E ( ρ S ⊗ τ E ) U S E † ] , \mathcal{E}(\rho) = \textrm{tr}_{E} \big[ U_{SE}(\rho_{S} \otimes \tau_{E}) U_{SE}^{\dagger} \big], E ( ρ ) = tr E [ U SE ( ρ S ⊗ τ E ) U SE † ] , where the subscript S S S E E E
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 ρ τ \tau τ
Properties
dim τ E ≤ d 2 \textrm{dim} ~ \tau_{E} \leq d^{2} dim τ E ≤ d 2 dim ρ = d \textrm{dim} ~ \rho = d dim ρ = d
Tracing out the system instead of the enviroment gives the complementary, E c \mathcal{E}^{c} E c
E c ( ρ ) = tr S [ U S E ( ρ S ⊗ τ E ) U S E † ] . \mathcal{E}^{c}(\rho) = \textrm{tr}_{S} \big[ U_{SE}(\rho_{S} \otimes \tau_{E}) U_{SE}^{\dagger} \big]. E c ( ρ ) = tr S [ U SE ( ρ S ⊗ τ E ) U SE † ] . Let E : H S → H S ′ \mathcal{E}: \mathcal{H}_{S} \rightarrow \mathcal{H}_{S'} E : H S → H S ′ ρ ∈ H S \rho~\in~\mathcal{H}_{S} ρ ∈ H S
There exists a set of M M M { K i } i = 1 M \{ K_{i} \}_{i=1}^{M} { K i } i = 1 M
E ( ρ ) = ∑ i = 1 M K i ρ K i † , \mathcal{E}(\rho) = \sum_{i=1}^{M} K_{i} \rho K_{i}^{\dagger}, E ( ρ ) = i = 1 ∑ M K i ρ K i † , where
∑ i = 1 M K i † K i = I . \sum_{i=1}^{M} K_{i}^{\dagger}K_{i} = \mathbb{I}. i = 1 ∑ M K i † K i = I . This condition ensures that the quantum channel is trace preserving.
The Kraus decomposition allows one to apply channels without having to consider the environment.
Properties
The Kraus decomposition is not unique. Although, if there are two sets of operators that both describe the same channel, those sets of operators are unitarily equivalent.
M ≤ dim H S × dim H S ′ M \leq \dim \mathcal{H}_S \times \dim \mathcal{H}_{S'} M ≤ dim H S × dim H S ′
Let E : H S → H S ′ \mathcal{E}: \mathcal{H}_{S} \rightarrow \mathcal{H}_{S'} E : H S → H S ′ ρ ∈ H S ~\rho~\in~\mathcal{H}_{S} ρ ∈ H S ∣ Φ ⟩ S S ∈ H S ⊗ H S \ket{\Phi}_{SS}~\in~\mathcal{H}_{S} \otimes \mathcal{H}_{S} ∣ Φ ⟩ SS ∈ H S ⊗ H S Schmit rank state with dim H S ≈ dim H S ′ \textrm{dim} ~ \mathcal{H}_{S} \approx \textrm{dim} ~ \mathcal{H}_{S'} dim H S ≈ dim H S ′
The Choi-Jamiolkowski isomorphism (Jamiołkowski (1972) Choi (1975)
J S ′ S E = ( N S ⊗ I S ) ( ∣ Φ ⟩ ⟨ Φ ∣ S S ) ∈ H S ′ ⊗ H S , \mathcal{J}^{\mathcal{E}}_{\rm S'S} = (\mathcal{N}_{\rm S} \otimes \mathcal{I}_{\rm S}) \big( \ket{\Phi}\bra{\Phi}_{\rm SS} \big)~\in~\mathcal{H}_{S'} \otimes \mathcal{H}_{S}, J S ′ S E = ( N S ⊗ I S ) ( ∣ Φ ⟩ ⟨ Φ ∣ SS ) ∈ H S ′ ⊗ H S , where I S \mathcal{I}_{\rm S} I S H S \mathcal{H}_{S} H S
Typically, the full Schmit rank state state is taken to be a maximally entangled state in a fixed orthonormal basis,
∣ Φ ⟩ A B = 1 d ∑ i = 0 d − 1 ∣ i ⟩ A ∣ i ⟩ B . \ket{\Phi}_{AB} = \frac{1}{\sqrt{d}} \sum_{i=0}^{d-1} \ket{i}_{A} \ket{i}_{B}. ∣ Φ ⟩ A B = d 1 i = 0 ∑ d − 1 ∣ i ⟩ A ∣ i ⟩ B . From the Choi-state, the action of E \mathcal{E} E ρ \rho ρ
