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A quantum operation/quantum channel chan: * *chan \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast \to \mathscr{H} \otimes \mathscr{H}^\ast is called a noisy operation [by Horodecki, Horodecki & Oppenheim 2003, but beware that this can be ambiguous] or a unistochastic channel [Życzkowski & Bengtsson 2004] if it has an environmental representation where the environment/bath system \mathscr{B} is in its maximally mixed quantum state:

chan(ρ)=tr (U(ρ1dim()id maximallymixedenvironment)U ). chan(\rho) \;\;=\;\; \mathrm{tr}^{\mathscr{B}} \Big( U \big( \rho \otimes \underset{ \mathclap{ {maximally \; mixed} \atop environment } }{ \underbrace{ \frac{1}{dim(\mathscr{B})} id_{\mathscr{B}} } } \big) U^\dagger \Big) \,.


The theorem (here) that every endo-quantum channel has an environmental representation is sometimes advertized with the addendum that “… and such that the environment may be chosen to be in a pure state”. But existing general proofs actually produce only such environmental representations. For environments in non-pure states it is not clear that they can environmentally represent all quantum channels, and for noisy/unistochastic channels it is not to be expected that they exhaust all quantum channels.

But Müller-Hermes & Perry 2019 show that at least all unital quantum channels on single qbits can be realized as noisy/unistochastic channels (with a bath of size at least 4).


(DQC1 model)
But the idea of exploring quantum computation/quantum information theory on (a few or just single) clean qbits coupled to an environment in a maximally mixed state goes back already to Knill & Laflamme 1998 (motivated by discussion of spin resonance qbits), who referred to this model of computation as deterministic quantum computation with one quantum bit (abbreviated “DQC1”, now also used for the corresponding quantum complexity class, first studied by Shor 2008, and often referred as the “one clean qbit”-model).

The DQC1 model is believed to be a very restricted, but still genuinely quantum, computing model – a sub-universal quantum computing model whose power is something between classical computation and universal quantum computation. [FKMNTT18, p. 2]

The communities using these different terminologies for closely related ideas may have been somewhat disconnected. A proposal to look at DQC1 in terms of quantum channels seems to not appear not before Xuereb, Campbell, Goold & Xuereb 2023; Fu, He, Li & Luo 2023 (and the “unistochastic” terminology is not used there).



Any mixed unitary quantum channel for a uniform probability distribution, i.e. one of the form

ρ1Card(S)s:SU sρU s ,, \rho \;\mapsto\; \tfrac{1}{Card(S)} \underset{s \colon S}{\sum} \, U_s \cdot \rho \cdot U_s^\dagger, \;\;\;\; \,,

for unitaries

s:SU s: 1 2 s\,\colon\, S \;\;\;\;\;\; \vdash \;\;\;\;\;\; U_s \,\colon\, \mathscr{H}_1 \to \mathscr{H}_2

is unistochastic, as evidently exhibited by the following coupling unitary

U tot: 1S 2S U_tot \;\colon\; \mathscr{H}_1 \otimes \underset{S}{\oplus} \mathbb{C} \to \mathscr{H}_2 \otimes \underset{S}{\oplus} \mathbb{C}
(1)U tot1Card(S)s:SU s|ss|. U_{tot} \;\;\coloneqq\;\; \tfrac{1}{Card(S)} \underset{s \colon S}{\sum} \, U_s \otimes \left\vert s \right\rangle \left\langle s \right\vert \mathrlap{\,.}

(cf. Müller-Hermes & Perry 2019, proof of Cor. 1.4)


On a single qbit, every mixed unitary quantum channel is a unistochastic channel.

