Spectral Entropy in Loschmidt Echo Analysis

• Loschmidt Echo Spectra Entropy is a framework that quantifies return probabilities and energy distributions by analyzing Loschmidt amplitudes in quantum many-body systems.
• It integrates multiple spectral representations—such as Fourier, energy, and transfer-matrix spectra—to elucidate critical phenomena, irreversibility, and dynamical phase transitions.
• Experimental and computational protocols, from NMR polarization echoes to projected echo setups, leverage these diagnostics to extract effective Hilbert-space dimensions and entropy rates.

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 $L(t)=|Z(it)|^2$ (Karch et al., 28 Jan 2025, Andraschko et al., 2013). 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 (Zangara et al., 2017, Ermann et al., 2022, Zhou et al., 7 Apr 2025).

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 $Z(z_k)=0$ (Andraschko et al., 2013). 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 (Zangara et al., 2017).

In interacting-spin NMR, the experimentally observed quantity is often not the global overlap but a local polarization echo. For $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$0 spin-$G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$1 degrees of freedom at infinite temperature plus a local excess polarization at site $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$2, the Loschmidt operator is

$G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$3

and the local recovered signal is

$G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$4

so that the local LE is a normalized autocorrelation function rather than a full wave-function overlap (Zangara et al., 2017).

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 $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$5 is

$G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$6

while the entropy-extraction procedure uses the spatial average of the logarithm of local return probabilities,

$G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$7

Here $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$8 is the probability to find the initial bitstring of length $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$9 in site-resolved snapshots of the time-evolved state (Karch et al., 28 Jan 2025).

Projected variants extend the same logic to entropic measurements. In a bipartition $\mathcal{L}(t)=|G(t)|^2$0, with two copies $\mathcal{L}(t)=|G(t)|^2$1 of the bath, the projected Loschmidt echo is

$\mathcal{L}(t)=|G(t)|^2$2

and the central identity is

$\mathcal{L}(t)=|G(t)|^2$3

This makes the second Rényi entropy a Loschmidt-echo quantity (Zhou et al., 7 Apr 2025).

A different “Loschmidt echo spectrum” appears in spin-1 Bose-Einstein condensates, where one resolves overlaps with excited states of the initial Hamiltonian:

$\mathcal{L}(t)=|G(t)|^2$4

with long-time average

$\mathcal{L}(t)=|G(t)|^2$5

In that setting, the survival probability is the $\mathcal{L}(t)=|G(t)|^2$6 component (Niu et al., 2022).

2. Spectral representations of Loschmidt dynamics

One spectral viewpoint starts from the decomposition of the initial state in the post-quench energy eigenbasis,

$\mathcal{L}(t)=|G(t)|^2$7

with continuous spectral function

$\mathcal{L}(t)=|G(t)|^2$8

Then $\mathcal{L}(t)=|G(t)|^2$9 is the characteristic function of the energy distribution in the diagonal ensemble, and the long-time average of the full Loschmidt echo satisfies

$Z(z)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$0

assuming non-degenerate spectrum and long-time dephasing of off-diagonal terms (Karch et al., 28 Jan 2025).

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

$Z(z)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$1

In that language, a Gaussian decay of $Z(z)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$2 at short times implies a Gaussian spectral line with width $Z(z)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$3, while exponential attenuation produces a Lorentzian with half-width at half-maximum $Z(z)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$4. The observed Gaussian-to-exponential crossover in polarization echoes therefore maps to a crossover from Gaussian to Lorentzian spectral features (Zangara et al., 2017).

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)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$5

with dominant eigenvalue $Z(z)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$6 giving

$Z(z)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$7

Non-analyticities arise when the leading eigenvalue crosses in modulus with a subleading eigenvalue,

$Z(z)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$8

which is also the Fisher-zero matching condition (Andraschko et al., 2013).

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 $Z(z)=\langle\Psi_0|e^{-zH}|\Psi_0\rangle$9 is measured as the half-distance around the mean phase that contains $L(t)=|Z(it)|^2$0 of the probability. In the weak perturbation regime, $L(t)=|Z(it)|^2$1 and the LE decays as $L(t)=|Z(it)|^2$2 with $L(t)=|Z(it)|^2$3; for specific coherent perturbations, oscillations in $L(t)=|Z(it)|^2$4 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 (Andraschko et al., 2013). Other works define explicit entropies of either the LE signal or the underlying energy distribution (Zangara et al., 2017, Karch et al., 28 Jan 2025).

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

Construction	Definition	Role
Return rate	$L(t)=|Z(it)|^2$5	Decay per degree of freedom
Spectral Shannon entropy	$L(t)=|Z(it)|^2$6	Spread of energy weights
Rényi-2 entropy	$L(t)=|Z(it)|^2$7	Effective dimension $L(t)=|Z(it)|^2$8
LE signal spectral entropy	$L(t)=|Z(it)|^2$9, $p_n=|c_n|^2$0	Spread of Fourier power
Energy-distribution entropy	$p_n=|c_n|^2$1	Broadening in resolved LE spectrum

In many-spin NMR, the “return rate” or dynamical free-energy density is defined by

$p_n=|c_n|^2$2

and a spectral entropy of the LE signal is introduced as

$p_n=|c_n|^2$3

As perturbation strength grows or as initial-state complexity increases, $p_n=|c_n|^2$4 decays faster, $p_n=|c_n|^2$5 broadens and flattens, and $p_n=|c_n|^2$6 increases; in the perturbation-independent decay regime, $p_n=|c_n|^2$7 reaches a ceiling determined by the intrinsic reversible bandwidth set by $p_n=|c_n|^2$8 (Zangara et al., 2017).