E ( ρ ) = d tr S [ ( I S ⊗ ρ t ) J S ′ S N ] , \mathcal{E}(\rho) = d ~ \textrm{tr}_{\rm S}\left[ \left(\mathbb{I}_{\rm S} \otimes \rho^{t}\right) \mathcal{J}^{\mathcal{N}}_{\rm S'S} \right], E ( ρ ) = d tr S [ ( I S ⊗ ρ t ) J S ′ S N ] , where ( ⋅ ) t (\cdot)^{t} ( ⋅ ) t I S \mathbb{I}_{\rm S} I S H S \mathcal{H}_{S} H S
Useful Channels ¶
Qubits
Let ρ ∈ H 2 \rho \in \mathcal{H}^2 ρ ∈ H 2 ρ \rho ρ
D p ph ( ρ ) = p ρ + ( 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}. D p ph ( ρ ) = pρ + ( 1 − p ) n ∑ ∣ n ⟩ ⟨ n ∣ ρ ∣ n ⟩ ⟨ n ∣ . With respect to the standard basis this becomes
D p ph ( ρ ) = p ρ + ( 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) D p ph ( ρ ) = pρ + ( 1 − p ) ( ∣ 0 ⟩ ⟨ 0 ∣ ρ ∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣ ρ ∣ 1 ⟩ ⟨ 1 ∣ ) The Kraus operators are
K 0 = 1 − p I , K 1 = p ∣ 0 ⟩ ⟨ 0 ∣ , K 2 = 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*} K 0 = 1 − p I , K 1 = p ∣ 0 ⟩ ⟨ 0 ∣ , K 2 = p ∣ 1 ⟩ ⟨ 1 ∣ . The Choi-state is
J ph = 1 2 ( 1 + p ) ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ + 1 2 ( 1 − p ) ∣ Φ 01 + ⟩ ⟨ Φ 01 + ∣ , \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}}, J ph = 2 1 ( 1 + p ) ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ + 2 1 ( 1 − p ) ∣ Φ 01 + ⟩ ⟨ Φ 01 + ∣ , where
∣ Φ 00 + ⟩ = ∣ 0 ⟩ ∣ 0 ⟩ + ∣ 1 ⟩ ∣ 1 ⟩ 2 , ∣ Φ 01 + ⟩ = ∣ 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*} ∣ Φ 00 + ⟩ ∣ Φ 01 + ⟩ = 2 ∣ 0 ⟩ ∣ 0 ⟩ + ∣ 1 ⟩ ∣ 1 ⟩ , = 2 ∣ 0 ⟩ ∣ 0 ⟩ − ∣ 1 ⟩ ∣ 1 ⟩ . d-dimensional
The d-dimensional dephasing channel has Choi-state:
J ph = α ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ + ( 1 − α ) d − 1 ∑ c = 1 d − 1 ∣ Φ 0 c + ⟩ ⟨ Φ 0 c + ∣ , \mathcal{J}^{\textrm{ph}} = \alpha \vert \Phi_{00}^{+} \rangle \langle \Phi_{00}^{+} \vert + \frac{(1-\alpha)}{d-1} \sum_{c=1}^{d-1} \vert \Phi_{0c}^{+} \rangle \langle \Phi_{0c}^{+} \vert, J ph = α ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ + d − 1 ( 1 − α ) c = 1 ∑ d − 1 ∣ Φ 0 c + ⟩ ⟨ Φ 0 c + ∣ , where α = ( p + 1 ) / d \alpha=(p+1)/d α = ( p + 1 ) / d ∣ Φ a b + ⟩ = ( I ⊗ W a b ) ∣ Φ 00 + ⟩ \ket{\Phi_{ab}^+} = (\mathbb{I} \otimes W_{ab}) \ket{\Phi_{00}^+} ∣ Φ ab + ⟩ = ( I ⊗ W ab ) ∣ Φ 00 + ⟩ W a b W_{ab} W ab Heisenberg-Weyl operators
Using the definition of the Choi-state, it can be seen that the Choi-state of a dephasing channel, D p p h ( ⋅ ) \mathcal{D}^{\rm ph}_{p}(\cdot) D p ph ( ⋅ )
J ph ≔ ( D p p h ( ⋅ ) ⊗ I ) ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ = p ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ + ( 1 − p ) d ∑ n = 0 d − 1 ∣ n n ⟩ ⟨ n n ∣ = 1 d ∑ n = 0 d − 1 ∣ n n ⟩ ⟨ n n ∣ + p d ∑ n ≠ m ∣ n n ⟩ ⟨ m m ∣ . \begin{align*}
\mathcal{J}^{\textrm{ph}} &\coloneqq (\mathcal{D}^{\rm ph}_{p}(\cdot) \otimes \mathcal{I}) \vert \Phi_{00}^{+} \rangle \langle \Phi_{00}^{+} \vert \\
&= p \vert \Phi_{00}^{+} \rangle \langle \Phi_{00}^{+} \vert + \frac{(1-p)}{d} \sum_{n=0}^{d-1} \ket{nn}\bra{nn} \\
&= \frac{1}{d} \sum_{n=0}^{d-1} \ket{nn}\bra{nn} + \frac{p}{d} \sum_{n \neq m} \ket{nn}\bra{mm}.
