---
title: Spectral Entropy in Loschmidt Echo Analysis
url: https://www.emergentmind.com/topics/loschmidt-echo-spectra-entropy
type: topic
---

# Spectral Entropy in Loschmidt Echo Analysis

Loschmidt echo spectra entropy refers to the family of spectral and entropy-like diagnostics built from the Loschmidt amplitude, its return probability, and related quasi-local or projected variants. For a quench from an initial state $|\psi_0\rangle$, the standard amplitude and echo are $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$ and $\mathcal{L}(t)=|G(t)|^2$; in complex time one also writes $Z(z)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$ with $L(t)=|Z(it)|^2$ [2501.16995][1312.4165]. This suggests a family of constructions rather than a single standardized observable: depending on context, the relevant “spectrum” may mean an energy distribution $p_n=|c_n|^2$, a transfer-matrix spectrum, a Fourier spectrum of the time trace, a Schmidt spectrum, or a local/subsystem return-probability spectrum; correspondingly, the relevant “entropy” may mean a return rate, a Shannon or Rényi entropy of spectral weights, an energy-distribution entropy, a second Rényi entropy obtained from a projected echo, or generalized temporal entropies [1702.01475][2201.02600][2504.05237].

## 1. Core definitions and operational variants

The full-system Loschmidt echo is the probability that a quantum many-body system returns to its initial state following dynamical evolution. In thermodynamic-limit studies of quenches, a standard complex-time formulation is
$$
Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad
f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),
$$
with real-time return rate
$$
l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),
$$
and Fisher zeros defined by $Z(z_k)=0$ [1312.4165]. A closely related notation, used in several many-body settings, is
$$
r(t)= -\frac{1}{N}\ln L(t),
$$
which plays the role of a dynamical free-energy density [1702.01475].

In interacting-spin NMR, the experimentally observed quantity is often not the global overlap but a local polarization echo. For $N$ spin-$1/2$ degrees of freedom at infinite temperature plus a local excess polarization at site $1$, the Loschmidt operator is
$$
U_{LE}(t)=U_-(t/2)U_+(t/2),\qquad
U_\pm(t/2)=e^{-i(\pm H_0+\Sigma)t/2},
$$
and the local recovered signal is
$$
M_{1,1}(t)=\frac{\operatorname{tr}[S_1^z(t)S_1^z(0)]}{\operatorname{tr}[S_1^z(0)S_1^z(0)]},
$$
so that the local LE is a normalized autocorrelation function rather than a full wave-function overlap [1702.01475].

A further experimentally accessible generalization is the subsystem Loschmidt echo. For a translationally invariant initial product state, the subsystem quantity for a contiguous block of size $N$ is
$$
\mathcal{L}_N(t)=\frac{1}{L-N+1}\sum_{i=1}^{L-N+1}\left\langle\prod_{j=i}^{i+N-1}\hat P_j\right\rangle,
$$
while the entropy-extraction procedure uses the spatial average of the logarithm of local return probabilities,
$$
\ln \mathcal{L}_N(t)=\frac{1}{L-N+1}\sum_{i=1}^{L-N+1}\ln\left\langle\prod_{j=i}^{i+N-1}\hat P_j\right\rangle.
$$
Here $\mathcal{L}_N(t)$ is the probability to find the initial bitstring of length $N$ in site-resolved snapshots of the time-evolved state [2501.16995].

Projected variants extend the same logic to entropic measurements. In a bipartition $H=H_A\otimes H_B$, with two copies $B_1,B_2$ of the bath, the projected Loschmidt echo is
$$
M(t;m_1,m_2)\equiv
\left|
\langle \psi_0,B_0,m_2|
U_{A,B_1}^\dagger(t)U_{A,B_2}(t)
|\psi_0,m_1,B_0\rangle
\right|^2,
$$
and the central identity is
$$
\operatorname{Tr}\rho_A^2(t)=\sum_{m_1,m_2}M(t;m_1,m_2),\qquad
S_2(A,t)=-\ln\!\left[\sum_{m_1,m_2}M(t;m_1,m_2)\right].
$$
This makes the second Rényi entropy a Loschmidt-echo quantity [2504.05237].

