bibkey: andreadakis2024scrambling authors: Faidon Andreadakis; Emanuel Dallas; Paolo Zanardi year: 2024 title: “Long-time Quantum Scrambling and Generalized Tensor Product Structures” doi: 10.1103/PhysRevA.109.052424 url: https://arxiv.org/abs/2312.13386v2 claim: “For an algebra A = direct sum over J = 1..d_Z of 1_{n_J} tensor L(C^{d_J}) on a space of dimension d = sum_J n_J d_J, Proposition 1 (ii) gives the long-time average of the A-OTOC of a Hamiltonian satisfying the non-resonance condition as 1 - (1/d) sum over X in {A, A’} of [Tr(R^(0),X R^(1),X’) - (1/2) Tr(R_D^(0),X R_D^(1),X’)], with R^(0),X_lk = ||P_X(|phi_k><phi_l|)||_2^2 and R^(1),X_kl = <P_X(Pi_k), P_X(Pi_l)> for the eigenbasis phi_k. Proposition 2 proves that the product eigenbasis minimizes it over Hamiltonians that are block diagonal in the sectors, with minimum 1 - (sum_J d_J + sum_J n_J - d_Z)/d, and over all such Hamiltonians when d_Z = 1. The Conjecture of Section IV states that this value is the minimum over the whole family of isomorphic algebras, attained by the algebra whose distinguished basis is the eigenbasis.” strata_touched:

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

Long-time Quantum Scrambling and Generalized Tensor Product Structures

Faidon Andreadakis, Emanuel Dallas, Paolo Zanardi, Phys. Rev. A 109, 052424 (2024), arXiv:2312.13386v2 (quant-ph). Quotations are from the arXiv v2 source, with the source’s macros \mathds, \Tr, \ket, \bra and \cref kept as printed.

The sector structure (Eqs. (2) and (3)):

$\mathcal{H}={⨁}_{J=1}^{d_Z}{\mathcal{H}}_J,{\mathcal{H}}_J\cong {\mathbb{C}}^{n_J}\otimes {\mathbb{C}}^{d_J}.$

and

$\mathcal{A}\cong {⨁}_{J=1}^{d_Z}{\mathbb{I}}_{n_J}\otimes \mathcal{L}({\mathbb{C}}^{d_J}),{\mathcal{A}}^{\prime }\cong {⨁}_{J=1}^{d_Z}\mathcal{L}({\mathbb{C}}^{n_J})\otimes {\mathbb{I}}_{d_J}.$

The projections (Section II):

In terms of \cref{structure_A}, we have \mathbb{P}_\mathcal{A} (\cdot )=\oplus_J \frac{\mathds{1}_{n_J}}{n_J} \otimes \Tr_{n_J}(\cdot ) and, similarly, \mathbb{P}_{\mathcal{A}^\prime} (\cdot )=\oplus_J \Tr_{d_J}(\cdot )\otimes \frac{\mathds{1}_{d_J}}{d_J}.

Proposition 1 (ii), Eq. (6), with the line break of the source display removed:

\overline{G_{\mathcal{A}}(\mathcal{U}_t)}^{NRC} = 1-\frac{1}{d}\left(\sum_{\mathcal{X}=\{\mathcal{A},\mathcal{A}^\prime\}}\Tr( R^{(0),\mathcal{X}}R^{(1),\mathcal{X}^\prime} ) - \right. \left. -\frac{1}{2}\Tr( R_D^{(0),\mathcal{X}}R_D^{(1),\mathcal{X}^\prime} )\right),

where, for algebra $\mathcal{X}$, we define $R_{lk}^{(0),\mathcal{X}}:={\Vert {\mathbb{P}}_{\mathcal{X}}\left(|{\varphi }_k⟩⟨{\varphi }_l|\right)\Vert }_2^2$, $R_{kl}^{(1),\mathcal{X}}:=⟨{\mathbb{P}}_{\mathcal{X}}\left({\Pi }_k\right),{\mathbb{P}}_{\mathcal{X}}\left({\Pi }_l\right)⟩$, and for matrix $M$, $M_D:=diag(M)$.

The display prints a minus sign at the end of its first line and another at the start of its second. Read with a single minus sign, Eq. (6) at the product basis equals the minimum of Proposition 2, Eq. (8); with a plus sign it would lie below that minimum by $2d_{\mathcal{Z}}/d$.

The duality (Section IV):

Note that, in general, for \cref{def_aotoc}, $G_{\mathcal{A}}(\mathcal{W}{\mathcal{U}}_t{\mathcal{W}}^{†})=G_{{\mathcal{W}}^{†}(\mathcal{A})}({\mathcal{U}}_t)$, where $\mathcal{W}$ is some unitary channel. This means that optimizing over unitary families of Hamiltonians is dual to optimizing over unitary families of algebras.

The Conjecture (Section IV, unnumbered):

Let $H$ be an NRC Hamiltonian and ${\mathcal{A}}_f$ be the equivalence class of isomorphic algebras of the form \mathcal{A}\cong \oplus_{J=1}^{d_\mathcal{Z}} \mathds{1}_{n_J} \otimes \mathcal{L}(\mathbb{C}^{d_J} ). Then, the algebra corresponding to a gTPS (\cref{structure_H}) for which the Hamiltonian eigenstates have the form $\{|{\psi }_J⟩\otimes |{\varphi }_J⟩|J=1,\dots d_{\mathcal{Z}};{\psi }_J=1,\dots ,n_J;{\varphi }_J=1,\dots ,d_J\}$ minimizes the $\mathcal{A}$-OTOC LTA over the family of isomorphic algebras. The minimum value is ${\overline{G_{\mathcal{A}}}}_{min}^{NRC}=1-\frac{1}{d}\left({\sum }_J{d}_J+{\sum }_J{n}_J-d_{\mathcal{Z}}\right).$

Section IV A (numerical evidence):

We make use of the algebra-Hamiltonian duality of the NRC LTA by fixing an algebra $\mathcal{A}$ and searching over the space of Hamiltonians for a violation of the conjecture. As seen in \cref{NRC}, the NRC LTA depends only on the Hamiltonian eigenbasis, which can be represented by a unitary matrix with eigenstates as the columns.

The encoding fixes $\mathcal{A}$ in its distinguished basis, indexes ${\mathbb{C}}^d$ by the pairs $(J,(a,b))$ with $a<n_J$, $b<d_J$, takes the eigenbasis to be the columns of a unitary $U$, and reads the minimum as the least element of the values of Eq. (6) over the unitary group; the number of sectors $d_{\mathcal{Z}}$ is written $r$.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.109.052424 (Phys. Rev. A 109, 052424, 2024).
• URL: https://arxiv.org/abs/2312.13386v2 (v2, 2024-01-31, the latest version; source main_revised.tex, md5 f0749ffbb93cd5151d595e05150e95ec): Eqs. (2)–(3) (l. 303–309), the projections (l. 311), Proposition 1 (ii) (l. 353–358), the duality (l. 406), Proposition 2 (l. 407–419), the Conjecture (l. 425–430) and Section IV A (l. 547–549). The conjecture environment is unnumbered; the equations are numbered in one sequence, so Eq. (6) is \label{NRC}.