1 Introduction↩︎

Quantum coherence is one of the central resources that distinguishes quantum mechanics from classical physics. It underlies the advantage of many quantum technologies, ranging from computation to metrology, and has been formalized through the resource theory of coherence [1], [2]. In this standard framework, coherence is quantified at the level of density matrices, with measures such as the relative entropy of coherence \(\mathcal{C}(\rho)\) capturing the degree to which a quantum state is a superposition in a chosen basis.

While powerful, this perspective overlooks a different layer of structure: the correlations that can exist within a collection of pure states—-an ensemble—-whose average is the density matrix. While two ensembles may have the same density matrix, they can differ sharply in how their pure states are distributed. Standard coherence measures do not distinguish such cases, leaving ensemble-level coherence phenomena outside their scope.

Geometric quantum mechanics (GQM) [3]–[20] provides a natural setting in which to formulate a theory of coherence for ensembles. In a chosen basis, the complex projective space of pure state \(\mathbb{C}P^{D-1}\) is coordinatized by probabilities \(p_k\) (encoding measurement-accessible statistics in the chosen basis) and phases \(\phi_k\) (carrying the complementary, measurement-inaccessible information). This accessible–inaccessible dichotomy suggests a fundamental question: how much information about the phases is statistically tied to the probabilities across an ensemble?

In this work we answer this question by introducing the probability–phase mutual information \(I(P;\Phi)\) and showing that it is a measure of coherence for ensembles of pure states. Our first result establishes that \(I(P;\Phi)\) satisfies all six axioms [2] of coherence measures (non-negativity, monotonicity, strong monotonicity, convexity, additivity, and pure-state normalization), thereby qualifying as a coherence measure for ensembles. Our second result establishes the relationship to the standard relative entropy of coherence \(\mathcal{C}(\rho)\) through the coherence surplus, revealing how ensembles, and their coherence, relate to density-matrix coherence.

The paper is organized as follows. In Section 2 we provide a brief summary of geometric quantum mechanics and establish the mathematical aspects of probability-phase mutual information. In Section 3 we develop the resource theory of coherence for ensembles, defining incoherent states and free operations, and proving that \(I(P;\Phi)\) satisfies all standard coherence axioms. In Section 4 we present analytical and numerical examples demonstrating the behavior of \(I(P;\Phi)\) across diverse geometric quantum states and physical situations. In Section 5 we discuss the relationship between \(I(P;\Phi)\) and the standard relative entropy of coherence \(\mathcal{C}(\rho)\). We then introduce the coherence surplus and highlight their complementary operational meanings. Finally, Section 6 concludes with a summary of results and future directions. Detailed proofs of the coherence axioms and the non-negativity of the coherence surplus are provided in Appendix 7 and 8, respectively.

2 Probability-Phase Mutual Information↩︎

Geometric quantum mechanics (GQM) [3]–[20] provides a differential-geometric framework for quantum systems. For a quantum system with Hilbert space \(\mathcal{H}\) of dimension \(D\), the space of pure states is the complex projective space \(\mathcal{P}(\mathcal{H}) \sim \mathbb{C}P^{D-1}\). This identification removes the redundancy inherent in the standard formulation—vectors differing only by normalization and global phase represent the same physical state.

The complex projective space \(\mathbb{C}P^{D-1}\) is a Kähler manifold, equipped with both a Riemannian geometry–providing the Fubini-Study metric–and a symplectic geometry–providing a closed non-degenerate symplectic two-form[21], [22]. The existence of a symplectic structure is the hallmark of Hamiltonian mechanics, and it underlies a deep geometric parallel between classical and quantum mechanics that, while known [3], [5], [6], [10] goes often underappreciated.

In this setting, probability and phase coordinates arise naturally after a basis’ choice \(\{\ket{e_k}\}_{k=0}^{D-1}\) in the Hilbert space \(\mathcal{H}\). They provide a natural parametrization of \(\mathbb{C}P^{D-1}\) through \(Z^k = \sqrt{p_k} e^{i\phi_k}\), where \(\ket{\psi} = \sum_{k=0}^{D-1} Z^k \ket{e_k}\) and we set \(\phi_0 = 0\) without loss of generality  [23]. A point in the manifold of quantum states is therefore identified by \(p = (p_1, \ldots, p_{D-1}) \in \Sigma_{D-1}\), where \(\Sigma_{D-1}\) is the \((D-1)\)-dimensional probability simplex defined by \(p_k \geq 0\) and \(\sum_{k=0}^{D-1} p_k = 1\) (with \(p_0 = 1 - \sum_{k=1}^{D-1} p_k\)), and \(\phi = (\phi_1, \ldots, \phi_{D-1}) \in \mathbb{T}_{D-1}\), the \((D-1)\)-dimensional torus with \(\phi_k \in [0, 2\pi)\). This choice diagonalizes the symplectic structure and reveals the canonical nature of \((p,\phi)\) coordinates.

Ensembles are introduced by promoting the pure state of a quantum system to a random variable \(\Psi\), distributed according to some probability measure \(\mu_\Psi\) on \(\mathcal{P}(\mathcal{H})\). We call \(\mu_{P,\Phi}\) a geometric quantum state [23]. While the measure is invariant under changes of coordinates, here we adopt a coordinate-specific notation that better suites our goal to address the statistical correlations between probabilities and phases: we introduce random variables \((P,\Phi)\), with \(\Psi(P,\Phi) = \sqrt{P}e^{i\Phi}\) and realizations \((p,\phi) \in \mathbb{C}P^{D-1}\), distributed according to \(\mu_{P,\Phi}\).