A different “Loschmidt echo spectrum” appears in spin-1 Bose-Einstein condensates, where one resolves overlaps with excited states of the initial Hamiltonian:
$$
L_n(t)=|\langle \psi_n|\psi(t)\rangle|^2,
$$
with long-time average
$$
\mathcal{L}_n=\lim_{T\to\infty}\frac{1}{T}\int_0^Tdt\,L_n(t).
$$
In that setting, the survival probability is the $n=0$ component [2211.15142].

## 2. Spectral representations of Loschmidt dynamics

One spectral viewpoint starts from the decomposition of the initial state in the post-quench energy eigenbasis,
$$
|\psi_0\rangle=\sum_n c_n|E_n\rangle,\qquad p_n=|c_n|^2,
$$
with continuous spectral function
$$
S(E)=\sum_n |c_n|^2\delta(E-E_n).
$$
Then $G(t)$ is the characteristic function of the energy distribution in the diagonal ensemble, and the long-time average of the full Loschmidt echo satisfies
$$
\overline{\mathcal{L}(t)}=\sum_n |c_n|^4\equiv \mathrm{IPR}=e^{-S_2}=1/d_{\mathrm{eff}},
$$
assuming non-degenerate spectrum and long-time dephasing of off-diagonal terms [2501.16995].

A second spectral viewpoint takes the time trace itself as the object to be Fourier analyzed. In many-spin NMR, the windowed Fourier transform of a time-domain LE signal defines
$$
S(\omega)=\left|\int_0^T w(t)\,M(t)e^{i\omega t}\,dt\right|^2.
$$
In that language, a Gaussian decay of $M(t)$ at short times implies a Gaussian spectral line with width $\sim 1/T_2$, while exponential attenuation produces a Lorentzian with half-width at half-maximum $\gamma\simeq 1/T_3$. The observed Gaussian-to-exponential crossover in polarization echoes therefore maps to a crossover from Gaussian to Lorentzian spectral features [1702.01475].

In dynamical quantum phase transition theory, the relevant spectrum is the spectrum of a boundary transfer matrix. Using a Trotter–Suzuki decomposition, the Loschmidt amplitude becomes
$$
Z(z)=\lim_{M\to\infty}\operatorname{Tr}_L[\overline T]^{L/2},
$$
with dominant eigenvalue $\overline\Lambda_0(z)$ giving
$$
f(z)= -\frac12 \lim_{M\to\infty}\ln \overline\Lambda_0(z),\qquad
l(t)= -\frac12 \ln |\overline\Lambda_0(it)|^2.
$$
Non-analyticities arise when the leading eigenvalue crosses in modulus with a subleading eigenvalue,
$$
|\overline\Lambda_0(z_0)|=|\overline\Lambda_k(z_0)|,
$$
which is also the Fisher-zero matching condition [1312.4165].

Chaotic single-particle studies introduce yet another spectral object, the local density of states. For perturbed cat maps, the LDOS is defined in terms of overlaps between unperturbed and perturbed eigenstates, and its width $\sigma$ is measured as the half-distance around the mean phase that contains $70\%$ of the probability. In the weak perturbation regime, $\sigma\propto \chi^2$ and the LE decays as $M(t)\approx e^{-\Gamma t}$ with $\Gamma\approx \sigma$; for specific coherent perturbations, oscillations in $\sigma(\chi)$ imprint on the Loschmidt-echo decay rate [0908.0702].

## 3. Entropy constructions and effective dimensions

Several entropy-like quantities are used in direct connection with Loschmidt echoes, but they are not interchangeable. Some works explicitly state that they do not define a specific “entropy of the Loschmidt echo spectrum” and instead focus on the rate function or on spectral crossing conditions [1312.4165]. Other works define explicit entropies of either the LE signal or the underlying energy distribution [1702.01475][2501.16995].

Before comparing them, it is useful to distinguish the main constructions.

| Construction | Definition | Role |
|---|---|---|
| Return rate | $r(t)=-(1/N)\ln L(t)$ | Decay per degree of freedom |
| Spectral Shannon entropy | $S_{\mathrm{spec}}=-\sum_n p_n\ln p_n$ | Spread of energy weights |
| Rényi-2 entropy | $S_2=-\ln\sum_n p_n^2$ | Effective dimension $d_{\mathrm{eff}}$ |
| LE signal spectral entropy | $S_{\mathrm{spec}}=-\sum_k p_k\ln p_k$, $p_k=S(\omega_k)/\sum_jS(\omega_j)$ | Spread of Fourier power |
| Energy-distribution entropy | $S_P(t)=-\int d\epsilon\,P_t(\epsilon)\ln P_t(\epsilon)$ | Broadening in resolved LE spectrum |