\end{align*} J ph : = ( D p ph ( ⋅ ) ⊗ I ) ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ = p ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ + d ( 1 − p ) n = 0 ∑ d − 1 ∣ nn ⟩ ⟨ nn ∣ = d 1 n = 0 ∑ d − 1 ∣ nn ⟩ ⟨ nn ∣ + d p n = m ∑ ∣ nn ⟩ ⟨ mm ∣ . By comparison of matrix elements, one can see that this can be rewritten as
J ph = α ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ + ( 1 − α ) d − 1 ∑ c = 1 d − 1 ∣ Φ 0 c + ⟩ ⟨ Φ 0 c + ∣ , \begin{align*}
\mathcal{J}^{\textrm{ph}} = \alpha \ket{\Phi_{00}^+}\bra{\Phi_{00}^{+}} + \frac{(1-\alpha)}{d-1} \sum_{c=1}^{d-1} \ket{\Phi_{0c}^+}\bra{\Phi_{0c}^{+}},
\end{align*} J ph = α ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ + d − 1 ( 1 − α ) c = 1 ∑ d − 1 ∣ Φ 0 c + ⟩ ⟨ Φ 0 c + ∣ , such that α = ( p + 1 ) / d \alpha = (p+1)/d α = ( p + 1 ) / d
∑ c = 1 d − 1 ∣ Φ 0 c + ⟩ ⟨ Φ 0 c + ∣ = ∑ c = 1 d − 1 ( I ⊗ W 0 , c ) ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ ( I ⊗ W 0 , c † ) = ∑ c = 1 d − 1 ( I A ⊗ ∑ n = 1 d − 1 Ω c n ∣ n ⟩ ⟨ n ∣ ) ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ ( I ⊗ ∑ m = 0 d − 1 Ω − c m ∣ m ⟩ ⟨ m ∣ ) = ∑ c = 1 d − 1 ∑ n , m = 0 d − 1 Ω c ( n − m ) ( I ⊗ ∣ n ⟩ ⟨ n ∣ ) ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ ( I ⊗ ∣ m ⟩ ⟨ m ∣ ) = d − 1 ∑ c = 1 d − 1 ∑ n , m = 0 d − 1 ∑ i , j = 0 d − 1 Ω c ( n − m ) ( I ⊗ ∣ n ⟩ ⟨ n ∣ ) ∣ i i ⟩ ⟨ j j ∣ ( I ⊗ ∣ m ⟩ ⟨ m ∣ ) = d − 1 ∑ n , m = 0 d − 1 [ ∑ c = 1 d − 1 Ω c ( n − m ) ] ∣ n n ⟩ ⟨ m m ∣ . \begin{align*}
&\sum_{c=1}^{d-1} \ket{\Phi_{0c}^{+}}\bra{\Phi_{0c}^{+}} \\
&= \sum_{c=1}^{d-1} \big( \mathbb{I} \otimes W_{0,c} \big) \ket{\Phi_{00}^+}\bra{\Phi_{00}^{+}} \big( \mathbb{I} \otimes W^\dagger_{0,c} \big) \\
&= \sum_{c=1}^{d-1} \big( \mathbb{I}_A \otimes \sum_{n=1}^{d-1} \Omega^{cn} \ket{n}\bra{n} \big) \ket{\Phi_{00}^{+}}\bra{\Phi_{00}^{+}} \big( \mathbb{I} \otimes \sum_{m=0}^{d-1} \Omega^{-cm} \ket{m}\bra{m} \big) \\
&= \sum_{c=1}^{d-1} \sum_{n,m=0}^{d-1} \Omega^{c(n-m)} \big( \mathbb{I} \otimes \ket{n}\bra{n} \big) \ket{\Phi_{00}^{+}}\bra{\Phi_{00}^{+}} \big( \mathbb{I} \otimes \ket{m}\bra{m} \big) \\
&= d^{-1} \sum_{c=1}^{d-1} \sum_{n,m=0}^{d-1} \sum_{i,j=0}^{d-1} \Omega^{c(n-m)} \big( \mathbb{I} \otimes \ket{n}\bra{n} \big) \ket{ii}\bra{jj} \big( \mathbb{I} \otimes \ket{m}\bra{m} \big) \\
&=d^{-1} \sum_{n,m=0}^{d-1} \bigg[ \sum_{c=1}^{d-1} \Omega^{c(n-m)} \bigg] \ket{nn}\bra{mm}.