When \(\mu_{P,\Phi}\) is absolutely continuous with respect to the Fubini-Study volume measure \(dV_{FS}\) on \(\mathcal{P}(\mathcal{H})\), it can be expressed through a probability density \(q(p,\phi)\) such that: \[\begin{align} \mu_{P,\Phi}(A) = \int_A q(p,\phi) \, dV_{FS} \nonumber \end{align}\] for any measurable set \(A \subseteq \mathcal{P}(\mathcal{H})\).

In probability-phase coordinates, the Fubini-Study volume element factorizes as \[\begin{align} dV_{FS} = \prod_{k=1}^{D-1} \frac{dp_k \, d\phi_k}{2} \label{eq:fubini-study-element} \end{align}\tag{1}\] This is formally identical to the position/momentum factorization of the phase space volume element \(\prod_i dp_i \, dq_i\) in classical statistical mechanics, where \((p,q)\) are canonically conjugate momentum-position pairs.

Given a joint measure \(\mu_{P,\Phi}\), the marginals can be obtained through partial integration: \[\begin{align} \mu_P(B) &= \int_{B \times \mathbb{T}_{D-1}} \mu_{P,\Phi}(dp,d\phi) \nonumber \\ \mu_\Phi(C) &= \int_{\Sigma_{D-1} \times C} \mu_{P,\Phi}(dp,d\phi) \nonumber \end{align}\] where \(B \subseteq \Sigma_{D-1}\) and \(C \subseteq \mathbb{T}_{D-1}\) are subsets of the probability simplex and the phase hyper-torus. A product measure \(\mu_P \cdot \mu_\Phi\) represents the case where probabilities and phases are statistically independent.

To properly define the probability-phase mutual information in a geometric quantum state, we employ the scaling framework originally developed by Renyi [24] and extended to quantum systems in Ref. [25]. Calling \(Z^\epsilon\) a coarse-graining of the state space at scale \(\epsilon\), and \(H(Z^\epsilon)\) the Shannon entropy of the resulting discrete distribution, one has \[H(Z^\epsilon) \sim_{\epsilon \to 0} \mathfrak{D}(-\log \epsilon) + H_G \label{eq:entropy95scaling}\tag{2}\] where \(\mathfrak{D}\) is the quantum information dimension and \(H_G\) is the geometric entropy. For measures with support on a finite number of points \(\mu = \sum_{k=1}^{N}q_k \delta_{\chi_\alpha}\) we have \(\mathfrak{D}=0\) and \(H_G = - \sum_{\alpha=1}^N q_\alpha \log q_\alpha\). Measures with full support on \(\mathcal{P}(\mathcal{H})\) exhibit \(\mathfrak{D} = 2(D-1)\), and the geometric entropy reduces to the differential entropy: \[\begin{align} H_{2(D-1)}[\mu_{P,\Phi}] = -\int_{\mathcal{P}(\mathcal{H})} \!\!\! q(p,\phi) \log q(p,\phi) \, dV_{FS}~.\nonumber \end{align}\]

The marginal entropies follow by applying the scaling method defined in Ref.[25] to the marginal measures. Defining \(Z^\epsilon_p\) and \(Z^\epsilon_\phi\) to be coarse grained variables in \(\Sigma_{D-1}\) and \(\mathbb{T}_{D-1}\) respectively, we have \[\begin{align} & H(Z^\epsilon_P) \sim_{\epsilon \to 0} \mathfrak{D}_P(-\log \epsilon) + H_P \\ &H(Z^\epsilon_{\Phi}) \sim_{\epsilon \to 0} \mathfrak{D}_\Phi(-\log \epsilon) + H_{\Phi} \end{align}\] This leads to the definition of mutual information dimension and probability-phase mutual information from the resulting scaling: \[\begin{align} I_\epsilon(P;\Phi) & := H(Z^\epsilon_P) + H(Z^\epsilon_{\Phi}) - H(Z^{\epsilon}_{P,\Phi}) \tag{3} \\ & \sim_{\epsilon \to 0} \mathfrak{D}_I (-\log \epsilon) + I(P;\Phi)~ \tag{4} \end{align}\] where \(\mathfrak{D}_I \equiv \mathfrak{D}_P+\mathfrak{D}_\Phi - \mathfrak{D}_{P,\Phi}\) and, similarly, \(I(P,\Phi) \equiv H_P + H_{\Phi} - H_{P,\Phi}\). The mutual information dimension \(\mathfrak{D}_I\) measures the dimensionality surplus in the support of \(\mu_P \mu_{\Phi}\) versus the true joint \(\mu_{P,\Phi}\) while \(I(P;\Phi)\) is the probability-phase mutual information: it measures the statistical dependence between probability and phase variables in the ensemble described by \(\mu_{P,\Phi}\). Equivalently, it can be expressed as the relative entropy: \[\begin{align} I(P;\Phi) = D_{\mathrm{KL}}(\mu_{P,\Phi} || \mu_P \cdot \mu_\Phi) \label{eq:mutual95information95relative95entropy} \end{align}\tag{5}\] quantifying how far the joint distribution deviates from statistical independence.

This concludes the mathematical exposition of geometric quantum states and their probability-phase mutual information. Before establishing the formal resource theory, we address a key conceptual question in the next subsection.

2.1 Why probability-phase mutual information?↩︎

We now address a fundamental question: why is \(I(P;\Phi)\) the natural quantity for characterizing coherence at the ensemble level?

