Irreversibility-Susceptibility Method (ISM)

• The irreversibility-susceptibility method (ISM) is a set of experimental and theoretical techniques that quantify a system’s tendency to lose information and become irreversible.
• It uses ac-susceptibility in superconductors and OTOC measurements in quantum systems to extract key parameters like depinning energy and critical current densities.
• ISM’s robust and scalable protocol enhances precision in mapping phase diagrams and analyzing quantum scrambling while overcoming material and computational limitations.

The irreversibility-susceptibility method (ISM) refers to a class of precise experimental and theoretical techniques for quantifying irreversibility and information loss in physical systems. ISM appears in distinct contexts: (1) in the study of vortex dynamics and flux-creep in superconductors via ac-susceptibility, and (2) in the measurement of out-of-time-order correlators (OTOCs) as a probe of quantum information scrambling. The common principle is to extract a quantitative measure—interpreted as a “susceptibility” of the system to irreversibility—using sensitive response or recovery-based protocols. ISM has had significant impact in both condensed matter and quantum information science, enabling fine-grained mapping of phase diagrams or scrambling dynamics with minimal system requirements.

1. Irreversibility–Susceptibility Method in Superconductors

In the context of type-II superconductors, ISM is a powerful tool for mapping the irreversibility line $H_{\mathrm{irr}}(T)$ and extracting the depinning energy scale $U_0(H)$ and the zero-field, zero-temperature critical current density $J_{c0}(0)$ from ac-susceptibility measurements. The prototypical implementation, presented by Prando et al. for optimally doped SmFeAsO$_{0.8}$F$_{0.2}$, employs un-oriented polycrystalline powder samples in a SQUID-based ac-susceptometer with the following protocol (Prando et al., 2011):

• Ac drive field $H_{\mathrm{ac}}$ swept over $0.0675 \times 10^{-4}\,\mathrm{T}$ to $1.5 \times 10^{-4}\,\mathrm{T}$, parallel to a dc field $H$ up to 5 T.
• Excitation frequency $\nu_m$ scanned from 37 Hz to 1488 Hz.
• Temperature $T$ swept through $T_c \sim 52\,\mathrm{K}$ at fixed $H$ and $\nu_m$, measuring $\chi'(T)$ (real part) and $\chi''(T)$ (imaginary part) with $\Delta T \lesssim 0.1\,\mathrm{K}$ resolution.
• Phase diagram construction: The dissipation-peak temperature $T_p$ (maximum in $\partial\chi'/\partial T$) defines the irreversibility point; $H_{\mathrm{irr}}(T)$ is mapped by plotting $H$ versus $T_p$ at lowest $\nu_m$.

Experimentally, $T_p$ is independent of $H_{\mathrm{ac}}$, shifts down with increasing $H$, and is lower for slower $\nu_m$. The region below $H_{\mathrm{irr}}(T)$ exhibits nonzero $\chi''$ (irreversible, glassy vortex dynamics); above lies a reversible regime.

2. Thermally Activated Flux-Creep and Extraction of Pinning Energies

Near $T_p$, the flux-line lattice's characteristic relaxation time $\tau_p$ is resonant with the drive: $2\pi\, \nu_m \tau_p = 1$. Assuming thermally activated creep,

$$\tau(T, H) = \tau_0 \exp\left[ \frac{U_0(H)}{k_B T} \right]$$

with $\tau_0 \sim 10^{-12}$–$10^{-10}$ s. This yields

$$\ln\left(\frac{\nu_m}{\nu_0}\right) = -\frac{U_0(H)}{k_B T_p}$$

enabling $U_0(H)$ extraction from the slope of $\ln \nu_m$ vs $1/T_p$ at fixed $H$. Above $H \sim 0.5\,\mathrm{T}$, $U_0(H) \propto 1/H$, reflecting a crossover from single-vortex to bundle creep. This scaling agrees with Tinkham’s two-fluid model. The explicit proportionality reads:

$$\frac{U_0(T,H)}{t} = \left[ K \frac{J_{c0}(0)}{H} \right] g(t), \quad t = T/T_c,\, g(t) = 4(1-t)^{3/2}$$

Plotting $[U_0 / (t g(t))]$ vs $1/H$ gives a line whose slope yields $J_{c0}(0)$, found to be $2.25(5)\times 10^7\, \mathrm{A/cm}^2$ in the referenced system (Prando et al., 2011).

3. Experimental Strengths and Applicability in Vortex Physics

Key strengths of this ISM approach are:

• Nearly isothermal determination of $U_0(H)$ (temperature stability within $1\,\mathrm{K}$).
• Sharp separation of intragranular losses by using the $\partial\chi'/\partial T$ criterion, avoiding ambiguities from intergranular weak-link effects.
• Use of readily prepared powder samples—eliminating the need for large single crystals—and the ability to apply the protocol across a wide class of high-$T_c$, highly anisotropic superconductors.

