nLab unistochastic quantum channel

Idea

A quantum operation/quantum channel $chan \,\colon\, \mathscr{H} \otimes \mathscr{H}^\ast \to \mathscr{H} \otimes \mathscr{H}^\ast$ 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:

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

Remark

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).

Remark

(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).

Examples

Example

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

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

for unitaries

$$s\,\colon\, S \;\;\;\;\;\; \vdash \;\;\;\;\;\; U_s \,\colon\, \mathscr{H}_1 \to \mathscr{H}_2$$

is unistochastic, as evidently exhibited by the following coupling unitary

$$U_tot \;\colon\; \mathscr{H}_1 \otimes \underset{S}{\oplus} \mathbb{C} \to \mathscr{H}_2 \otimes \underset{S}{\oplus} \mathbb{C}$$

(1)$$U_{tot} \;\;\coloneqq\;\; \tfrac{1}{Card(S)} \underset{s \colon S}{\sum} \, U_s \otimes \left\vert s \right\rangle \left\langle s \right\vert \mathrlap{\,.}$$

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

Proposition

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

(Müller-Hermes & Perry 2019, Cor 1.4)

Proof

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

$$\rho \;\mapsto\; p_1 \, U_1 \cdot \rho \cdot U_1^\dagger \;+\; p_2 \, U_2 \cdot \rho \cdot U_2^\dagger$$

then we want to find $U'_i$ with

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

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

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

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

(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 $S$: Since $U_1^\dagger \cdot U_2$ is evidently a normal operator, the spectral theorem applies to show that we may find $S$ such that $D$ above is a diagonal matrix:

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

This way we are now reduced to finding unitary operators $W_i$, such that

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

Plugging in the simple ansatz

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

shows that this works for

(4)$$\tfrac{1}{2} \big( e^{2 \pi \mathrm{i} \phi_1} + e^{2 \pi \mathrm{i} \phi_2} \big) \;\; = \;\; p_1 \,+\, p_2 \, D_11 \overline{D_22} \mathrlap{\,.}$$

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

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

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

Example

The bit-flip quantum channel

(6)$$\array{ \mathllap{ flip_p \;\colon\; } QBit \otimes QBit^\ast &\longrightarrow& QBit \otimes QBit^\ast \\ \rho &\mapsto& (1-p)\,\rho \;+\; p \, X \cdot \rho \cdot X }$$

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

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

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

which gives

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

and the condition (4) in this case is

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

This is solved for

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

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

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

for

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

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

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

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

References

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

• Emanuel Knill, Raymond Laflamme, On the Power of One Bit of Quantum Information, Phys. Rev. Lett. 81 (1998) 5672-5675 [arXiv:quant-ph/9802037, doi:10.1103/PhysRevLett.81.5672]

(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:

• David Poulin, Raymond Laflamme, Gerard J. Milburn, Juan Pablo Paz, Testing integrability with a single bit of quantum information, Phys. Rev. A 68 22302 (2003) [arXiv:quant-ph/0303042, doi:10.1103/PhysRevA.68.022302]

• David Poulin, Robin Blume-Kohout, Raymond Laflamme, Harold Ollivier, around (2) of: Exponential speed-up with a single bit of quantum information: Testing the quantum butterfly effect, Phys. Rev. Lett. 92 177906 (2004) [arXiv:quant-ph/0310038, doi:10.1103/PhysRevLett.92.177906]

• Peter W. Shor, Stephen P. Jordan, Estimating Jones polynomials is a complete problem for one clean qubit, Quantum Information and Computation 8 (2008) 681 [arXiv:0707.2831, doi:10.5555/2017011.2017012]

• Dan Shepherd, Computation with Unitaries and One Pure Qubit [arXiv:quant-ph/0608132]

• Chris Cade, Ashley Montanaro, The Quantum Complexity of Computing Schatten $p$-norms [arXiv:1706.09279]

See also:

• Wikipedia, One Clean QBit.

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

• Tomoyuki Morimae, Keisuke Fujii, Harumichi Nishimura, Power of one non-clean qubit, Phys. Rev. A 95 042336 (2017) [arXiv:1610.07244, doi:10.1103/PhysRevA.95.042336]

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

• Alexander Müller-Hermes, Christopher Perry, All unital qubit channels are 4-noisy operations, Letters in Mathematical Physics 109 (2019) 1–9 [doi:10.1007/s11005-018-1104-x, arXiv:1802.01337]

Discussion of DQC1 in the language of quantum channels:

• Jake Xuereb, Steve Campbell, John Goold, André Xuereb, DQC1 as an Open Quantum System, Phys. Rev. A 107 042222 (2023) [arXiv:2209.03947, doi:10.1103/PhysRevA.107.042222]

• Shuangshuang Fu, Jiayu He, Xiaohui Li and Shunlong Luo, Uncertainties and coherence in DQC1, Physica Scripta, 98 4 (2023) [doi:10.1088/1402-4896/acc5ba]

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

• Michal Horodecki, Pawel Horodecki, Jonathan Oppenheim, Reversible transformations from pure to mixed states, and the unique measure of information, Phys. Rev. A 67 062104 (2003) [doi:10.1103/PhysRevA.67.062104, arXiv;quant-ph/0212019]

• picked up by Müller-Hermes & Perry 2019

and the terminology “unistochastic channels” was introduced in:

• Karol Życzkowski, Ingemar Bengtsson, p. 13 of: On Duality between Quantum Maps and Quantum States, Open Systems & Information Dynamics 11 01 (2004) 3-42 [doi:10.1023/B:OPSY.0000024753.05661.c2]

• Ingemar Bengtsson, Karol Życzkowski, p. 259 of: Geometry of Quantum States — An Introduction to Quantum Entanglement, Cambridge University Press (2006) [doi:10.1017/CBO9780511535048, ResearchGate]

• Marcin Musz, Marek Kuś, Karol Życzkowski, Unitary quantum gates, perfect entanglers, and unistochastic maps, Phys. Rev. A 87 (2013) 022111 [doi:10.1103/PhysRevA.87.022111]