\end{align*} c = 1 ∑ d − 1 ∣ Φ 0 c + ⟩ ⟨ Φ 0 c + ∣ = c = 1 ∑ d − 1 ( I ⊗ W 0 , c ) ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ ( I ⊗ W 0 , c † ) = c = 1 ∑ d − 1 ( I A ⊗ n = 1 ∑ d − 1 Ω c n ∣ n ⟩ ⟨ n ∣ ) ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ ( I ⊗ m = 0 ∑ d − 1 Ω − c m ∣ m ⟩ ⟨ m ∣ ) = c = 1 ∑ d − 1 n , m = 0 ∑ d − 1 Ω c ( n − m ) ( I ⊗ ∣ n ⟩ ⟨ n ∣ ) ∣ Φ 00 + ⟩ ⟨ Φ 00 + ∣ ( I ⊗ ∣ m ⟩ ⟨ m ∣ ) = d − 1 c = 1 ∑ d − 1 n , m = 0 ∑ d − 1 i , j = 0 ∑ d − 1 Ω c ( n − m ) ( I ⊗ ∣ n ⟩ ⟨ n ∣ ) ∣ ii ⟩ ⟨ jj ∣ ( I ⊗ ∣ m ⟩ ⟨ m ∣ ) = d − 1 n , m = 0 ∑ d − 1 [ c = 1 ∑ d − 1 Ω c ( n − m ) ] ∣ nn ⟩ ⟨ mm ∣ . It can then be seen, using the sum of a geometric series, that
∑ c = 1 d − 1 Ω c ( n − m ) = { − 1 i f n ≠ m ( d − 1 ) i f n = m \sum_{c=1}^{d-1} \Omega^{c(n-m)} =
\begin{cases}
-1 & ~\mathrm{if}~n \neq m \\
(d-1) & ~\mathrm{if}~n=m
\end{cases} c = 1 ∑ d − 1 Ω c ( n − m ) = { − 1 ( d − 1 ) if n = m if n = m Therefore,
∑ c = 1 d − 1 ∣ Φ 0 c + ⟩ ⟨ Φ 0 c + ∣ = d − 1 d ∑ n = m d − 1 ∣ n n ⟩ ⟨ n n ∣ − 1 d ∑ n ≠ m ∣ n n ⟩ ⟨ m m ∣ . \begin{split}
\sum_{c=1}^{d-1} \ket{\Phi_{0c}^{+}}\bra{\Phi_{0c}^{+}} &= \frac{d-1}{d} \sum_{n=m}^{d-1} \ket{nn}\bra{nn} - \frac{1}{d} \sum_{n \neq m}\ket{nn}\bra{mm}.
\end{split} c = 1 ∑ d − 1 ∣ Φ 0 c + ⟩ ⟨ Φ 0 c + ∣ = d d − 1 n = m ∑ d − 1 ∣ nn ⟩ ⟨ nn ∣ − d 1 n = m ∑ ∣ nn ⟩ ⟨ mm ∣ . Inputting this into the above gives
J ph = α d ∑ n , m = 0 d − 1 ∣ n n ⟩ ⟨ m m ∣ + 1 − α d − 1 [ d − 1 d ∑ n = m d − 1 ∣ n n ⟩ ⟨ n n ∣ − 1 d ∑ n ≠ m d − 1 ∣ n n ⟩ ⟨ m m ∣ ] = [ α d + 1 − α d ] ∑ n = m d − 1 ∣ n n ⟩ ⟨ n n ∣ + [ α d − 1 − α d ( d − 1 ) ] ∑ n ≠ m d − 1 ∣ n n ⟩ ⟨ m m ∣ A B = 1 d ∑ n = m ∣ n n ⟩ ⟨ n n ∣ + p d ∑ n ≠ m d − 1 ∣ n n ⟩ ⟨ m m ∣ , \begin{split}
\mathcal{J}^{\textrm{ph}} &= \frac{\alpha}{d} \sum_{n,m=0}^{d-1} \ket{nn}\bra{mm} + \frac{1-\alpha}{d-1} \biggl[ \frac{d-1}{d} \sum_{n=m}^{d-1} \ket{nn}\bra{nn} - \frac{1}{d} \sum_{n \neq m}^{d-1}\ket{nn}\bra{mm} \biggl] \\
&= \bigg[ \frac{\alpha}{d} + \frac{1-\alpha}{d} \bigg] \sum_{n=m}^{d-1} \ket{nn}\bra{nn} + \bigg[ \frac{\alpha}{d} - \frac{1-\alpha}{d(d-1)} \bigg] \sum_{n \neq m}^{d-1}\ket{nn}\bra{mm}_{AB} \\