Quantum coherence fundamentally arises from superposition—the ability of quantum states to exist in combinations of basis states with both definite probability amplitudes and relative phases. In the geometric framework, these two aspects are separated into probability coordinates \(P\) (accessible through measurements in the reference basis) and phase coordinates \(\Phi\) (inaccessible to such measurements).

When probabilities and phases are statistically independent—that is, when \(\mu_{P,\Phi} = \mu_P \cdot \mu_\Phi\)—knowledge of the probabilities yields no information about the phases. This represents the absence of systematic correlations between accessible and inaccessible degrees of freedom. Conversely, when \(I(P;\Phi)\) is large, phases and probabilities are highly correlated.

Consider a simple example: an ensemble of qubit states \(\ket{\psi} = \sqrt{1-p}\ket{0} + \sqrt{p}e^{i\phi}\ket{1}\). If \(\phi\) is chosen uniformly regardless of \(p\), then \(I(P;\Phi)=0\) and the ensemble exhibits no systematic phase structure. If instead \(\phi = 2\pi p + \epsilon\) (with small noise \(\epsilon\)), then probabilities and phases are linearly correlated, yielding finite \(I(P;\Phi)\) and reflecting structured interference patterns across the ensemble.

This motivates our central question: does \(I(P;\Phi)\) satisfy the formal requirements of a coherence measure? In the next section, we establish this by proving that \(I(P;\Phi)\) satisfies six axioms, from resource theory [2], that qualify it as a measure of coherence for ensembles of pure states.

3 Resource theory of coherence for ensembles of pure states↩︎

Resource theories [26] provide a rigorous framework for quantifying physical resources that cannot be freely created under a specified set of constraints. The resource theory of coherence, developed extensively in recent years [1], [27]–[32], identifies quantum coherence as a resource relative to a specific basis \(\left\{ \ket{e_k}\right\}_{k=0}^{D-1}\). In the standard formulation, a density matrix \(\rho\) is considered incoherent if it is diagonal in the reference basis \(\rho_{kj}=\bra{e_k}\rho\ket{e_j}=\rho_{kk}\delta_{jk}\). The set of all incoherent density matrices is called \(\mathcal{I}\). Incoherent operations are CPTP maps \(\Lambda\) that cannot create coherence from incoherent states, thus preserving \(\mathcal{I}\): \(\mathcal{F}:= \left\{ \rho \in \mathcal{I} \Rightarrow \Lambda[\rho] \in \mathcal{I}\right\}\). Within this context, an important operation is dephasing: \(\Delta[\rho] = \sum_{k=0}^{D-1}\rho_{kk}\ket{e_k}\bra{e_k}\). This takes any density matrix \(\rho\) and it extracts its incoherent part \(\Delta[\rho]\). An important measure of coherence that emerges from this approach is the relative entropy of coherence: \(\mathcal{C}(\rho):=D^Q_{KL}(\rho \vert \vert \Delta[\rho])=S(\Delta[\rho]) - S(\rho)\), where \(D_{KL}^Q(\rho || \sigma)\) is the quantum relative entropy between two density matrices and \(S(\rho)\) is the von Neumann entropy.

3.1 A geometric approach to quantum coherence in ensembles↩︎

We leverage the geometric setting to systematically extend the resource theory of coherence to ensembles of pure states. This requires defining three new notions: geometric dephasing, incoherent ensembles and free operations.

In geometric quantum mechanics, the density matrix is replaced by the geometric quantum state—an ensemble of pure states described by the measure \(\mu_{P,\Phi}\). Additionally, quantum operations are replaced by geometric channels \(\mathcal{K}\): conditional probability measures \(\mathcal{K}(Z|\Gamma)\). This is equivalent to the definition of channels in classical information theory. Specifically, a geometric channel \(\mathcal{K}\) is a linear operation mapping a generic probability measure \(\mu_n\) on \(\mathbb{C}P^n\) into another probability measure \(\sigma_m\) on \(\mathbb{C}P^m\): \(\mathcal{K}[\mu_n] = \sigma_m\).

When written in terms of densities \(d \mu_n = q_n(\Gamma) dV_{FS}^{\Gamma}\) and \(d\sigma_m = \tilde{q}_m(Z)dV_{FS}^Z\) this takes the familiar form of a kernel, or conditional probability density: \[\begin{align} \tilde{q}_m(Z) = \int_{\mathbb{C}P^{n}}\!\!\! dV_{FS}^\Gamma~~\mathcal{K}(Z \vert \Gamma) q_n(\Gamma) \nonumber \end{align}\] We now have everything we need to build the three core elements of our resource theory of coherence for ensembles. First, we define geometric dephasing \(\Delta_G\), as the operation which maps any geometric quantum state \(\mu_{P,\Phi}\) into the product of its probability and phases marginals: \[\Delta_G[\mu_{P,\Phi}] = \mu_{P}~\cdot~\mu_{\Phi} \nonumber\] This translation shifts the focus from coherence as superposition within individual quantum states to coherence as statistical correlation within quantum ensembles.

Paralleling the standard resource theory of quantum coherence, \(\mu_{P,\Phi}\) is considered incoherent when it has a form compatible with the output of the dephasing channel \(\Delta_G\). That is, when probability and phase variables are statistically independent \(\mu_{P,\Phi} = \mu_P \cdot \mu_\Phi\). This condition ensures that knowledge of probabilities provides no information about the phases, and vice versa. The set of incoherent ensembles is thus: \[\begin{align} \mathcal{I}_G = \{\mu_P \cdot \mu_\Phi \}~, \label{eq:free95states} \end{align}\tag{6}\] where \(\mu_P\) is a measure on \(\Sigma_{D-1}\) and \(\mu_\Phi\) is a measure on \(\mathbb{T}_{D-1}\).

