Non-Markovian Quantum Dynamics

Non-Markovian onset at g*tau_c > 0.1

Trace distance > 1% at g*tau_c > 0.1

MET: embedding dim = Hankel rank

Tight bound on N_G (Edge B)

Memory kernel K(tau) = sum a_k exp(-mu_k tau). The Markovian regime corresponds to a delta limit (dashed). Slow modes (small Re(mu)) sustain information backflow longer.

MET Theorem (2.1, PROVED): the original system S + environment E with rational memory kernel K(tau) is exactly equivalent to Lindblad dynamics in the enlarged space S+anc. of dimension = Hankel rank of the kernel.

Coherence rho_01(t) for the three regimes. Below threshold P_c = 0.098 (g*tau_c < 0.1): exponential decay (Markovian). Above: oscillations and backflow characteristic of non-Markovian dynamics.

Geometric N_G by regime (bars) and zero-frequency Choi matrix eigenvalue (strip). Green: CP-divisible. Dark blue: strong backflow (slow mode, mean N_G = 1.67 vs 0.19 for fast mode).

H2 threshold exceeded: NM detectable (SNR > 3)

ILLUSTRATIVE MODEL - Deterministic demonstration based on thresholds reported in the publication. Not a live quantum simulation.

Weakly NM (0.1 < g*tau_c < 1)

Quantum physics · Open systems · Non-Markovian dynamics

Non-Markovian Quantum Dynamics via Nakajima-Zwanzig

Exact projection-operator formalism for open quantum systems without presupposed Lindblad structure

The Lindblad-GKSL equations (Gorini-Kossakowski-Sudarshan 1976, Lindblad 1976) form the cornerstone of open quantum dynamics under the Markov approximation. They assume the system-bath coupling is so weak and bath memory so short that the system evolution does not depend on its past history. This assumption, conceptually simple, is structurally incorrect for environments with non-negligible temporal correlations.

This work starts from the exact Nakajima-Zwanzig (NZ) equation and shows that any dynamics with rational memory kernel admits an exact Markovian representation in an enlarged space. This result the Markov Embedding Theorem (MET) unifies non-Markovianity with open systems theory under a single four-cone geometric structure.

Using the projection method (operators P and Q = 1 - P), the dynamics of subsystem S is isolated from the bath E. The exact equation for rho_S(t) is:

The kernel K_NZ(t,s) encodes the full history of system-bath correlations. In Prony (rational) form: K(tau) = sum_k a_k exp(-mu_k tau), with Re(mu_k) > 0, this kernel admits an exact finite representation. The Markov limit corresponds to K(tau) to mathcalL_GKSL * delta(tau) as tau_c to 0 (Theorem 2.2).

If K(tau) is a rational kernel of Hankel rank r,

there exists an enlarged system (S + r ancillas) such that:

Minimal enlarged space dimension = Hankel rank of kernel

The Hankel rank of the kernel determines the minimal dimension of the enlarged space required. For a single-mode kernel (K = a exp(-mu tau)), a single ancilla suffices. For r modes, r ancillas are necessary and sufficient. This result generalizes the Stinespring dilation theorem to non-Markovian dynamics.

Corollary 2.2 (numerically verified): the Hankel rank extraction algorithm identifies the minimal embedding dimension with 396/400 concordant trials in numerical validation (99% success rate).

Admissibility condition and asymptotic Markovian rate

The operator K(0) = integral_0^inf K(tau) dtau is the asymptotic Markovian rate. The admissibility condition requires K(0) to be a valid GKSL semigroup generator. When this condition is satisfied, NZ dynamics correctly converges to the Markov limit as tau_c to 0.

The measure N_G quantifies the reverse information flow (backflow) from the bath to the system. Formally: N_G = sup_S_0 (1)/(V(0)) integral [dV/dt]_+ dt, where V(t) is the trace volume of an initial state ensemble S_0 and [x]_+ = max(0,x) denotes the positive part.

N_G corrects the Lorenzo-Plastina-Paternostro (2013) measure which conflated P-divisibility and volume-monotonicity. The four-cone structure (Edge A of the treatise, proved in Theorem 5.1) establishes: semigroup contained in CP-div contained in P-div contained in volume-monotone contained in all CPTP dynamics. N_G is faithful to CP-divisibility, not P-divisibility.

Theorem 3.3: CP-divisibility via Choi matrix

Dynamics is CP-divisible if and only if:

C_K(tau): instantaneous Choi matrix of kernel K(tau)

Necessary and sufficient condition (no approximation)

The tight bound on N_G (Edge B, 400 validation trials): N_G le C * sum_k | a_k| / Re(mu_k), with C ~ 1.55 (tightest empirical bound). Slow modes (Re(mu_k) small) generate more backflow: mean N_G = 1.67 for slow modes vs 0.19 for fast modes. N_G decreases monotonically with min_k Re(mu_k).

G_int = integral | rho_exact(t) - rho_Markov(t)|_1 dt quantifies the cumulative error of the Markov approximation over the full trajectory. G_int is small when gtau_c < 0.1 (Markovian regime) and becomes significant above threshold P_c = 0.098. For pure dephasing (exact case of Theorem 3.5), gamma(t) < 0 implies exactly non-Markovian.

The paper formulates 10 explicitly falsifiable hypotheses with numerical thresholds, testable on superconducting qubits (IBM, Google, Rigetti). Each hypothesis specifies the target platform, experiment type and rejection threshold. The six main hypotheses are summarized below; H7-H10 concern quantum ensembles, model discrimination and scalability.

H1: non-Markovian onset at gtau_c > 0.1 (P_c = 0.098 numerically validated), SNR > 3, p < 0.01

H2: trace distance > 1% when gtau_c > 0.1 (experimentally observable threshold)

H3: N_G distinguishes CP-div from P-div with DeltaBIC > 10 (confirmed DeltaBIC = 42)

H5 (MET): embedding dimension = Hankel rank; Corollary 2.2: 396/400 concordant trials

H6 (Edge B): tight bound N_G le C * sum | a_k| / Re(mu_k), C ~ 1.55, 0 violations / 400 trials

Theorem 5.1: four-face unification (PROVED/VERIFIED-NUM)

Theorem 5.1 establishes the four-cone structure of quantum divisibility. It is partially proved analytically and partially verified numerically over 597 random trials.

H1 confirmed: 98% of trials with SNR = 8.3 show onset at P_c = 0.098

H2 confirmed: P_c = 0.098, trace distance > 1% confirmed beyond threshold

H3 confirmed: DeltaBIC = 42 (well above the threshold of 10)

H4 confirmed: DS = 0.15 bits (> 0.1 bits required)

H5 partial (85%): R = 1.45; non-concordant cases correspond to kernels near rank degeneracy

H6 confirmed: d_L = 0.23 (zero violation distance), C ~ 1.55 over 400 trials

Corollary 2.2: 396/400 concordant trials for Hankel rank extraction

Theorem 5.1 face (iv): 0 violations in 597 random trials

The numerical validation code is published on GitHub: https://github.com/hopenmind/metcore. It includes Python implementations of the MET algorithm, Hankel rank extraction, N_G computation, Choi matrix, and all 10 hypothesis testing protocols. The test suite reproduces all reported validation results.

This paper is published open access on Zenodo under CC BY-NC-SA 4.0 license. Validation code is available under MIT license on GitHub.

Open access · CC BY-NC-SA 4.0 · HOPE 'N MIND SASU · 2026