For full-system quenches, the most direct spectral entropy is attached to the energy-distribution weights $p_n=|c_n|^2$9. One defines the spectral Shannon entropy

$$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),$$0

the Rényi entropies

$$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),$$1

and especially

$$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),$$2

Because $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),$$3, the second Rényi entropy is directly related to the effective dimension:

$$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),$$4

This is an exact long-time relation for the full Loschmidt echo under the non-degeneracy and dephasing assumptions (Karch et al., 28 Jan 2025).

Spin-1 Bose-Einstein condensates use an energy-resolved entropy associated with the resolved Loschmidt echo spectrum. With

$$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),$$5

the time-dependent entropy is

$$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),$$6

and the long-time averaged version is defined analogously from $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),$$7. In that system, $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),$$8 grows fastest and saturates earliest at the critical quench, while $$Z(z)=\langle \Psi_0|e^{-zH}|\Psi_0\rangle,\qquad f(z)=-\lim_{N\to\infty}\frac{1}{N}\ln Z(z),$$9 increases below the critical quench, saturates above it, and shows a local decrease near the critical point (Niu et al., 2022).

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

$$l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),$$0

where $$l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),$$1 is the bipartite von Neumann entropy of one particle after a forward–backward sequence. For noninteracting product initial states, $$l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),$$2 and a perfect time reversal returns the initial separable state at $$l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),$$3, whereas perturbations produce a finite entanglement echo. Numerically, one finds linear growth at small $$l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),$$4,

$$l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),$$5

with $$l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),$$6 in the Fermi-golden-rule regime (Ermann et al., 2022).

The same study resolves the Schmidt decomposition,

$$l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),$$7

and the entanglement entropy

$$l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),$$8

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 (Ermann et al., 2022).

Subsystem Loschmidt echoes provide a different bridge between LE observables and thermodynamic entropy. In the long-time regime, one defines

$$l(t)=f(it)+f(-it)=-\lim_{N\to\infty}\frac{1}{N}\ln L(t),$$9

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

$Z(z_k)=0$0

When $Z(z_k)=0$1 is extensive in $Z(z_k)=0$2, the slope determines the per-site accessible Hilbert-space dimension,

$Z(z_k)=0$3

This was used experimentally to distinguish ergodic and fragmented regimes (Karch et al., 28 Jan 2025).

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

$Z(z_k)=0$4

For the von Neumann case, the CFT prediction is

$Z(z_k)=0$5

while for general $Z(z_k)=0$6,

$Z(z_k)=0$7

These generalized temporal entropies are determined by the same transfer-matrix spectrum that controls finite-time corrections to the Loschmidt echo (Carignano et al., 2024).

Projected-echo protocols tie these threads together operationally. Because

$Z(z_k)=0$8

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 (Zhou et al., 7 Apr 2025).

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 $Z(z_k)=0$9 remains tied to the intrinsic reversible dynamics scale $r(t)= -\frac{1}{N}\ln L(t),$0 and saturates to $r(t)= -\frac{1}{N}\ln L(t),$1 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 $r(t)= -\frac{1}{N}\ln L(t),$2 dominates the irreversibility (Zangara et al., 2017).

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 (Andraschko et al., 2013).

Subsystem measurements reveal the same phenomenon locally. In a one-dimensional Bose–Hubbard experiment, the subsystem rate function

$r(t)= -\frac{1}{N}\ln L(t),$3

exhibits a cusp-like feature signaling a dynamical quantum phase transition at $r(t)= -\frac{1}{N}\ln L(t),$4, and the cusp is sharpened by genuine higher-order connected correlations rather than by low-order local observables alone (Karch et al., 28 Jan 2025).

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 (Niu et al., 2022).

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

$r(t)= -\frac{1}{N}\ln L(t),$5

with period $r(t)= -\frac{1}{N}\ln L(t),$6. These singularities are captured by an ensemble of independent two-level systems, and their harmonic content defines a line-rich spectral structure concentrated at $r(t)= -\frac{1}{N}\ln L(t),$7; interactions generate a later-time crossover to faster decay of the cusp singularities (Benini et al., 2020).

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 $r(t)= -\frac{1}{N}\ln L(t),$8; remove DC offset and detrend; apply a window $r(t)= -\frac{1}{N}\ln L(t),$9 such as Hann or exponential; compute the discrete Fourier transform $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$00 and define $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$01; normalize the spectral weights $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$02; compute $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$03; and, in dynamically prepared LE experiments, compute the return rate

$G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$04

for $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$05 (Zangara et al., 2017).

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 $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$06 gives direct access to Fisher zeros rather than only real-time dynamics (Andraschko et al., 2013).

An alternative computational route is the linear differential equation approach to the Loschmidt amplitude. Approximating $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$07 by a polynomial in $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$08 yields a finite-order ODE for

$G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$09

with initial conditions fixed by moments $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$10. The paper argues that for an $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$11-dimensional Hamiltonian, a generic cutoff at $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$12 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 (Vogl, 2024).

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 (Karch et al., 28 Jan 2025).

Projected-echo protocols extend LE metrology to Rényi entropies. In superconducting qubit platforms and cavity-QED trapped ultracold gases, one composes forward evolution on $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$13 with backward evolution on $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$14, projects onto the reference states on $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$15 and $G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$16, and uses the resulting projected LE sum to obtain

$G(t)=\langle\psi_0|e^{-iHt}|\psi_0\rangle$17

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 (Zhou et al., 7 Apr 2025).

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.