&= \frac{1}{d} \sum_{n=m} \ket{nn}\bra{nn} + \frac{p}{d} \sum_{n \neq m}^{d-1}\ket{nn}\bra{mm},
\end{split} J ph = d α n , m = 0 ∑ d − 1 ∣ nn ⟩ ⟨ mm ∣ + d − 1 1 − α [ d d − 1 n = m ∑ d − 1 ∣ nn ⟩ ⟨ nn ∣ − d 1 n = m ∑ d − 1 ∣ nn ⟩ ⟨ mm ∣ ] = [ d α + d 1 − α ] n = m ∑ d − 1 ∣ nn ⟩ ⟨ nn ∣ + [ d α − d ( d − 1 ) 1 − α ] n = m ∑ d − 1 ∣ nn ⟩ ⟨ mm ∣ A B = d 1 n = m ∑ ∣ nn ⟩ ⟨ nn ∣ + d p n = m ∑ d − 1 ∣ nn ⟩ ⟨ mm ∣ , if α = ( p + 1 ) / d \alpha=(p+1)/d α = ( p + 1 ) / d
Qubits
Let ρ ∈ H 2 \rho \in \mathcal{H}^2 ρ ∈ H 2 ρ \rho ρ
D p pol ( ρ ) = p ρ + ( 1 − p ) tr [ ρ ] I / 2. \mathcal{D}^{\textrm{pol}}_p(\rho) = p \rho + (1-p) \textrm{tr}\big[\rho\big] \mathbb{I}/2. D p pol ( ρ ) = pρ + ( 1 − p ) tr [ ρ ] I /2. The Kraus operators are
K 0 = 1 4 + 3 p 4 I , K 1 = 1 − p 4 X , K 2 = 1 − p 4 Y , K 3 = 1 − p 4 Z . \begin{align*}
K_0 &= \sqrt{ \frac{1}{4} + \frac{3p}{4}} \mathbb{I}, ~ ~ K_1 = \sqrt{ \frac{1-p}{4}} X, \\
K_2 &= \sqrt{ \frac{1-p}{4}} Y, ~ ~ K_3 = \sqrt{ \frac{1-p}{4}} Z.
\end{align*} K 0 K 2 = 4 1 + 4 3 p I , K 1 = 4 1 − p X , = 4 1 − p Y , K 3 = 4 1 − p Z . The Choi-state is the so-called isotropic state,
J pol = p ∣ Φ + ⟩ ⟨ Φ + ∣ + ( 1 − p ) I / 4 , \mathcal{J}^{\textrm{pol}} = p \ket{\Phi^+} \bra{\Phi^+}+ (1-p) \mathbb{I}/4, J pol = p ∣ Φ + ⟩ ⟨ Φ + ∣ + ( 1 − p ) I /4 , where ∣ Φ + ⟩ \ket{\Phi^+} ∣ Φ + ⟩
∣ Φ + ⟩ = ∣ 0 ⟩ ∣ 0 ⟩ + ∣ 1 ⟩ ∣ 1 ⟩ 2 , \begin{align*}
\ket{\Phi^{+}} &= \frac{\ket{0}\ket{0}+\ket{1} \ket{1}}{\sqrt{2}},
\end{align*} ∣ Φ + ⟩ = 2 ∣ 0 ⟩ ∣ 0 ⟩ + ∣ 1 ⟩ ∣ 1 ⟩ , d-dimensional
The depolarising channel acting on a d d d ρ ∈ H d \rho \in \mathcal{H}^d ρ ∈ H d
D p pol ( ρ ) = p ρ + ( 1 − p ) I / d , \mathcal{D}^{\textrm{pol}}_p(\rho) = p \rho + (1-p)\mathbb{I}/d, D p pol ( ρ ) = pρ + ( 1 − p ) I / d , with
J pol = p ∣ Φ + ⟩ ⟨ Φ + ∣ + ( 1 − p ) I / d 2 . \mathcal{J}^{\textrm{pol}} = p \ket{\Phi^+} \bra{\Phi^+}+ (1-p) \mathbb{I}/d^2. J pol = p ∣ Φ + ⟩ ⟨ Φ + ∣ + ( 1 − p ) I / d 2 .
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 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