bibkey: dirr2008relative authors: Gunther Dirr; Uwe Helmke; Martin Kleinsteuber; Thomas Schulte-Herbrüggen year: 2008 title: “Relative C-numerical ranges for applications in quantum control and quantum information” doi: 10.1080/03081080701535898 url: https://arxiv.org/abs/math-ph/0702005v1 claim: “Conjecture 3.6: Every element of E(tloc) is similar to a block-shift matrix via an element of Πex_loc := Πloc·Πout.” strata_touched:

• D5/S3/Quantum/Information/LocalPermutationBlockShiftRefutation license: citation-only triage: anchor

Relative C-numerical ranges for applications in quantum control and quantum information

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DOI: https://doi.org/10.1080/03081080701535898

Source: https://arxiv.org/abs/math-ph/0702005v1

Linear and Multilinear Algebra 56, 27–51. The page locators below refer to the arXiv v1 pagination.

Conjecture 3.6

Page 26, continuing on page 27:

Every element of $E(t_{\mathrm{loc}})$ is similar to a block-shift matrix via an element of ${\Pi }_{\mathrm{loc}}^{\mathrm{ex}}:={\Pi }_{\mathrm{loc}}\cdot {\Pi }_{\mathrm{out}}$. Here, ${\Pi }_{\mathrm{loc}}$ denotes the subgroup of all (signed) local permutations, i.e. ${\Pi }_{\mathrm{loc}}$ consists of all matrices of the form $P_1\otimes \cdots \otimes P_n$ with $P_k\in \{\pm I_2,\pm \left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]\}$ for $k=1,\dots ,n$, and ${\Pi }_{\mathrm{out}}$ stands for the subgroup generated by $I_2\otimes \cdots \otimes I_2\otimes P_{\mathrm{out}}\otimes I_2\otimes \cdots \otimes I_2\in U(2^n)$, $n\ge 2$, where the factor $P_{\mathrm{out}}$ appears in all possible positions.

The signed swap is $P_{\mathrm{out}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 1\end{array}\right]$. The tensor basis has its first factor as the most significant bit. The local torus consists of ${\sum }_{k}i{\lambda }_k{Z}_k$, with real ${\lambda }_k$ and $Z=\mathrm{diag}(1,-1)$. Equation (48) expresses membership of a nonzero root space as $[\Delta ,A]=i\phi A$, where $\Delta \in t_{\mathrm{loc}}$ and $\phi \ne 0$.

Corollaries 2.18 and 2.19

Page 15, Corollary 2.18:

Let $t$ be a torus algebra of $\mathfrak{k}$. The relative $C$-numerical range of $A\in {\mathbb{C}}^{N\times N}$, $A\ne 0$ is rotationally symmetric for all $C\in {\mathbb{C}}^{N\times N}$ if and only if there exists $U\in K$ such that $UA{U}^{†}$ is in $E(t)$.

Page 15, Corollary 2.19:

The classical $C$-numerical range of $A$ is rotationally symmetric for all $C\in {\mathbb{C}}^{N\times N}$ if and only if $A$ is unitarily similar to a block-shift matrix $M$, i.e. $M=(M_{kl}{)}_{1\le k,l\le m}$ is of block form such that all the $M_{kk}$ are square blocks and $M_{kl}=0$ if $l+1\ne k$.

The latter allows arbitrary unitaries. Refuting the restriction to extended local permutations does not refute either corollary.

Remark 2

Page 27, Remark 2(a):

Moreover, ${\Pi }_{\mathrm{loc}}$ and ${\Pi }_{\mathrm{out}}$ commute and hence ${\Pi }_{\mathrm{loc}}^{\mathrm{ex}}:={\Pi }_{\mathrm{loc}}\cdot {\Pi }_{\mathrm{out}}={\Pi }_{\mathrm{out}}\cdot {\Pi }_{\mathrm{loc}}$ is isomorphic to the direct product of ${\Pi }_{\mathrm{loc}}$ and ${\Pi }_{\mathrm{out}}$.

Therefore, we expect ${\Pi }_{\mathrm{loc}}^{\mathrm{ex}}$ to be the smallest subgroup of permutations satisfying the above conjecture.

Elementwise commutation fails for the displayed signed generators: for $L=J\otimes I_2$, $(L{P}_{\mathrm{out}}{)}_{01}=-1$ whereas $(P_{\mathrm{out}}L{)}_{01}=0$. This calculation is separate from the settling module. On supports, local bit flips and coordinate swaps compose as $x\mapsto \pi (x)\oplus a$; moving a swap past a flip relabels the flipped coordinate. This statement does not assert elementwise commutation or a direct product of the signed matrix groups.

Refutation and boundaries

The settling module uses three qubits and $A={\sum }_{j=0}^2|{\bar{e}}_j⟩⟨e_j|$, $\Delta =i{\sum }_{k=0}^2{Z}_k$. It proves $[\Delta ,A]=-2iA$ and proves that no matrix in ${\Pi }_{\mathrm{loc}}^{\mathrm{ex}}$ makes $A$ block-shift. Its affine support action sends some supported row above its column in Kronecker order, whereas every contiguous block-shift matrix is strictly lower triangular.

The same paper argument extends to $n\ge 3$ with frequency $-2(n-2)$; this uniform extension is not a theorem of the settling module. For $n=2$ that frequency is zero. The source’s Cases 1–16 are consistent with the conjecture in that dimension; the module does not re-prove that classification. The question of a smallest sufficient signed-permutation subgroup remains open.