(Müller-Hermes & Perry 2019, Cor 1.4)

By Exp. one is reduced to showing that on QBits every mixed unitary quantum channel equals one with uniformly distributed unitaries (this is M-H & P 19, Thm. 1.2). We further spell out the case where there are two unitaries to start with (M-H & P 19, Lem. 1.1):

So let the mixed unitary channel be given by

ρp 1U 1ρU 1 +p 2U 2ρU 2 \rho \;\mapsto\; p_1 \, U_1 \cdot \rho \cdot U_1^\dagger \;+\; p_2 \, U_2 \cdot \rho \cdot U_2^\dagger

then we want to find U iU'_i with

p 1U 1(-)U 1 +p 2U 2(-)U 2 =12U 1(-)U 1 +12U 2(-)U 2 . p_1 \, U_1 \cdot (\text{-}) \cdot U_1^\dagger \;+\; p_2 \, U_2 \cdot (\text{-}) \cdot U_2^\dagger \;\; = \;\; \tfrac{1}{2} \, U'_1 \cdot (\text{-}) \cdot U'_1^\dagger \;+\; \tfrac{1}{2} \, U'_2 \cdot (\text{-}) \cdot U'_2^\dagger \,.

First, by replacing (-)S(-)S (\text{-}) \mapsto S \cdot (\text{-}) S^\dagger — for a unitary SS to be specified in a moment —, this is equivalent to

p 1U 1S(-)S U 1 +p 2U 2S(-)S U 2 =12U 1S(-)S U 1 +12U 2S(-)S U 2 . \begin{array}{r} p_1 \, U_1 \cdot S \cdot (\text{-}) \cdot S^\dagger \cdot U_1^\dagger \\ +\; p_2 \, U_2 \cdot S \cdot (\text{-}) \cdot S^\dagger \cdot U_2^\dagger \end{array} \;\; = \;\; \begin{array}{r} \tfrac{1}{2} \, U'_1 \cdot S \cdot (\text{-}) \cdot S^\dagger \cdot U'_1^\dagger \\ + \; \tfrac{1}{2} \, U'_2 \cdot S \cdot (\text{-}) \cdot S^\dagger \cdot U'_2^\dagger \mathrlap{\,.} \end{array}

and then conjugating both sides by S U 1 S^\dagger \cdot U^\dagger_1 gives that this is equivalent to

(2)p 1(-)+p 2S U 1 U 2SD(-)S U 2 U 1SD =12S U 1 U 1SW 1(-)S U 1 U 1SW 1 +12S U 1 U 2SW 2(-)S U 2 U 1SW 2 p_1 \, (\text{-}) \;+\; p_2 \, \underset{D}{ \underbrace{ S^\dagger U_1^\dagger \cdot U_2 \cdot S } } \cdot (\text{-}) \cdot \underset{D^\dagger}{ \underbrace{ S^\dagger \cdot U_2^\dagger \cdot U_1 \cdot S } } \;\; = \;\; \begin{array}{r} \tfrac{1}{2} \, \underset{ W_1 }{ \underbrace{ S^\dagger \cdot U_1^\dagger \cdot U'_1 \cdot S } } \cdot (\text{-}) \cdot \underset{ W_1^\dagger }{ \underbrace{ S^\dagger \cdot U'_1^\dagger \cdot U_1 \cdot S } } \\ + \; \tfrac{1}{2} \, \underset{ W_2 }{ \underbrace{ S^\dagger \cdot U_1^\dagger \cdot U'_2 \cdot S } } \cdot (\text{-}) \cdot \underset{W_2^\dagger}{ \underbrace{ S^\dagger \cdot U'_2^\dagger \cdot U_1 \cdot S } } \end{array}

At this point we fix SS: Since U 1 U 2U_1^\dagger \cdot U_2 is evidently a normal operator, the spectral theorem applies to show that we may find SS such that DD above is a diagonal matrix:

(3)SU 1 U 2S =diag(D 11,D 22)=D. S \cdot U_1^\dagger \cdot U_2 \cdot S^\dagger \;\;=\;\; diag\big( D_{11}, \, D_{22} \big) \,=\, D \,.

This way we are now reduced to finding unitary operators W iW_i, such that

p 1(-)+p 2D(-)D =12W 1(-)W 1 +12W 2(-)W 2 . p_1 \, (\text{-}) \;+\; p_2 \, D \cdot (\text{-}) \cdot D^\dagger \;\; = \;\; \tfrac{1}{2} \, W_1 \cdot (\text{-}) \cdot W_1^\dagger \;+\; \tfrac{1}{2} \, W_2 \cdot (\text{-}) \cdot W_2^\dagger \,.