In many-spin NMR, the “return rate” or dynamical free-energy density is defined by
$$
r(t)= -\frac{1}{N}\ln L(t),
$$
and a spectral entropy of the LE signal is introduced as
$$
S_{\mathrm{spec}}= -\sum_k p_k\ln p_k,\qquad
p_k=\frac{S(\omega_k)}{\sum_j S(\omega_j)}.
$$
As perturbation strength grows or as initial-state complexity increases, $M(t)$ decays faster, $S(\omega)$ broadens and flattens, and $S_{\mathrm{spec}}$ increases; in the perturbation-independent decay regime, $S_{\mathrm{spec}}$ reaches a ceiling determined by the intrinsic reversible bandwidth set by $H_0$ [1702.01475].

For full-system quenches, the most direct spectral entropy is attached to the energy-distribution weights $p_n=|c_n|^2$. One defines the spectral Shannon entropy
$$
S_{\mathrm{spec}}=-\sum_n p_n\ln p_n,
$$
the Rényi entropies
$$
S_\alpha=\frac{1}{1-\alpha}\ln\sum_n p_n^\alpha,
$$
and especially
$$
S_2=-\ln\sum_n p_n^2.
$$
Because $\overline{\mathcal L(t)}=\sum_n p_n^2$, the second Rényi entropy is directly related to the effective dimension:
$$
d_{\mathrm{eff}}=\frac{1}{\sum_n p_n^2}=e^{S_2}.
$$
This is an exact long-time relation for the full Loschmidt echo under the non-degeneracy and dephasing assumptions [2501.16995].

Spin-1 Bose-Einstein condensates use an energy-resolved entropy associated with the resolved Loschmidt echo spectrum. With
$$
P_t(\epsilon)=\sum_n L_n(t)\,\delta(\epsilon-\epsilon_n),
$$
the time-dependent entropy is
$$
S_P(t)= -\int d\epsilon\,P_t(\epsilon)\ln P_t(\epsilon),
$$
and the long-time averaged version is defined analogously from $P_{\delta\kappa}(\epsilon)=\sum_n\mathcal L_n\delta(\epsilon-\epsilon_n)$. In that system, $S_P(t)$ grows fastest and saturates earliest at the critical quench, while $S_P(\delta\kappa)$ increases below the critical quench, saturates above it, and shows a local decrease near the critical point [2211.15142].

A plausible implication is that “Loschmidt echo spectra entropy” is best viewed as a framework rather than a single formula: one entropy tracks Fourier broadening of a time trace, another tracks diagonal-ensemble occupation statistics, and another tracks energy-resolved spreading in an excited-state basis.

## 4. Entanglement, subsystem, and temporal entropy extensions

Entanglement-based extensions replace the return probability of the full state by the return properties of reduced states or by entropy measured after an echo sequence. In the quantum Chirikov standard map, the “Loschmidt echo of entanglement” is defined as
$$
G(t_r)\equiv S(2t_r),
$$
where $S$ is the bipartite von Neumann entropy of one particle after a forward–backward sequence. For noninteracting product initial states, $S(0)=0$ and a perfect time reversal returns the initial separable state at $2t_r$, whereas perturbations produce a finite entanglement echo. Numerically, one finds linear growth at small $t_r$,
$$
G(t_r)=\alpha t_r,
$$
with $\alpha=B(\Delta U)^2$ in the Fermi-golden-rule regime [2201.02600].

The same study resolves the Schmidt decomposition,
$$
|\psi(t)\rangle=\sum_i \alpha_i(t)|u_i(t)\rangle\otimes |v_i(t)\rangle,
$$
and the entanglement entropy
$$
S(t)= -\sum_i \alpha_i^2(t)\log_2\alpha_i^2(t).
$$
Its main findings concern unusual features of the entropy of entanglement and the spectrum of Schmidt decomposition under chaos, interactions, and absorption: interacting, symmetrically absorbed systems produce antisymmetric limit states with pairwise-degenerate singular values, while mixed-phase regimes produce broad Schmidt spectra and large long-lived entropy plateaus [2201.02600].