Free operations in this resource theory are those that preserve the factorized structure of incoherent states. Specifically, a channel \(\mathcal{K}\) acting on geometric quantum states is free if it can be decomposed as \(\mathcal{K} = \mathcal{K}_P \cdot \mathcal{K}_\Phi\), where \(\mathcal{K}_P\) acts only on the probability variables and \(\mathcal{K}_\Phi\) acts only on the phase variables. The set of free operations is therefore: \[\begin{align} \mathcal{F}_G = \{\mathcal{K}_P \cdot \mathcal{K}_\Phi\} \nonumber \end{align}\]

With this setup, we now apply the framework established by Streltsov, Adesso, and Plenio [2]. For \(I(P;\Phi)\) to qualify as a coherence measure, it must satisfy the following six properties. We now explicitly state the properties, and comment on their physical relevance in our resource theory.

(C1) Non-negativity: \(I(P;\Phi) \geq 0\) with equality if and only if \(\mu_{P,\Phi} \in \mathcal{I}_G\). This follows directly from the non-negativity of mutual information, see Eq.(5 ), and the fact that \(D_{KL}(\mu||\nu) = 0\) if and only if \(\mu = \nu\).

(C2) Monotonicity: \(I(P;\Phi)\) does not increase under free operations. For any \(\mathcal{K} \in \mathcal{F}_G\), we have \(I(\mathcal{K}[\mu_{P,\Phi}]) \leq I(\mu_{P,\Phi})\). This guarantees that when we act with free operations resources can only be depleted, not created.

(C3) Strong monotonicity: \(I(P;\Phi)\) does not increase on average under selective free operations. This property guarantees that measurements do not create ensemble coherence.

(C4) Convexity: \(I(P;\Phi)\) is a convex function of the geometric quantum state, guaranteeing that classically mixing two ensembles can only deplete coherence.

(C5) Uniqueness for pure states: For pure states, described by a Dirac measure with support on a single point \(\mu^{\mathrm{pure}}_{P,\Phi} = \delta_{p_0,\phi_0}\), the probability-phase mutual information vanishes: \[I(P;\Phi) = H(\Delta_G[\delta_{p_0,\phi_0}]) = 0 \nonumber\] This property distinguishes \(I(P;\Phi)\) fundamentally from density-matrix coherence measures. A pure state \(\ket{\psi}\) in superposition can have \(\mathcal{C}(\rho) > 0\), yet when viewed as an ensemble (a single point in state space), it necessarily has \(I(P;\Phi) = 0\). The probability-phase mutual information therefore captures coherence that emerges only at the ensemble level—it quantifies correlations across a statistical distribution of pure states, not superposition within individual states. This is the additional resource that is invisible to density matrices and that becomes relevant for when density matrices are mixed rather than pure.

(C6) Additivity: For non-interacting systems described by \(\mu_{P,\Phi} \cdot \nu_{P',\Phi'}\), we have \(I(P,P';\Phi,\Phi') = I(P;\Phi) + I(P';\Phi')\). This guarantees that we do not create our resource, mutual information, by simply putting together non-interacting systems.

Detailed proofs that \(I(P,\Phi)\) satisfies all these properties are given in Appendix 7. The net result establishes \(I(P;\Phi)\) as an operationally meaningful measure of coherence for ensembles of pure states. A discussion to compare it with density-matrix-based measures of coherence, like the relative entropy of coherence [2], is given in Section 5.

4 Examples↩︎

We have concluded the technical development of the probability-phase mutual information, and established its validity as a coherence measure for ensembles. The next subsections show how to compute \(I(P,\Phi)\), both analytically and numerically, in a variety of cases: Product measures, Canonical ensemble, Gaussian measure, Spiral ensemble, and Deep thermalization. These illustrate a wide variety of probability-phase correlations that can arise in geometric quantum states, from the complete absence of correlations to maximal correlations in coherent ensembles.

For geometric quantum states where analytical calculations are intractable, we employ a direct numerical approach that leverages the discrete partitioning of complex projective spaces detailed in Ref.[25]:

1. Discretization: Partition the state space into cells of linear size \(\epsilon\)

2. Histogram estimation: Coarse grain \(\mu_{P,\Phi}\) to compute empirical probabilities of the joint and marginals

3. Entropy calculation: Apply the scaling analysis to extract \(\mathfrak{D}_{P\Phi},\mathfrak{D}_P,\mathfrak{D}_\Phi,H_G,H_P,H_\Phi\)

4. Mutual information: Compute via Eq. 4 : \(I(P;\Phi) = H_P + H_G - H_{P\Phi}\)

This procedure has been validated against the analytical examples presented, showing convergence as \(\epsilon \to 0\) with the expected scaling behavior.

4.1 Example 1: Product Measures↩︎

We first consider a class of ensembles which share the common property that they can be written as product measures.

4.1.0.1 Dirac measure.

Consider an ensemble composed of a single pure state, corresponding to a concentration of measure on one point in the quantum state space. This is represented by the Dirac measure: \(\delta^{\Psi}_{\psi_0} = \delta^{P}_{p_0} \delta^{\Phi}_{\phi_0}\) where \(\delta_{p_0}^P\) is the Dirac measure on \(\Sigma_{D-1}\) with support on \(p_0\) and \(\delta_{\phi_0}^{\Phi}\) is the Dirac measure on \(\mathbb{T}_{D-1}\) with support on \(\phi_0\). The joint measure is clearly a product of its marginals, so the mutual information vanishes.

