TRACe Metrics Overview

• TRACe Metrics are a family of trace-based measures that rigorously quantify distances in quantum states, operator algebras, and probabilistic systems.
• They employ operator monotone functions and optimal transport techniques to ensure properties like monotonicity, additivity, and convexity under various physical operations.
• The metrics underpin practical applications such as system verification, quantum information processing, and noncommutative geometry by providing robust and computable distances.

A family of TRACe metrics arises across multiple domains, each tied by the use of traces—whether as quantum trace functionals, tracial state spaces, trace distances in coalgebraic process theory, or distance metrics defined over traces in C*-algebraic and probabilistic settings. These metrics instantiate a rigorous paradigm for quantifying distances or similarities in settings where the trace, broadly construed, encodes either the observable statistics, dynamics, or underlying geometry of quantum states, operator algebras, or systems with nondeterminism and probability.

1. Quantum TRACe Metrics: Monotone Quantum Metrics Beyond CPTP

TRACe metrics in the sense of (Yamagata, 2020) are the unique family of quantum monotone metrics on positive operators ρ ≥ 0 with Tr ρ ≤ 1 that are monotone under all completely positive, trace non-increasing (CPTNI) maps and additive noise. For a full matrix algebra B(H), the inner product is given for any operator monotone function $f:(0,\infty)\to(0,\infty)$ by

$$K_{\rho}(X,Y) = \operatorname{Tr}\left[X^*\,\bigl(R_{\rho}f(L_{\rho}R_{\rho}^{-1})\bigr)^{-1}Y\right]$$

or, equivalently, via the Kubo–Ando mean $m_h$ with $h(x)=x f(1/x)$: $$K_{\rho}(X,Y) = \operatorname{Tr}[X^*\,(R_{\rho}^{-1} m_h L_{\rho}^{-1})\,Y].$$ Key features include:

• Monotonicity under CPTNI + noise: $K$ contracts under all physical operations, including general forms of noise.
• Complete characterization: Every such metric arises uniquely from a static operator-monotone $f$; no trace-indexed parameterization is permitted.
• Direct-sum additivity: $$K_{\rho_1\oplus\rho_2}(X_1\oplus X_2, X_1\oplus X_2) = K_{\rho_1}(X_1,X_1) + K_{\rho_2}(X_2,X_2)$$.
• Convexity and monotonicity: For fixed $X$, $K_\rho(X,X)$ is convex and order-decreasing in $\rho$.
• One-point sum inequality: For any $\rho_1,\rho_2$, $$K_{\rho_1+\rho_2}(X_1+X_2,X_1+X_2)\leq K_{\rho_1}(X_1,X_1)+K_{\rho_2}(X_2,X_2)$$.
• Examples: Choosing $f(x)=1$ yields the "right" metric; $f(x)=x$ the "left" metric; $f(x)=(1+x)/2$ yields the Bures–Uhlmann metric.

The TRACe class thus restores ideal metric-theoretic properties absent in previous CPTP-based approaches.

2. Trace-Space and Optimal Transport Metrics in C*-Algebraic Contexts

In the context of nuclear, simple C*-algebras of real rank zero and stable rank one (Jacelon, 2021), the TRACe-metric refers to an optimal-transport-type metric on the tracial state space $T(A)$: $$d_{\mathrm{tr}}(\tau_1,\tau_2) = \sup\left\{ |\tau_1(a) - \tau_2(a)| : a=a^*, L(a)\leq 1 \right\}$$ where $L$ is a tracial Lipschitz seminorm. This leads to precise formulas for the distance between *-homomorphisms (unitary orbits) into such algebras, and admits explicit computation via Wasserstein-type metrics: $d_u(\varphi,\psi) = W_\infty(\varphi, \psi)$ for domains $C(X)$ admitting continuous transport, with generalization to noncommutative one-dimensional NCCW complexes. For interval domains, Schatten-p variants coincide with $L^p$ generalizations of Wasserstein metrics.

The notion of a tracially Lipschitz family $L$ of observables is crucial for stability and for the application of TRACe-metrics to ergodic theorems (almost-sure CLT for tracially Lipschitz elements).

