Probability-Phase Mutual Information

Abstract

Quantum coherence, is an exquisitely quantum phenomenon that depends on both probability amplitudes and relative phases. Standard coherence measures quantify superposition within density matrices but cannot distinguish ensembles that produce the same mixed state through different distributions of pure states. Building on the geometric formulation of quantum mechanics, we introduce the probability-phase mutual information $I(P;Φ)$. We show that it characterizes quantum coherence at the ensemble level and that ensemble coherence systematically exceeds density-matrix coherence, thus quantifying the structure lost when averaging over pure states. Eventually, its relevance for quantum thermodynamics and deep thermalization in condensed matter physics is highlighted by explicit examples: canonical ensembles reveal temperature-dependent probability-phase correlations absent from thermal density matrices, while a non-vanishing $I(P;Φ)$ signals the breakdown of deep thermalization.

Figures (5)

• Figure 1: Probability–phase mutual information $I(P;\Phi)$ for the canonical ensemble of a qubit with Hamiltonian $H=\sigma_z+g\sigma_x$, shown as a function of inverse temperature $\beta$ and coupling strength $g$. As expected, $I(P;\Phi)$ vanishes when $g=0$. At non-vanishing coupling, correlations emerge: the mutual information grows with increasing $\beta$ (lower temperature) and peaks at small $g$, reflecting stronger probability–phase dependence in ordered regimes.
• Figure 2: Probability–phase mutual information $I(P;\Phi)$ for a Gaussian ensemble defined by the Fubini–Study distance $D_{FS}(p,\phi; p_0, \phi_0)$ from a reference state $p_0=1/2$, $\phi_0 = \pi$ as a function of the width $\sigma$. For small $\sigma$, the ensemble is sharply localized (approaching a Dirac measure) and $I(P;\Phi)$ is near zero. Increasing $\sigma$ introduces nontrivial correlations between probabilities and phases, leading to a peak in mutual information before the ensemble approaches the uniform distribution at large $\sigma$, where $I(P;\Phi)\to 0$.
• Figure 3: Probability–phase mutual information $I(P;\Phi)$ for the spiral ensemble as a function of noise strength $\delta$. For $\delta \ll 1$, the ensemble is near-perfectly correlated along a spiral curve, yielding finite mutual information. Adding noise gradually washes out these correlations, producing a monotonic decrease of $I(P;\Phi)$. As $\delta \to \pi$, the distribution approaches the uniform (Haar) ensemble and the mutual information vanishes. The information dimension $D_I \approx 0$ for all $\delta \neq 0$, indicating that the support dimension of the joint distribution matches that of the marginals.
• Figure 4: Time evolution of probability–phase mutual information $I(P;\Phi)$ during deep thermalization of a bipartite quantum system. The initial rise reflects the creation of probability–phase correlations by coherent dynamics, followed by a transient overshoot and dip consistent with two-step relaxation behavior. At late times, $I(P;\Phi)$ saturates to a finite plateau rather than vanishing, indicating that the projected ensemble remains correlated and does not reach a Haar-uniform distribution within the simulated timescales.
• Figure 5: Commutative diagram relating ensemble and density-matrix coherence. The top row shows transformations at the ensemble level: geometric dephasing $\Delta_G$ factorizes the joint probability-phase measure, followed by phase uniformization. The bottom row shows the corresponding density matrix transformations, where $\sigma$ is the intermediate state after geometric dephasing. The vertical arrows represent the expectation operator $\mathbb{E}$ mapping ensembles to their density matrix. The diagram commutes at both ends, establishing the connection between probability-phase mutual information $I(P;\Phi)$ and the relative entropy of coherence $\mathcal{C}(\rho)$.

Theorems & Definitions (9)

• Definition 1: Coherence Surplus
• Theorem 1
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