Plugging in the simple ansatz

W i[e 2πiϕ i 0 0 1] W_i \;\coloneqq\; \left[ \array{ e^{2 \pi \mathrm{i} \phi_i } & 0 \\ 0 & 1 } \right]

shows that this works for

(4)12(e 2πiϕ 1+e 2πiϕ 2)=p 1+p 2D 11D 22¯. \tfrac{1}{2} \big( e^{2 \pi \mathrm{i} \phi_1} + e^{2 \pi \mathrm{i} \phi_2} \big) \;\; = \;\; p_1 \,+\, p_2 \, D_11 \overline{D_22} \mathrlap{\,.}

which always has a solution for the ϕ i\phi_i (M-H & P 19, p. 3).

Finally plugging back into (2) shows that the desired uniformly distributed unitaries may be taken to be:

(5)U iU 1S[e 2πiϕ i 0 0 1]S . U'_i \;\;\coloneqq\;\; U_1 \cdot S \cdot \left[ \array{ e^{2 \pi \mathrm{i} \phi_i} & 0 \\ 0 & 1 } \right] \cdot S^\dagger \,.


The bit-flip quantum channel

(6)flip p:QBitQBit * QBitQBit * ρ (1p)ρ+pXρX \array{ \mathllap{ flip_p \;\colon\; } QBit \otimes QBit^\ast &\longrightarrow& QBit \otimes QBit^\ast \\ \rho &\mapsto& (1-p)\,\rho \;+\; p \, X \cdot \rho \cdot X }

is unistochastic, by Exp. , since it is manifestly mixed unitary.

Explicitly, the unitary SS (3) in this case may be taken to be

S=12[1 1 1 1] S \;=\; \tfrac{1}{\sqrt{2}} \left[ \array{ 1 & 1 \\ -1 & 1 } \right]

which gives

D=12[1 1 1 1][0 1 1 0]X12[1 1 1 1]=[1 0 0 1], D \;\; = \;\; \tfrac{1}{\sqrt{2}} \left[ \array{ 1 & 1 \\ -1 & 1 } \right] \cdot \underset{ X }{ \underbrace{ \left[ \array{ 0 & 1 \\ 1 & 0 } \right] } } \cdot \tfrac{1}{\sqrt{2}} \left[ \array{ 1 & -1 \\ 1 & 1 } \right] \;\; = \;\; \left[ \array{ 1 & 0 \\ 0 & -1 } \right] \mathrlap{\,,}

and the condition (4) in this case is

12(e 2πiϕ 1+e 2πiϕ 2)=(12p). \tfrac{1}{2} \big( e^{2 \pi \mathrm{i} \phi_1} + e^{2 \pi \mathrm{i} \phi_2} \big) \;\; = \;\; (1 - 2 p) \,.

This is solved for

ϕ 1=ϕ 2=arccos(12p), \phi_1 = -\phi_2 = arccos(1-2p) \,,

so that the desired unitary operators as obtained from formula (5) are:

