trureturing — Discovering truth

bibkey: gschwendtnerbluhmwinter2021programmability
authors: Martina Gschwendtner, Andreas Bluhm, Andreas Winter
year: 2021
title: Programmability of covariant quantum channels
doi: 10.22331/q-2021-06-29-488
url: https://arxiv.org/abs/2012.00717v2
claim: Exact covariant programming has mixed optimum d_c; the Section 5 question asks whether pure programs attain the same dimension.
strata_touched:
  - D5/S3/Quantum/QuantumChannels/PureProgramQubitObstruction
  - D5/S3/Quantum/QuantumChannels/CovariantPureProgramDimensionRefutation
license: citation-only
triage: anchor

Verified locator

DOI: https://doi.org/10.22331/q-2021-06-29-488

Source: https://arxiv.org/abs/2012.00717v2

Quantum 5, 488. The page numbers below refer to the arXiv v2 PDF.

Covariance and processors

Page 2 defines states by $\mathcal{D}({\mathcal{H}}_1)=\{\rho \in \mathcal{B}({\mathcal{H}}_1)\mid \rho \ge 0,\mathrm{tr}(\rho )=1\}$.

A quantum channel is a completely positive trace-preserving map $T:\mathcal{B}({\mathcal{H}}_1)\to \mathcal{B}({\mathcal{H}}_2)$, where $d_1,d_2\in \mathbb{N}$.

Definition 2, page 3, states:

A unitary representation $U$ of a group $G$ on a finite-dimensional vector space ${\mathcal{H}}_1$ is called irreducible representation (irrep) if and only if the only invariant subspaces of $\{U_g\mid g\in G\}$ are $\{0\}$ and the whole space.

Definition 10, page 4, states:

We call $T$ $UV$-covariant if

$$T(U_{g}A{U}_g^{\ast })=V_{g}T(A)V_g^{\ast }\forall A\in \mathcal{B}({\mathcal{H}}_1),\forall g\in G.$$

Definition 12, page 5, states:

Let ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ be separable Hilbert spaces. Then we call $\mathcal{P}\in \mathrm{CPTP}({\mathcal{H}}_1\otimes {\mathcal{H}}_P,{\mathcal{H}}_2)$, with finite-dimensional ${\mathcal{H}}_P$, an $ϵ$-programmable quantum processor for a set $\mathcal{C}\subset \mathrm{CPTP}({\mathcal{H}}_1,{\mathcal{H}}_2)$ of channels ($ϵ$-PQP${}_{\mathcal{C}}$), if for every quantum channel $T\in \mathcal{C}$ there exists a state ${\pi }_T\in \mathcal{D}({\mathcal{H}}_P)$ such that

$$\frac{1}{2}\Vert \mathcal{P}(\cdot \otimes {\pi }_T)-T(\cdot ){\Vert }_{\diamond }\le ϵ.$$

For $\mathcal{C}={\mathcal{T}}_{UV}$ this is an $ϵ$-CPQP${}_{UV}$. At $ϵ=0$ the processor implements every covariant channel exactly; $d_P=\dim {\mathcal{H}}_P$ is its program dimension.

Dimension bounds and pure programs

Write the commutant of $\bar{U}\otimes V$ as $\mathcal{K}\cong {⨁}_k{1}_{b_k}\otimes \mathcal{B}({\mathbb{C}}^{n_k})$ and set $d_c={\sum }_k{n}_k$. Theorem 31 supplies the lower bound $d_P\ge d_c$ in the exact case. The teleportation construction of Theorem 25 attains $d_P=d_c$ with mixed programs. Purifying those programs gives the general upper bound $d_c^2$. Remark 13 (pure-mixed), page 6, states:

In the literature, where the set of channels $\mathcal{C}$ often consists of isometries or even unitaries, it is customary to impose that the program state ${\pi }_T$ is pure.

It discusses more refined pure-program constructions. Corollary 19 (abelian) attains the lower bound with pure programs when $\mathcal{K}$ is abelian.

Section 5, page 21, asks:

We leave it as an open question whether the upper bounds can be improved to match the lower ones also for pure program states.

The affirmative assertion is that every mixed-program dimension can also be attained with pure programs. The finite-group restriction already contains the counterexample $G=D_4$, its two-dimensional irreducible $U$, and trivial two-dimensional $V$. Here $\bar{U}\otimes V$ contains two copies of $\bar{U}$, giving $d_c=2$. All covariant channels then replace the input by a state. A mixed qubit program implements them, but a pure qubit program cannot produce the two basis states, their equal superposition, and $1/2$ through one fixed processor. The Lean result certifies this dimension-two obstruction. The pure dimension-three construction and the general multiplicity bounds require additional analytic arguments.