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QuTiP 5.3 Analysis, Steady States, and Spectra

Research and API verification date: 2026-07-23. Examples target qutip==5.3.0.

Analysis starts with invariants

For every reported state, record quantitative checks before interpreting an observable:

import numpy as np

def density_audit(rho, tolerance=1e-9):
    eigenvalues = np.asarray(rho.eigenenergies(), dtype=float)
    trace = complex(rho.tr())
    return {
        "is_hermitian": bool(rho.isherm),
        "trace_error": float(abs(trace - 1.0)),
        "minimum_eigenvalue": float(eigenvalues.min()),
        "positive_within_tolerance": bool(eigenvalues.min() >= -tolerance),
    }

Also check:

• state.dims matches every observable and the declared subsystem order;
• ket norm or density-matrix trace stays stable over time;
• Hermitian observables have negligible imaginary expectation;
• populations remain within tolerance of [0, 1];
• symmetry, conserved quantity, or analytic-limit checks hold where applicable;
• numerical tolerance is smaller than the effect being claimed.

Do not repair a state by clipping eigenvalues or renormalizing unless that post-processing is part of a documented method and its impact is reported.

Expectations and uncertainty

from qutip import expect, num, variance

n_op = num(N)
mean_n = expect(n_op, rho)
variance_n = variance(n_op, rho)

For solver output, dict-form e_ops gives named result.e_data:

result = mesolve(
    H,
    rho0,
    tlist,
    c_ops=c_ops,
    e_ops={"number": n_op, "energy": H},
)
number_vs_time = result.e_data["number"]

For Monte Carlo/stochastic results, report both ensemble means and sampling uncertainty. std_expect is trajectory spread, not automatically the standard error; a simple independent-trajectory standard error scales as std / sqrt(ntraj), subject to the solver's sampling design.

Entropy, purity, and distances

from qutip import entropy_linear, entropy_vn, fidelity, tracedist

von_neumann_nats = entropy_vn(rho)          # default natural-log base
von_neumann_bits = entropy_vn(rho, base=2)
linear_entropy = entropy_linear(rho)
purity = float((rho * rho).tr().real)
state_fidelity = fidelity(rho, sigma)
trace_distance = tracedist(rho, sigma)

Always state the logarithm base. Check the QuTiP definition before comparing fidelity values with a source that may square or unsquare the quantity.

For bipartite entropy:

rho_A = rho_AB.ptrace(0)  # keep subsystem 0
entanglement_entropy = entropy_vn(rho_A, base=2)

This is an entanglement entropy only when the global bipartite state and the chosen measure meet the necessary assumptions. For mixed states, reduced-state entropy also contains classical mixture.

Common specialized functions include concurrence, negativity, entropy_mutual, and partial_transpose. Verify their supported dimensions and argument definitions in the current API before applying them.

Steady-state calculation

Current signature:

steadystate(A, c_ops=[], *, method="direct", solver=None, **kwargs)

A may be a Hamiltonian or a Liouvillian. Available high-level methods include direct, eigen, svd, power, and propagator; linear-system solver choices are separate.

from qutip import liouvillian, operator_to_vector, steadystate

rho_ss = steadystate(H, c_ops, method="direct")
L = liouvillian(H, c_ops)
residual = (L * operator_to_vector(rho_ss)).norm()

Report:

• residual norm and normalization error;
• Hermiticity and minimum eigenvalue;
• method and linear solver;
• matrix/data representation and relevant tolerances;
• whether the zero eigenvalue is unique;
• comparison with long-time evolution from more than one initial state when uniqueness matters.

A small residual does not prove uniqueness or physicality. Degenerate steady spaces require analysis of the Liouvillian nullspace and initial-state dependence.

The svd method is dense and intended for small systems. Sparse/direct methods can still be memory intensive; monitor fill-in and compare methods on a reduced model.

For periodically driven systems, a static steadystate call is generally not the desired asymptotic object. Use an appropriate periodic/Floquet approach. In QuTiP 5.3, steadystate_fourier is the current name for the specialized cosine-driven Fourier solver; steadystate_floquet is deprecated.

Two-time correlations

Current stationary/transient two-operator API:

from qutip import correlation_2op_1t, correlation_2op_2t

corr_1t = correlation_2op_1t(
    H,
    rho0,
    taulist,
    c_ops,
    a_op,
    b_op,
    solver="me",
    options={"atol": 1e-10, "rtol": 1e-8},
)

corr_2t = correlation_2op_2t(
    H,
    rho0,
    tlist,
    taulist,
    c_ops,
    a_op,
    b_op,
)

For correlation_2op_1t, the quantity is ordered according to the function's documented (A(\tau)B(0))-style convention. Do not infer operator order from a variable name.