4.1.0.2 Uniform measure.

The uniform (Haar) ensemble is also a product measure. For this ensemble, every point in \(\mathcal{P}(\mathcal{H}) \sim \mathbb{C}P^{D-1}\) is equally probable. For a \(D\)-dimensional Hilbert space the uniform distribution is written in terms of the Fubini-Study volume \(V_{D-1}=\frac{\pi^{D-1}}{(D-1)!}\) [25]. Using Eq. 1 and the Fubini-Study volume, the uniform density in probability-phase coordinates is: \[q_{P,\Phi}(p,\phi) = \frac{(D-1)!}{(2\pi)^{D-1}} \nonumber\]

Integration over the probability simplex \(\Sigma_{D-1}\) and hyper-torus \(\mathbb{T}_{D-1}\) yields the marginals \(q_P(p) = (D-1)!\) and \(q_\Phi(\phi) = \frac{1}{(2\pi)^{D-1}}\), confirming the factorization and vanishing of mutual information.

4.1.0.3 Diagonal ensemble.

Given some isolated system initialized in \(\ket{\psi_0}\) and evolving with hamiltonian \(H = \sum_n E_n \ket{E_n} \bra{E_n}\) with non-degenerate energy gaps, the distribution of coefficients \(c_n(t)=\braket{E_n \vert \psi_t}=|c_n(0)|e^{-\frac{i}{\hbar}E_nt}\) resulting from time-evolution is, as obtained by von Neumann [33], the product of a Dirac measure in \(\Sigma_{D-1}\) and a uniform measure in phases: \[\mu_{DE} = \delta^P_{p_0} \cdot \frac{1}{(2\pi)^{D-1}} \nonumber\] where \(p_0\) fixes the probability coordinates in the energy basis to be \(p_0 = (|c_1^2|,\ldots,|c_{D-1}|^2)\). This has vanishing probability-phase mutual information in the energy basis. We note that \(\mu_{DE}\) is the ensemble whose average provides the stationary state commonly called diagonal ensemble: \(\mathbb{E}_{\mu_{DE}}[\psi] = \sum_{n}|c_n|^2 \ket{E_n}\bra{E_n}\).

4.1.0.4 Naive gaussian.

A final example of a product measure is the naive Gaussian measure: \[q(p,\phi) = \frac{1}{\mathcal{N}}\,\exp{\left[ -\frac{(p-p_0)^2}{2\sigma_p^2} - \frac{(\phi - \phi_0)^2}{2\sigma_\phi^2} \right]} \nonumber\] The marginal distributions are then gaussian in each coordinate with normalization constants satisfying \(\mathcal{N} = \mathcal{N}_p\mathcal{N}_\phi\) which requires the joint measure be a product of marginals.

In each of these cases, the factorization of the joint distribution means that the joint distribution is a product measure \(\mu_{P,\Phi} = \mu_p\cdot\mu_\phi\). Therefore, from Eq. 6 , \(\mu_{P,\Phi} \in \mathcal{I}_G\) for each of these ensembles and \(I(P,\Phi)=0\).

4.2 Example 2: Canonical Ensemble↩︎

Consider a canonical ensemble \(q_\beta(p,\phi)\) at inverse temperature \(\beta\) and with Hamiltonian \(H(g)\) where \(g\) is a coupling parameter: \[\begin{align} q_{\beta,g}(p,\phi) &= \frac{1}{Z_{\beta,g}}\exp{(-\beta h_g(p,\phi))} \nonumber \\ Z_{\beta,g}&=\int_{\Sigma_{D-1}}\!\!\! dp \int_{\mathbb{T}_{D-1}} \!\!\!d \phi~e^{- \beta h_g(p,\phi)}\nonumber \label{eq:canonical95ensemble} \end{align}\tag{7}\] where \(h_g(p,\phi) = \bra{p,\phi}H(g)\ket{p, \phi}\) is the expectation value of \(H(g)\) on a generic pure state \(\ket{p,\phi} = \sum_{j=1}^{D} \sqrt{p_j}e^{i\phi_j}\ket{e_j}\) parametrized with probability-phase coordinate in the basis \(\left\{\ket{e_j}\right\}_j\) that diagonalizes \(H(g=0)\): \(H(g=0)\ket{e_j} = e_j \ket{e_j}\).

For a qubit with Hamiltonian \(H = \sigma_z + g\sigma_x\), using the Pauli-z basis we have \[h_g(p,\phi) = 1 - 2p + 2g\sqrt{p(1-p)}\,\cos\phi \nonumber\] The marginal distributions are obtained by partial integration over the probability and phase degrees of freedom. In the case of no coupling (\(g=0\)) these can be evaluated analytically: the joint distribution factorizes and the mutual information vanishes.

In the general case \(g > 0\), the joint distribution does not factorize. We evaluate \(I(P;\Phi)\) numerically within the \(\beta\)-\(g\) parameter space. The results are shown in Figure 1.

For finite coupling \(g > 0\), we see that \(I(P;\Phi)\) increases with larger \(\beta\) (lower temperature). This indicates that a low temperature canonical ensemble suggests higher statistical dependence between probability-phase coordinates.

We highlight that the rational for the choice of the canonical ensemble goes beyond the need to illustrate the behavior of the probability-phase mutual information. Its relevance stems from the concept of conditional thermalization [34], [35], geometric quantum thermodynamics [36] and deep thermalization [37]–[43]. The discussion on this point is postponed to Section 4.5.

4.3 Example 3: Gaussian Measure↩︎

Consider a gaussian density defined using the Fubini-Study distance \(D_{FS}=D_{FS}(p,\phi;p_0,\phi_0)\) expressed in probability-phase coordinates. This density results from the extension of the standard Gaussian density on metric manifolds, intended as exponentials of squared distances. The distribution can be written as: \[\mathcal{G}_2(p,\phi) = \frac{1}{\mathcal{N}}\exp{\left[ -\frac{D_{FS}(p,\phi;p_0,\phi_0)^2}{2\sigma^2} \right]} \nonumber\] where the Fubini-Study distance for a single qubit is: \(D_{FS}(p,\phi;p_0,\phi_0) = \arccos{|\braket{\psi(p,\phi)|\psi(p_0,\phi_0)}|}\), with \(\bra{\psi(p,\phi)}\psi(p_0,\phi_0)\rangle = \sqrt{(1-p_0)(1-p)} + \sqrt{p_0p }~e^{i(\phi_0-\phi)}\)

Since \(D_{FS}\) does not factorize as a sum over \(p\) and \(\phi\), it introduces non-trivial correlation, leading to a non-zero mutual information. As we can observe in Figure 2, by tuning the variance \(\sigma\), one can observe the behavior of \(\mathcal{G}_2(p,\phi)\) as it transitions from a Dirac measure (\(\sigma \approx 0\)) to the uniform measure \(\sigma \approx \pi/2\).

4.4 Example 4: Spiral Ensemble↩︎

To demonstrate non-zero mutual information, and benchmark our numerical algorithm, we introduce the spiral ensemble for a qubit: \[\begin{align} p & \sim \text{Uniform}(0,1) \nonumber\\ \phi & = 2\pi p + \varepsilon, \quad \varepsilon \sim \text{Uniform}(-\delta,\delta) \nonumber \end{align}\] This parametric ensemble allows us to explore correlations between probability and phase coordinates. The noise parameter \(\delta \in [0,\pi]\) controls the strength of statistical correlation between \(P\) and \(\Phi\). With \(\delta=0\) leading to perfect linear correlations while \(\delta \gg 1\) leading to complete independence caused by noise.

To compute the corresponding mutual information, we utilize the numerical scaling approach described above. We extract the information dimension \(\mathfrak{D}\) and geometric entropy \(H_G\) using the scaling of the joint and marginal entropies, as in Eq.@eq:eq:entropy95scaling

We then combine the results into the mutual information dimension \(\mathfrak{D}_I\) and the geometric mutual information \(I(P;\Phi)\) according to 4 . We expect that for nonzero noise, \(\mathfrak{D}_I \rightarrow 0\) since there is no deficit in support between the joint and the marginals. Also, as \(\delta\) is increased, the spiral ensemble is broadened - causing it to approximate the Haar measure for sufficiently large \(\delta\). From this, we can interpret the monotonic decrease in \(I(P;\Phi)\) presented in Figure 3 as an interpolation from the highly structured spiral ensemble at low noise to the Haar uniform measure, explaining the trend of \(I(P;\Phi) \rightarrow 0\) for increasing \(\delta\).

4.5 Example 5: Ensemble Thermalization↩︎

In quantum thermalization the density matrix of an open quantum system relaxes to a steady state described by a thermal state, e.g, Gibbs’ canonical density matrix. Instead of framing the question at the density matrix level one can talk about ensemble thermalization, referring to the idea that the ensemble behind a density matrix acquires a thermal equilibrium statistics. This phenomenon has been studied under three different names. It is known with “Deep Thermalization” [37]–[43] within the condensed matter and quantum information community. And, it has been investigated under the name of “Universal Probability Distribution” [44], “Geometric thermalization” [36] or “Conditional Thermalization” [34], [35] in Quantum Thermodynamics. While framed under slightly different lights, the core idea behind all these advanced studies of quantum thermalization and the resulting thermodynamic behavior is the same.

We have a unitarily evolving bipartite quantum system in a pure state: \(\ket{\Psi_t} \in \mathcal{H}_A \cdot \mathcal{H}_B\). Measuring the complement \(B\) in an orthonormal basis \(\{\ket{z_B}\}_{z_B}\) and collecting the conditional pure states of \(A\), \[\begin{align} p_t(z_B) &= \braket{\Psi_t | (\mathbb{I}_A\cdot \ket{z_B}\!\bra{z_B}) |\Psi_t}, \nonumber \\ \ket{\Psi_t(z_B)} & = \frac{(\mathbb{I}_A\cdot \bra{z_B})\ket{\Psi_t}}{\sqrt{p_t(z_B)}}, \nonumber \end{align}\] leads to the projected ensemble \(\mathcal{E}_{\mathrm{proj}}(t)=\{p_t(z_B),\,\ket{\Psi_t(z_B)}\}_{z_B}\) on \(A\). This is a conditional ensemble and also the geometric quantum state on \(\mathcal{P}(\mathcal{H}_A)\). Deep thermalization posits that [43], in chaotic dynamics and at very high energy density (no conservation constraints), the geometric quantum state approaches the unitarily invariant (Haar) distribution on \(A\)—i.e., an infinite-temperature uniform ensemble.

In our geometric formalism, this simply means constant density with respect to the Fubini–Study (FS) volume element. As shown in Sec. 4.1, this leads to a vanishing mutual information. This gives a sharp, basis-resolved statement: deep thermalization \(\Rightarrow\) \(I(P,\Phi)=0\), no statistical dependence between probabilities and phases at the ensemble level.

Here we study an interesting example of the time-dependent \(I_t(P,\Phi)\) of a bipartite system in which the many-body system is a linear chain of \(L=14\) spin-1/2 systems interacting with the following long-range Hamiltonian \[\begin{align} H = -|J|\sum_{i>j}^L \frac{\sigma_z^i\sigma_z^j}{|i-j|^\alpha} - h\sum_i\sigma_x^i \nonumber \end{align}\] with open boundary conditions and parameters \(J=1\), \(\alpha=2\), \(h=-0.6\). Our system of interest \(A\) is made by a single qubit located at \(n=7\), while the environmental basis \(\ket{z_B} = \ket{s_1,\ldots,s_L}\), with \(s_i \in \left\{ 0,1\right\}~\forall i\), is the one that diagonalizes all the Pauli matrices in the \(z\) direction \(\sigma_z^i\). The resulting \(I_t(P;\Phi)\) is given in Figure 4. We observe: (i) a rapid rise from zero at very early times, reflecting a fast creation of probability–phase correlations by coherent dynamics; (ii) a non‑monotonic transient (overshoot and dip), consistent with the two‑step relaxation phenomenology seen in random circuit models (linear entanglement growth followed by exponential saturation) [37]; and (iii) saturation to a plateau at long times. The trend—suppression and flattening of \(I_t(P;\Phi)\)—is the signature expected as the projected ensemble approaches a high‑randomness regime. The nonzero late‑time offset indicates that, in this system and for this parameters choice, the asymptotic ensemble is not Haar uniform, at least within the timescale considered here.

5 Discussion↩︎

The probability–phase mutual information \(I(P;\Phi)\) and the relative entropy of coherence \(\mathcal{C}(\rho)\) quantify fundamentally different types of coherence that emerge at ensemble and density matrix levels. While \(\mathcal{C}(\rho)\) quantifies coherence by evaluating the relevance of the off-diagonal matrix elements in a given basis, \(I(P;\Phi)\) does it by addressing the question of how much information can about phases is accessible from the probabilities.

Crucially, this distinction is sharp for pure states: a \(\ket{\psi}\) in superposition has \(\mathcal{C}(\ket{\psi}\bra{\psi}) > 0\), yet when viewed as an ensemble—a Dirac measure concentrated at a single point—necessarily has \(I(P;\Phi) = 0\). No spread means no correlations. Therefore, \(I(P,\Phi)\) captures a form of coherence that is available only to mixed states.

This raises a natural question: when a mixed state exhibits both types of coherence, how do they relate? The same density matrix \(\rho\) can arise from ensembles with different correlation structures, so \(I(P;\Phi)\) and \(\mathcal{C}(\rho)\) can vary independently. Yet, they are related. The coherence surplus, which we define in the next subsection, establishes a quantitative connection between them, revealing how ensemble correlations contribute to density-matrix coherence.

5.1 Relating ensemble and density-matrix coherence via the coherence surplus↩︎

The relation between \(I(P,\Phi)\) and \(\mathcal{C}(\rho)\) becomes clear through the diagram in Figure 5, which illustrates a two-step transformation process. The top row depicts the ensemble-level transformations. Starting from a joint probability-phase measure \(\mu_{P,\Phi}\), geometric dephasing \(\Delta_G\) factorizes it into the product of marginals \(\mu_P \cdot \mu_\Phi\). A subsequent phase uniformization replaces the phase marginal with the uniform distribution, yielding \(\mu_P \cdot \mathrm{unif}_\Phi\). The bottom row shows the corresponding density-matrix transformations, connected by the expectation operator \(\mathbb{E}\) that maps ensembles to their density matrices: \[\begin{align} \rho & = \mathbb{E}_{\mu_{P\Phi}}[\psi] \\ \sigma &= \mathbb{E}_{\mu_P \cdot \mu_\Phi}[\psi]\\ \Delta[\rho] & = \mathbb{E}_{\mu_P \cdot \mathrm{unif}_\Phi}[\psi] = \Delta[\sigma] \end{align}\]

At the ensemble level, the relative entropy \(D_{\mathrm{KL}}(\mu_{P\Phi} \| \mu_P \cdot \mathrm{unif}_\Phi)\) quantifies how far the geometric quantum state is from being both incoherent and with uniform distribution of phases. At the density-matrix level, the Kullback-Leibler divergence \(\mathcal{C}(\rho) = D^Q_{\mathrm{KL}}(\rho \| \Delta[\rho])\) measures the relative entropy between \(\rho\) and the dephased state \(\Delta[\rho]\). The difference between these two levels—the coherence surplus—captures how much coherence structure the ensemble contains beyond what is visible in the density matrix:

Since the ensemble-level relative entropy decomposes as \(D_{\mathrm{KL}}(\mu_{P\Phi} \| \mu_P \cdot \mathrm{unif}_\Phi) = I(P;\Phi) + D_{\mathrm{KL}}(\mu_\Phi \| \mathrm{unif}_\Phi)\) via chain rule, we obtain the following theorem:

The proof, provided in Appendix 8, relies on the commutativity of the diagram in Figure 5 and the quantum data-processing inequality.