3. Trace-Based Metrics in Coalgebraic and Process-Theoretic Semantics

Trace metrics are systematically developed for quantitative and logical semantics of probabilistic and nondeterministic systems:

• Coalgebraic trace metrics (Baldan et al., 2015, Forster et al., 2023) arise via lifting functors and monads from $\mathit{Set}$ to $\mathit{PMet}$ (pseudometric spaces), admitting canonical Kantorovich/Wasserstein behavioral distances. Determinization via Eilenberg–Moore permits the definition of trace pseudo-metrics for both nondeterministic and probabilistic automata, with the distance between states $x,y$ given by the behavioral metric on their determinized representations.
• Logical characterization (Castiglioni et al., 2017) leverages a minimal modal logic $\mathcal{L}$, with the trace metric on processes given by the Hausdorff/Kantorovich liftings: $$d_{\mathrm{strong}}(s,t)=\mathrm{Haus}_{\psi\in\mathcal{L}(s),\psi'\in\mathcal{L}(t)} D(\psi,\psi')$$ where $D$ is the Kantorovich-lifted discrete formula metric. This equates the process metric with logical distinguishability of the corresponding sets of trace-distribution formulas.

4. Trace Metric Variants: Trace-by-Trace, Extremal, and Distributional

In the paper of nondeterministic probabilistic processes (Castiglioni, 2018), several trace-based metrics are introduced:

• Trace-distribution metric: Compares entire trace distributions (sup-over-resolutions, inf-over-matches) with discounting. Not non-expansive under composition.
• Trace-by-trace metric: Compares maximum difference for each trace individually (sup-over-traces of the per-trace value). Strictly non-expansive, congruent for parallel composition.
• Extremal-probabilities (supremal) metric: Considers only the maximal probability achievable for each trace in either process; strictly coarser and most congruent for compositional reasoning.

These metrics provide calibrated distinguishability, with precise interpolation between them determined by the policy class (deterministic vs. randomized schedulers). The spectrum is: $$m_{\mathrm{Tr,dis}} > m_{\mathrm{Tr,tbt}} > m_{\bigsqcup}$$ Under randomized scheduling the relations tighten, with supremal and randomized trace-by-trace coinciding.

5. Trace Metrics in Quantum Information and Noncommutative Geometry

Trace metrics in the quantum information context appear as:

• Trace-norm-based quantum correlations (Aaronson et al., 2013). The trace distance

$$D_{\mathrm{tr}}(\rho,\sigma) = \frac{1}{2} \lVert \rho - \sigma \rVert_1$$

is used to define quantum discord-type, classical, and total correlation metrics. For Bell-diagonal two-qubit states, closed-form expressions exist for all three, with the trace-metric discord exhibiting broader freezing (robust constancy under certain quantum noise) than relative-entropy or Hilbert–Schmidt-based versions.

• Trace-induced quantum compact metrics (Aguilar et al., 2019). For matrix algebras $M_n(\mathbb{C})$, Lip-norms

$$L_{n,k}(a) = \max\left\{ \|a-P_{1,n}(a)\|,\, k\|a-P_{k,n}(a)\| \right\}$$

induce noncommutative Gromov–Hausdorff propinquity metrics on the state space, distinguishing block diagonal structure. Separation results show that metrics connected via the trace cannot be deformed into one another for non-prime $n$.

6. Interrelations, Examples, and Distinctions

Across these instances, several unifying features emerge:

• Operator monotonicity, convexity, and additivity characterize quantum TRACe metrics as the most robust extension under general quantum operations.
• Optimal transport and Wasserstein metrics serve as both the structural and computational backbone for trace metrics on state spaces and operator algebraic traces.
• Coalgebraic perspectives connect trace metrics to logical separation and functorial constructions, enforcing robustness under system composition.
• Practical implications include metric-based verification, optimal matching of unitary orbits, and quantitative measures for regression, clustering, and dynamical analysis of both classical and quantum systems.

Trace-based metrics are thus anchored in foundational theory (operator monotone functions, categorical lifts, optimal transport), but are realized in concrete formulae, with well-defined quantitative and algebraic properties. Their expressiveness, congruence relations, and dynamic behavior under process evolution or physical noise highlight both the versatility and subtleties involved in the choice of trace-based metric for a given application or domain.