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Structures in higher-order quantum correlations due to non-spatial symmetries

Li Sun, Chong Chen, Ren-Bao Liu

TL;DR

This work reveals deep structures in higher-order quantum correlations arising from non-spatial symmetries. It derives generalized relations from time-translation invariance that constrain CTOCs and reduce the independent high-order Wightman correlations, and it establishes a framework to relate CTOCs to OTOCs via symmetry-induced mappings. The authors show that T- and S-symmetries connect OTOCs of different ranks, enabling access to certain rank-2 OTOCs through QNS without time-reversal, and they formulate a generalized fluctuation-dissipation theorem for higher-order correlations. Demonstrating with the transverse-field Ising model, they verify the selection rules, rank-conversion relations, and the higher-order FDT, highlighting practical paths to probe information scrambling through CTOCs.

Abstract

Quantum nonlinear spectroscopy (QNS) via a quantum sensor can access $2^{n-1}$ types of $n$-th order contour-time-ordered correlations (CTOCs) arising from different orderings of quantum operators, while classical nonlinear spectroscopy can detect only one in each order. QNS and its classical counterpart have similar spatial symmetry properties, but they are expected to have characteristically different non-spatial symmetry properties since different orderings of operators can behave differently under non-spatial transformations (such as exchange of operators). Here, we investigate how higher-order correlations extracted by QNS are constrained by non-spatial symmetries, including particle-hole (C), time-reversal (T), chiral (S) symmetry, and time translation symmetry. We find that the generalized C-symmetry imposes special selection rules on QNS, and the generalized T- and S-symmetry relate CTOCs to out-of-time-order correlations (OTOCs). The time translation symmetry leads to a generalized fluctuation-dissipation theorem for the spectra of higher-order CTOCs and OTOCs. This work discloses deep structures in higher-order quantum correlations due to non-spatial symmetries and provides access to certain types of OTOCs that are not directly observable.

Structures in higher-order quantum correlations due to non-spatial symmetries

TL;DR

This work reveals deep structures in higher-order quantum correlations arising from non-spatial symmetries. It derives generalized relations from time-translation invariance that constrain CTOCs and reduce the independent high-order Wightman correlations, and it establishes a framework to relate CTOCs to OTOCs via symmetry-induced mappings. The authors show that T- and S-symmetries connect OTOCs of different ranks, enabling access to certain rank-2 OTOCs through QNS without time-reversal, and they formulate a generalized fluctuation-dissipation theorem for higher-order correlations. Demonstrating with the transverse-field Ising model, they verify the selection rules, rank-conversion relations, and the higher-order FDT, highlighting practical paths to probe information scrambling through CTOCs.

Abstract

Quantum nonlinear spectroscopy (QNS) via a quantum sensor can access types of -th order contour-time-ordered correlations (CTOCs) arising from different orderings of quantum operators, while classical nonlinear spectroscopy can detect only one in each order. QNS and its classical counterpart have similar spatial symmetry properties, but they are expected to have characteristically different non-spatial symmetry properties since different orderings of operators can behave differently under non-spatial transformations (such as exchange of operators). Here, we investigate how higher-order correlations extracted by QNS are constrained by non-spatial symmetries, including particle-hole (C), time-reversal (T), chiral (S) symmetry, and time translation symmetry. We find that the generalized C-symmetry imposes special selection rules on QNS, and the generalized T- and S-symmetry relate CTOCs to out-of-time-order correlations (OTOCs). The time translation symmetry leads to a generalized fluctuation-dissipation theorem for the spectra of higher-order CTOCs and OTOCs. This work discloses deep structures in higher-order quantum correlations due to non-spatial symmetries and provides access to certain types of OTOCs that are not directly observable.

Paper Structure

This paper contains 4 sections, 4 theorems, 25 equations, 4 figures, 1 table.

Table of Contents

  1. Generalized relations for high-order quantum correlations due to time translation invariance.
  2. Obtaining the spectra of OTOCs from those of CTOCs
  3. FDT for third-order CTOCs in T- or S-symmetric systems
  4. Third-order correlations of TFIM

Key Result

Theorem 1

When the system is C-symmetric and the physical quantities of interest are C-symmetric/anti-symmetric, i.e., ${\mathcal{C}} { B}_{k_i} ^T {\mathcal{C}} ^{-1} =\alpha_i { B}_{k_i}$ for $\alpha_i=+/- 1$, respectively, the correlations in the Wightman basis satisfy where $\tilde{\sigma}\left(i\right)\equiv\sigma\left(n-i+1\right)$ denotes the reversed order of $\sigma$, and the physical correlatio

Figures (4)

  • Figure 1: Examples of the 5-th order correlations. (a) CTOC ${\rm Tr}\left[\hat{B}_2\hat{B}_3\hat{B}_5\hat{B}_4\hat{B}_1\hat{\rho}\right]$ (i.e., ${\rm Tr}\left[\hat{B}_5\hat{B}_4\hat{B}_1\hat{\rho}\hat{B}_2\hat{B}_3\right]$ by cyclic permutation in the trace), with a time-ordered sequence of operators $\hat{B}_5\hat{B}_4\hat{B}_1\hat{\rho}$ on the forward branch of the time-contour and an anti-time ordered sequence $\hat{B}_2\hat{B}_3$ on the backward branch. (b) Rank-3 OTOC ${\rm Tr}\left[\hat{B}_4\hat{B}_2\hat{B}_3\hat{B}_1\hat{B}_5\hat{\rho}\right]$, which needs three time-contours to arrange the operators. Here we set $t_1<t_2<\cdots <t_5$.
  • Figure 2: Wightman correlations and their relations under ${\mathcal{C}}$-, ${\mathcal{T}}$-, and ${\mathcal{S}}$-transforms. (a-d) CTOCs for operators in ascending and descending time sequences. (e,f) General rank-1 OTOCs (i.e., CTOCs). (g, h) Rank-2 OTOCs. Correlations in (a/c/e/g) are correspondingly related to those in (b/d/f/h) by ${\mathcal{C}}$-transform, (a/b/e/f) to (c/d/g/h) by ${\mathcal{T}}$-transform, and (a/b/e/f) to (d/c/h/g) by ${\mathcal{S}}$-transform, as indicated by the arrows marked by C, T, and S, correspondingly.
  • Figure 3: OTOCs under ${\mathcal{T}}$- and ${\mathcal{S}}$-transforms. (a) A 5th-order CTOC is transformed to a rank-2 OTOC. When there are operators on both the first forward and the last backward branches of the time-contour, after the ${\mathcal{T}}$-transform, two empty branches need to be added to ensure the time-contour always starts forward from and ends backward to the system state, which is assumed to be stationary so that ${\rho}\left(0\right)={\rho}\left(-\infty\right)$. (b) Conversion between rank-2 OTOCs by ${\mathcal{T}}$- and ${\mathcal{S}}$-transforms. (c) Conversion between rank-2 and rank-3 OTOCs by ${\mathcal{T}}$- and ${\mathcal{S}}$-transforms.
  • Figure 4: Third-order correlations in transverse-field Ising model.(a) Third-order CTOCs $C^{\boldsymbol \eta}_{321}$ of C-anti-symmetric observables as functions of $t_3$. (b) Real and imaginary parts of Wightman correlations $W_{213}$ and $W_{21^{\prime}3}$ of T-symmetric observables as functions of $t_3$ (with $t^{\prime}_1\equiv t_3+t_2-t_1$). In the calculation, the periodic boundary condition is chosen and parameters are such that $\lambda=1.5$, $\beta=1$, $N=8$, $t_1=0$ and $t_2=2$.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof