trureturing — Discovering truth

bibkey: hanliu2023twirling authors: Deguang Han and Kai Liu year: 2023 title: “Zero-Error Correctibility and Phase Retrievability for Twirling Channels” doi: 10.1016/S0034-4877(24)00012-0 url: https://arxiv.org/abs/2311.16349v1 claim: “Section 5 conjectures pr(Phi_pi) = max{beta(Phi_pi), k} when I_k <= d < I_(k+1), for d inequivalent irreducible summands with positive multiplicities; Theorem 3.4 proves beta(Phi_pi) = max_i m_i.” strata_touched:

• D5/S3/Quantum/QuantumChannels/TwirlingPhaseRetrievabilityCounterexample license: citation-only triage: anchor

Verified locator

DOI: https://doi.org/10.1016/S0034-4877(24)00012-0

Source: https://arxiv.org/abs/2311.16349v1

The arXiv version is dated 27 November 2023. The journal citation is Reports on Mathematical Physics 93(1) (2024), 87–102. Page numbers below refer to the arXiv PDF.

Definitions and conjecture

The twirling channel is defined on page 3:

We define a linear map ${\Phi }_{\pi ,\sigma }:B(K,H)\to B(K,H)$ by ${\Phi }_{\pi ,\sigma }(T)={\int }_G\pi (g)T\sigma (g^{-1})d\mu (g)$ and denote ${\Phi }_{\pi ,\sigma }$ by ${\Phi }_{\pi }$ when $\pi =\sigma$.

For finite groups, normalized Haar integration is the average over the group. Unitarity identifies the inverse with the matrix star.

Page 5 defines phase retrievability:

A subspace $M$ of $H$ is called phase retrievable under $\Phi$ if $\Phi$ is pure state injective on $M$, i.e., $\Phi (x\otimes x)=\Phi (y\otimes y)$ implies that $x\otimes x=y\otimes y$ for $x,y\in M$, or equivalently if there exists a POVM $\{F_j{\}}_{j\in \mathbb{J}}$ in $B(K)$ such that $\{P_M{\Phi }^{\ast }(F_j)P_M{\}}_{j\in \mathbb{J}}$ is a phase retrievable operator valued frame for $M$, where $P_M$ is the orthogonal projection onto $M$.

The phase retrievability index $pr({\Phi }_{\pi })$ is defined to be the largest integer $k$ such that there exits a $k$-dimensional subspace $M\subset H$ such that $M$ is phase retrievable under $\Phi$.

The spelling “exits” is retained from the source. Rank-one states are encoded by Matrix.vecMulVec x (star x), and the largest admissible dimension by the natural supremum.

Page 4 defines the frame length:

For an $n$-dimensional Hilbert space $H$, we will use ${\mathcal{I}}_n$ to denote the smallest integer $N$ such that there is a phase retrievable frame $\{x_j{\}}_{j=1}^N$ for $H$.

The encoding uses complex coordinate spaces, Fin n for a family of length $n$, and the natural infimum. A family that retrieves every rank-one state spans the space: a nonzero vector orthogonal to the family would have the same measurements as zero.

Lemma 3.1 on page 5 characterizes codes:

Let $\mathcal{C}$ be a subspace of $H$, and let $P$ be the orthogonal projections onto $\mathcal{C}$. Suppose $\Phi$ is a quantum channel with Kraus operators $\{E_i{\}}_{i=1}^r$. Then the following are equivalent:

Its items (i) and (ii) are:

$\mathcal{C}$ is a quantum code for $\Phi$.

there exists a Hermitian matrix $A=[a_{ij}]$ such that $P{E}_i^{\ast }{E}_{j}P=a_{ij}P$ holds for all $i,j$.

IsCode uses this compression condition. A self-adjoint idempotent with range $M$ is the orthogonal projection onto $M$. For the twirl, the Kraus operators are $\pi (g)/\sqrt{|G|}$, whose common positive normalization can be absorbed in the compression scalars. Those scalars are automatically Hermitian when the projection is nonzero; for the zero code one may choose them all zero. The index $\beta$ is the largest code dimension. Theorem 3.4 on page 7 proves $\beta ({\Phi }_{\pi })=\max \{m_1,\dots ,m_d\}$.

The representation conventions use invariant complex subspaces and an invertible complex-linear intertwiner. An irreducible representation acts on a nonzero space. The block representation is the direct sum of $I_{m_i}\otimes {\rho }_i$, indexed by Σ i, Fin (m i) × Fin (n i).

The final conjecture of Section 5, page 13, reads:

Conjecture. Let $\pi =m_1{\pi }_1\oplus \dots \oplus m_d{\pi }_d$ be a unitary representation of $G$ such that ${\pi }_i$ and ${\pi }_j$ are inequivalent irreducible representations for all $i\ne j$ and ${\mathcal{I}}_k\le d<{\mathcal{I}}_{k+1}$. Then $pr({\Phi }_{\pi })=\max \{\beta ({\Phi }_{\pi }),k\}$.

The public claim specializes this assertion to finite groups, includes positive multiplicities, and quantifies all decomposition data. A counterexample to this specialization refutes the compact-group conjecture.

Counterexample and boundary

For $C_3$, each of the three one-dimensional characters appears twice. Character orthogonality makes the twirl the three-block pinching. The encoding $(u,v,w)\mapsto ((u,v),(u,w),(v,w))/\sqrt{2}$ retains every entry of a $3\times 3$ operator in its three $2\times 2$ compressions. Its range therefore has phase retrievability dimension three. The code dimension is two, and ${\mathcal{I}}_1\le 3<{\mathcal{I}}_2$, giving $k=1$. These facts yield the module’s result : ¬ claim.

The rectangular encoding matrix uses Matrix.conjTranspose; square matrices retain star. Nonzero dimension is included in irreducibility. Both conventions preserve the mathematical meaning of the source.

The multiplicity-one character case proved immediately before the conjecture is compatible with this counterexample. The source’s code-dimension theorem and its phase-retrievability lower bounds also remain compatible. The module does not determine the exact index for general multiplicities or groups.