Limitations include the empirical nature of the $1/H$ scaling and the two-fluid/Arrhenius framework, with manysample/grain effects only treated phenomenologically.

ISM protocols of this form are widely adopted for mapping $H_{\mathrm{irr}}(T)$ and extracting $J_c$ in cuprate and iron-based superconductors, establishing a general phenomenology for type-II vortex matter (Prando et al., 2011).

4. ISM for Out-of-Time-Order Correlators and Quantum Scrambling

The ISM was recently generalized to probe quantum information scrambling via OTOC measurement, as implemented by Emori and Tajima on quantum hardware (Emori et al., 27 Dec 2025). In this context, ISM defines the OTOC $C_\beta(t)$ as the “irreversibility susceptibility”:

$$C_\beta(t) = \lim_{\theta\to 0} \frac{1 - \langle \sigma^x_Q \rangle_{\text{out}}}{2\theta^2}$$

where $\sigma^x_Q$ is the $x$-component of an ancilla qubit coupled weakly to a thermalized system via $V$ with strength $\theta$, with the ancilla prepared in $\vert +\rangle$. The OTOC is thus cast as a second-order “susceptibility” to a weak perturbation under time-reversed evolution.

The ISM protocol consists of:

1. Initialization: System register $S$ in $\rho_\beta$, ancilla $Q$ in $\vert +\rangle$.
2. Weak coupling: Apply $U_V(\theta) = \exp[-i\theta Z_Q \otimes V_S]$.
3. Scrambling: Forward $U(t)$, operator $W$ at $t=0$, backward $U^\dagger(t)$.
4. Recovery coupling: Apply $U_V^\dagger(\theta)$.
5. Measurement: Read out $\sigma^x_Q$, average over many repeats.

This circuit realizes the channel-recovery error whose $\theta\to 0$ susceptibility matches the OTOC-defined quantum irreversibility (Emori et al., 27 Dec 2025, Emori et al., 2023).

5. Comparative Performance and Implementation in Quantum Systems

The ISM was benchmarked against the rewinding time method (RTM) and weak-measurement method (WMM) for measurement of OTOC in XXZ spin chains on the Quantinuum reimei emulator (Emori et al., 27 Dec 2025). The following features were notable:

• Reduced overhead: Only a single ancilla and a pair of weak-coupling gates, with no need for ancillary time-reversal controls.
• Superior bias robustness: ISM matched matrix-product predictions for OTOCs with deviations $<10\%$ across interaction strengths $\Delta$.
• Shot-noise sensitivity: Signal is suppressed by $\theta^2$; many measurements are required for statistical significance.
• Scalability: Thermal state preparation via variational quantum algorithms remains the computational bottleneck for large systems.

RTM exhibits systematic underestimation at late times for larger $\Delta$, while WMM shows overestimation, highlighting the noise resilience of ISM.

Protocol	Ancillas required	Unique limitation
ISM	1	Strong shot noise, high state prep cost
RTM	1	Time-reversed evolution, coherence errors
WMM	4	Crosstalk in repeated weak coupling

6. Related Irreversibility Quantifiers: Subjectivity Measures

While not named “ISM,” certain irreversibility metrics directly generalize ISM-like susceptibility to stochastic and quantum channels. Liu, Aw, and Scarani introduced “Bayesian subjectivity” as the average difference in inverted channels given variation of prior:

$$I_c^s(\Phi) = \iint \left\| \hat\Phi_{\pi_1} - \hat\Phi_{\pi_2} \right\|_\lambda d\pi_1 d\pi_2,\qquad I_q^s(F) = \iint \left\| \widehat{F}_{\gamma_1} - \widehat{F}_{\gamma_2} \right\|_\diamond d\gamma_1 d\gamma_2$$

This “susceptibility” vanishes for reversible maps, is maximal for erasures, and rises jointly with volume contraction and purification of fixed points (Liu et al., 15 Mar 2025). While the context differs, the operational logic—measuring system “fragility” to inversion and recovery—parallels ISM’s core principle.

7. Broader Impact and Applicability

The ISM framework has demonstrated utility in probing:

• Vortex melting, irreversibility lines, and critical current determination in superconducting materials (Prando et al., 2011).
• Quantum information scrambling and operator spreading in many-body spin chains, offering efficient and robust OTOC estimators (Emori et al., 27 Dec 2025).
• Theoretical characterizations of irreversibility and information loss in channels and stochastic processes, where susceptibility to recovery defines meaningful physical metrics (Liu et al., 15 Mar 2025).

ISM’s design—exploiting controlled, weak perturbations and recovery analysis rather than brute-force time-reversal—renders it widely adaptable and experimentally feasible for both condensed matter and quantum information settings. A plausible implication is the emergence of ISM (in both nomenclature and concept) as a standard approach to quantifying and operationalizing irreversibility in next-generation quantum devices and strongly correlated materials.