The positivity of \(\delta_{\mathcal{C}}\) quantifies the coherence discrepancy between the ensemble and density-matrix level of description. We highlight three physical consequences that descend from this, which are of immediate interest.

Coherence bound. The positivity of \(\delta_{\mathcal{C}}\) immediately yields a novel upper bound to the relative entropy of coherence: \[\mathcal{C}(\rho) \leq I(P;\Phi) + D_{\mathrm{KL}}(\mu_\Phi \| \mathrm{unif}_\Phi) \label{eq:global-bound}\tag{8}\] This reveals the fundamental asymmetry between ensembles and density matrices: the ensemble always contains at least as much coherence structure as the density matrix. This is physically intuitive—the geometric quantum state captures the full distribution of pure states, while the density matrix retains only their average. Correlations visible at the ensemble level may be washed out by averaging, but cannot be created by it. The coherence surplus quantifies this information loss.

Classical-quantum decomposition. The coherence surplus naturally identifies to independent contributions to coherence. The mutual information \(I(P;\Phi)\) characterizes how much phase information is accessible from observing the probabilities, classically accessible: even though phases themselves cannot be directly measured in the reference basis, their statistical relationship to probabilities creates an indirect information channel. In contrast, the “phase non-uniformity” \(D_{\mathrm{KL}}(\mu_\Phi||\mathrm{unif}_\Phi)\) characterizes phase structure that remains completely inaccessible to any observer restricted to the reference basis, reflecting the non-commutative nature of quantum measurements. An ensemble that lacks both must necessarily produce a diagonal density matrix.

Entropy Gap Bound. Recalling Definition 1 and expanding the relative entropies as entropy differences, the coherence surplus leads to a counterintuitive bound between the two entropy gaps: \[\begin{align} H_G(\mu_P\mathrm{unif}_\Phi) - S(\Delta[\rho]) \ge H_G(\mu_{P\Phi}) - S(\rho)~. \nonumber \end{align}\] Since the von Neumann entropy is the entropy of the eigenensemble of \(\rho\), the quantities on both sides of these inequality are positive, as proven in [25], [45]. However, the existence of a specific order between them is non-trivial and it is due to the non-negativity of \(\delta_{\mathcal{C}}\). The bound indicates that geometric dephasing \(\Delta_G\)—-operating at the ensemble level—-adds more classical noise to \(\mu_{P\Phi}\) than the standard dephasing \(\Delta\) does to the density matrix \(\rho\), thus increasing the gap between geometric and von Neumann entropy.

6 Conclusions↩︎

We have introduced the probability–phase mutual information \(I(P;\Phi)\) and, employing a resource-theoretic approach, established it as a coherence measure for ensembles of pure quantum states. We have shown that \(I(P;\Phi)\) quantifies a form of quantum coherence that exists only at the ensemble level—correlations across distributions of pure states rather than superpositions within individual states. The coherence surplus \(\delta_{\mathcal{C}} = I(P;\Phi) + D_{\mathrm{KL}}(\mu_\Phi || \mathrm{unif}_\Phi) - \mathcal{C}(\rho) \geq 0\) formalizes the relationship between \(I(P,\Phi)\), bounding how much ensemble structure survives the averaging into density matrices.

Our framework is directed to the emerging trend of a “physics of ensembles” complementary to the standard density-matrix formalism, and addressing phenomena at the ensemble level: deep thermalization [38], [42] manifests as \(I(P;\Phi) \to 0\) when projected ensembles approach the Haar distribution; geometric thermalization [36] reveals temperature-tuned probability-phase correlations absent from thermal density matrices. More broadly, any scenario where the same mixed state arises from qualitatively different preparation procedures—adaptive measurements, engineered reservoirs, conditional dynamics—becomes accessible to ensemble-level analysis.

Ultimately, the probability-phase mutual information provides a powerful tool for an ensemble-first description of quantum coherence—one that complements density matrices by revealing the statistical architecture underlying mixed-state coherence.

7 Proof of C1 - C6 properties↩︎

We provide detailed proofs that the probability-phase mutual information \(I(P;\Phi)\) satisfies the six core properties required for a coherence measure in the ensemble framework.

7.1 Proof of C1: Non-negativity↩︎

Property C1: \(I(P;\Phi) \geq 0\) with equality if and only if \(\mu_{P,\Phi} \in \mathcal{I}_G\).

7.2 Proof of C2: Monotonicity↩︎

Property C2: For any free operation \(\mathcal{K} \in \mathcal{F}_G\), we have \(I(\mathcal{K}[\mu_{P,\Phi}]) \leq I(\mu_{P,\Phi})\).

7.3 Proof of C3: Strong Monotonicity↩︎

Property C3: \(I(P;\Phi)\) does not increase on average under selective free operations.

7.4 Proof of C4: Convexity↩︎

Property C4: \(I(P;\Phi)\) is a convex function of the geometric quantum state.

7.5 Proof of C5: Uniqueness for Pure States↩︎

Property C5: For pure states, \(I(P;\Phi) = H(\Delta_G[\delta_{p,\phi}]) = 0\).

7.6 Proof of C6: Additivity↩︎

Property C6: For non-interacting systems, \(I(P,P';\Phi,\Phi') = I(P;\Phi) + I(P';\Phi')\).

These proofs establish that the probability-phase mutual information \(I(P;\Phi)\) satisfies all the required properties of a coherence measure in the ensemble framework, validating its use as a quantifier of quantum coherence for geometric quantum states.

8 Proof of Coherence accounting Theorem↩︎

We provide a proof of Theorem 1.

References↩︎