Subsystem Loschmidt echoes provide a different bridge between LE observables and thermodynamic entropy. In the long-time regime, one defines
$$
S_N^{\mathrm{eff}}\equiv -\ln \overline{\mathcal L_N}.
$$
Under ETH and for subsystem sizes larger than the correlation length in the measurement basis, the diagonal ensemble is equivalent to a grand-canonical ensemble for the subsystem, and
$$
S_N^{\mathrm{eff},(i)}\le S_N(\beta,\mu),\qquad
S_N^{\mathrm{eff},(i)}\approx S_N(\beta,\mu)-\frac{\beta^2}{2}\delta E_i^2.
$$
When $S_N^{\mathrm{eff}}$ is extensive in $N$, the slope determines the per-site accessible Hilbert-space dimension,
$$
(\dim \mathcal H_{\mathrm{eff}})^{1/L}\approx
\exp\!\left(\frac{d}{dN}\left[-\ln \overline{\mathcal L_N}\right]\right).
$$
This was used experimentally to distinguish ergodic and fragmented regimes [2501.16995].

A more formal temporal-entropy construction arises after quenches to conformal critical points. Defining reduced transition matrices from the dominant left and right eigenvectors of the spatial transfer matrix, one introduces temporal Rényi entropies
$$
S_n^{(T)}(t)=\frac{1}{1-n}\ln \operatorname{Tr}[\rho_T(t)^n].
$$
For the von Neumann case, the CFT prediction is
$$
S_{\mathrm{gen}}(t)=
s_0+\frac{i\pi c}{12}+
\frac{c}{6}\ln\!\left(\frac{2T}{\pi}\sin\frac{\pi t}{T}\right)+
\sum_j C_j\,T^{-2x_j}+\cdots,
$$
while for general $n$,
$$
S_n^{(T)}(t)=
s_n+\frac{c}{12}\left(1+\frac{1}{n}\right)
\ln\!\left(\frac{2T}{\pi}\sin\frac{\pi t}{T}\right)+
\sum_j C_j^{(n)}\,T^{-2x_j}+\cdots.
$$
These generalized temporal entropies are determined by the same transfer-matrix spectrum that controls finite-time corrections to the Loschmidt echo [2405.14706].

Projected-echo protocols tie these threads together operationally. Because
$$
\operatorname{Tr}\rho_A^2(t)=\sum_{m_1,m_2}M(t;m_1,m_2),
$$
the second Rényi entropy can be measured by a Loschmidt-echo-type forward/backward sequence without random-noise averaging, and the same framework also gives an averaged OTOC identity [2504.05237].

## 5. Irreversibility, criticality, and singular spectral structure

In many-spin NMR, Loschmidt-echo decay is the operative definition of irreversibility. The central experimental observation is that the polarization-echo decay time $T_3$ remains tied to the intrinsic reversible dynamics scale $T_2$ and saturates to $T_3\simeq 4T_2$ in the limit of vanishing perturbation strengths. This perturbation-independent decay is interpreted as a hallmark of emergent irreversibility: intrinsic many-spin dynamics can amplify arbitrarily small non-inverted interactions such that the reversible interaction scale $T_2$ dominates the irreversibility [1702.01475].

Critical phenomena enter through several distinct spectral mechanisms. In the XXZ chain, non-analyticities in the return rate are caused by crossings in the spectrum of the boundary transfer matrix, or equivalently by Fisher-zero lines pinching the imaginary axis. Crossing an equilibrium phase transition is neither necessary nor sufficient for such non-analyticities: the decisive criterion is spectral, namely the crossing in modulus of leading and subleading transfer-matrix eigenvalues [1312.4165].

Subsystem measurements reveal the same phenomenon locally. In a one-dimensional Bose–Hubbard experiment, the subsystem rate function
$$
f_N(t)= -\frac{1}{N}\ln \mathcal L_N(t)
$$
exhibits a cusp-like feature signaling a dynamical quantum phase transition at $t_c\simeq 0.9\,\hbar/J$, and the cusp is sharpened by genuine higher-order connected correlations rather than by low-order local observables alone [2501.16995].

Excited-state quantum phase transitions produce an energy-resolved variant of the same idea. In a spin-1 BEC, both the time-evolved and long-time averaged Loschmidt echo spectrum undergo a remarkable change as the system passes through the critical point of the ESQPT. At the critical quench strength, the resolved spectrum spreads rapidly, the associated energy-distribution entropy grows fastest, and the long-time averaged energy distribution shows a sharp peak at the critical energy [2211.15142].