Passing state0=None requests a steady-state initial condition only for supported constant systems with collapse operators. Compute and audit the steady state explicitly when provenance matters.

Current three-operator entry points include:

from qutip import correlation_3op, correlation_3op_1t, correlation_3op_2t

QuTiP 5.3 added max_t_plus_tau and mapping controls to selected two-time and three-operator routines. The old correlation_4op_1t recipe is not a current public API; express a four-operator quantity through the documented three-operator interfaces when mathematically appropriate, or derive a tested regression workflow.

Correlation checks:

• operator ordering and adjoints;
• transient versus stationary definition;
• normalized versus unnormalized coherence;
• regression-theorem assumptions;
• convergence of both tlist and taulist;
• tail decay before finite-window transforms.

Direct stationary spectrum

Current signature:

spectrum(H, wlist, c_ops, a_op, b_op, solver="es")

import numpy as np
from qutip import spectrum

wlist = np.linspace(-5.0, 5.0, 1001)
S = spectrum(H, wlist, c_ops, a_op, b_op, solver="es")

The function computes the Fourier transform of a steady-state correlation. Supported solver strategies include exponential-series ("es"), pseudo-inverse ("pi"), and generic linear solve ("solve").

QuTiP 5 removed public spectrum_ss and spectrum_pi. Select the strategy with the solver argument to spectrum; do not call the removed functions.

Audit:

• stationarity and steady-state uniqueness;
• angular-frequency units;
• operator order;
• whether the spectrum is symmetrized, one-sided, or normally ordered;
• negative-frequency interpretation and thermal detailed balance;
• frequency window/resolution;
• convergence across solver strategies near singular points.

FFT of a sampled correlation

Current signature:

spectrum_correlation_fft(tlist, y, inverse=False)

from qutip import spectrum_correlation_fft

frequencies, spectrum_values = spectrum_correlation_fft(taulist, corr)

Before trusting peaks:

1. require a uniform, strictly increasing taulist;
2. verify the correlation has decayed at the end of the window;
3. double the time window to test frequency resolution;
4. halve the timestep to test aliasing and high-frequency content;
5. compare window functions and disclose any window applied outside QuTiP;
6. check forward/inverse sign and normalization conventions against an analytic signal;
7. avoid interpreting zero-padding as additional physical resolution.

Use a direct spectrum calculation as a cross-check when its steady-state assumptions apply.

Liouvillian and eigenvalue diagnostics

eigenvalues = L.eigenenergies()
gap_candidates = sorted(
    (-value.real for value in eigenvalues if value.real < -1e-12)
)

Liouvillian spectra are non-Hermitian in general. Eigenvalue conditioning, degeneracy, and sparse solver targeting can make naive sorting misleading. Verify left/right eigenvector conventions and residuals before interpreting a spectral gap.

For Hamiltonians:

energies, states = H.eigenstates()
ground_energy, ground_state = H.groundstate()

Track basis and units, handle degeneracy explicitly, and sweep truncation before claiming spectral convergence.

Convergence matrix

Vary one numerical control at a time, then perform selected joint checks:

Control	Typical comparison
Hilbert cutoff	observables and boundary occupation
output grid	interpolated trace/peak/FFT quantities
atol, rtol	endpoint and maximum trajectory differences
integrator	representative observable and invariant differences
simulation duration	steady-state distance and correlation tail
frequency range/spacing	peak location, area, and edge sensitivity
trajectories	mean, uncertainty, and seed sensitivity
sec_cutoff	positivity and observable stability
HEOM depth/exponents	reduced state and target observable

Define acceptance thresholds before looking at the final comparison. Report absolute and relative differences and handle near-zero denominators explicitly.

Portable result audit

../scripts/result_audit.py reads only bounded strict JSON. It checks schema, version, finite values, monotonic time grids, population bounds, analytic reference error when available, convergence deltas, and whether assumptions, seeds, and solver stats were recorded. It does not load QuTiP result files or other Python-object serialization.

../scripts/steady_state_spectrum_planner.py produces a bounded plan for steady-state and direct/FFT spectrum checks without running a model.

Sources (verified 2026-07-23)

• Solver, correlation, spectrum, and steady-state API
• Steady-state guide
• Correlation guide
• Quantum-object API
• QuTiP 5.3.0 release notes
• QuTiP 5 changelog