U 1/2 =12[1 1 1 1][e ±iϕ 0 0 1]12[1 1 1 1] =12[1+e ±iϕ 1e ±iϕ 1e ±iϕ 1+e ±iϕ] =12e ±iϕ/2[e iϕ/2+e ±iϕ/2 e iϕ/2e ±iϕ/2 e iϕ/2e ±iϕ/2 e iϕ/2+e ±iϕ/2] =e ±iϕ/2[cos(ϕ/2) ±isin(ϕ/2) ±isin(ϕ/2) cos(ϕ/2)]. \begin{array}{l} U_{1/2} \\ \;=\; \tfrac{1}{\sqrt{2}} \left[ \array{ 1 & 1 \\ -1 & 1 } \right] \cdot \left[ \array{ e^{ \pm \mathrm{i} \phi } & 0 \\ 0 & 1 } \right] \cdot \tfrac{1}{\sqrt{2}} \left[ \array{ 1 & -1 \\ 1 & 1 } \right] \\ \;=\; \tfrac{1}{2} \left[ \array{ 1 + e^{ \pm \mathrm{i} \phi } & 1 - e^{ \pm \mathrm{i} \phi } \\ 1 - e^{ \pm \mathrm{i} \phi } & 1 + e^{ \pm \mathrm{i} \phi } } \right] \\ \;=\; \tfrac{1}{2} e^{ \pm \mathrm{i} \phi/2 } \left[ \array{ e^{ \mp \mathrm{i} \phi/2 } + e^{ \pm \mathrm{i} \phi/2 } & e^{ \mp \mathrm{i} \phi/2 } - e^{ \pm \mathrm{i} \phi/2 } \\ e^{ \mp \mathrm{i} \phi/2 } - e^{ \pm \mathrm{i} \phi/2 } & e^{ \mp \mathrm{i} \phi/2 } + e^{ \pm \mathrm{i} \phi/2 } } \right] \\ \;=\; e^{ \pm \mathrm{i} \phi/2 } \left[ \array{ cos(\phi/2) & \pm\mathrm{i}\,sin(\phi/2) \\ \pm\mathrm{i}\,sin(\phi/2) & cos(\phi/2) } \right] \mathrlap{\,.} \end{array}


ϕarccos(12p). \phi \,\coloneqq\, arccos(1-2p) \,.

Hence a uniformly mixed unitary channel representation of the bit-flip quantum channel (6) is:

(7)flip p(ρ) = 12[cos(ϕ/2) isin(ϕ/2) isin(ϕ/2) cos(ϕ/2)]ρ[cos(ϕ/2) +isin(ϕ/2) +isin(ϕ/2) cos(ϕ/2)] + 12[cos(ϕ/2) +isin(ϕ/2) +isin(ϕ/2) cos(ϕ/2)]ρ[cos(ϕ/2) isin(ϕ/2) isin(ϕ/2) cos(ϕ/2)], whereϕarccos(12p). \begin{array}{rcl} flip_p(\rho) &=& \tfrac{1}{2} \, \left[ \array{ cos(\phi/2) & -\mathrm{i}\,sin(\phi/2) \\ -\mathrm{i}\,sin(\phi/2) & cos(\phi/2) } \right] \cdot \rho \cdot \left[ \array{ cos(\phi/2) & +\mathrm{i}\,sin(\phi/2) \\ +\mathrm{i}\,sin(\phi/2) & cos(\phi/2) } \right] \\ &+& \tfrac{1}{2} \, \left[ \array{ cos(\phi/2) & +\mathrm{i}\,sin(\phi/2) \\ +\mathrm{i}\,sin(\phi/2) & cos(\phi/2) } \right] \cdot \rho \cdot \left[ \array{ cos(\phi/2) & -\mathrm{i}\,sin(\phi/2) \\ -\mathrm{i}\,sin(\phi/2) & cos(\phi/2) } \right] \mathrlap{\,,} \\ && \text{where}\; \phi \,\equiv\, arccos(1-2p) \text{.} \end{array}

With (1) this uniformly mixed unitary presentation immediately gives the unistochastic presentation of the bit-flip channel.


DQC1 computation

The exploration of the possibilities of quantum computing/quantum information theory with (a few or even just single) “clean” qubits coupled to a maximally mixed state environment goes back to

(motivated by the practical reality of NMR spin resonance qbits) who called this model of computation deterministic quantum computation with one quantum bit (DQC1).

The model was further discussed in:

See also:

Discussion where even the single system qbit is not fully coherent, either:

Proof that all unital quantum channels on single qubits are unistochastic (noisy operations) for a bath of size at least 4:

Discussion of DQC1 in the language of quantum channels:

Discussion of classical simulation (or not) of the DQC1 model

  • Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, Seiichiro Tani, Impossibility of Classically Simulating One-Clean-Qubit Computation, Phys. Rev. Lett. 120 200502 (2018) [arXiv:1409.6777, doi:10.1103/PhysRevLett.120.200502]

Noisy operations/unistochastic channels

The terminology of “noisy operations” is due to

and the terminology “unistochastic channels” was introduced in:

Last revised on October 12, 2023 at 15:23:26. See the history of this page for a list of all contributions to it.