Localized phases generate a different singular structure. In strongly localized fermionic systems quenched from a Néel or charge-density-wave state, the disorder-averaged rate function develops periodic cusp singularities at
$$
t_m=(2m+1)\pi/J,
$$
with period $2\pi/J$. These singularities are captured by an ensemble of independent two-level systems, and their harmonic content defines a line-rich spectral structure concentrated at $\omega=nJ$; interactions generate a later-time crossover to faster decay of the cusp singularities [2008.12719].

A plausible synthesis is that LE “spectra entropy” serves, across these settings, as a compact description of how singular return structures are redistributed: by transfer-matrix eigenvalue crossings in DQPTs, by critical-energy accumulation in ESQPTs, or by harmonic concentration and subsequent interaction-induced broadening in localized phases.

## 6. Computation, extraction, and measurement protocols

Time-domain traces are the common experimental input for most LE spectral-entropy analyses. For NMR polarization echoes or dynamically prepared LE protocols, the proposed workflow is: acquire the time-domain LE signal $M(t)$; remove DC offset and detrend; apply a window $w(t)$ such as Hann or exponential; compute the discrete Fourier transform $\widetilde M(\omega)$ and define $S(\omega)=|\widetilde M(\omega)|^2$; normalize the spectral weights $p_k=S(\omega_k)/\sum_jS(\omega_j)$; compute $S_{\mathrm{spec}}=-\sum_k p_k\ln p_k$; and, in dynamically prepared LE experiments, compute the return rate
$$
r(t)= -\frac{1}{N}\ln M_{1,1}(t,t_p)
$$
for $t_p\gtrsim \tau_{eq}$ [1702.01475].

Transfer-matrix methods provide the corresponding computational machinery in integrable and nonintegrable one-dimensional systems. The boundary transfer matrix can be constructed in the thermodynamic limit using the light-cone renormalization group algorithm with an iterative Lanczos solver, while general complex $z$ gives direct access to Fisher zeros rather than only real-time dynamics [1312.4165].

An alternative computational route is the linear differential equation approach to the Loschmidt amplitude. Approximating $H$ by a polynomial in $H$ yields a finite-order ODE for
$$
G(t)=\langle \psi_0|e^{-iHt}|\psi_0\rangle,
$$
with initial conditions fixed by moments $G^{(k)}(0)=(-i)^k\langle \psi_0|H^k|\psi_0\rangle$. The paper argues that for an $n$-dimensional Hamiltonian, a generic cutoff at $n^\ast=n-1$ already yields exact results, and in random-matrix tests the LDE approach often offers advantages over Taylor and cumulant expansions even when truncated at finite order [2404.04921].

Quantum-gas microscopy makes subsystem Loschmidt echoes directly measurable as probabilities of recovering the initial bitstring on contiguous local blocks. In the long-time regime this allows direct extraction of the entropy density and of the thermodynamic-limit accessible Hilbert-space dimension per site, while in the short-time regime it resolves cusp singularities associated with DQPTs [2501.16995].

Projected-echo protocols extend LE metrology to Rényi entropies. In superconducting qubit platforms and cavity-QED trapped ultracold gases, one composes forward evolution on $A\cup B_2$ with backward evolution on $A\cup B_1$, projects onto the reference states on $A$ and $B_1$, and uses the resulting projected LE sum to obtain
$$
S_2(A,t)= -\ln\!\left[\sum_{m_1,m_2}M(t;m_1,m_2)\right].
$$
Because the protocol does not rely on random-noise averaging and also connects to averaged OTOCs, it places LE-based entropy measurements within the same operational toolbox as echo spectroscopy [2504.05237].

Overall, the computational and experimental literature converges on a common principle: Loschmidt echoes convert reversibility, spectral structure, and entropic spreading into observables that can be analyzed either in time, in frequency, in energy, or on subsystems. The exact entropy attached to that analysis depends on which spectrum is being resolved and on whether the target is global overlap, local refocusing, subsystem return probability, Schmidt-weight redistribution, or diagonal-ensemble complexity.

Source: https://www.emergentmind.com/topics/loschmidt-echo-spectra-entropy