Quantum Information Copy Time (QICT): Spectral Foundations, Canonical Filtered-Certification Protocols, and Transport Benchmarks

1. Scope and Standing Assumptions

We separate definition-level statements (spectral QICT and its finiteness/regularization) from model-level inputs (transport coefficients, Lyapunov exponent, matching scale), which must be supplied by a chosen microscopic theory.

Assumption 1 

(Finite-volume definition and controlled limits). All objects are defined first in finite volume (or a finite-dimensional truncation) where traces exist and generators are closed operators on dense domains. Thermodynamic limits are taken after introducing an explicit infrared (IR) regulator (Sec. Section 4).

Assumption 2 

(Faithful stationary state (thermal: KMS)). The reference state $\rho$ is faithful and stationary; in thermal applications $\rho \propto e^{-\beta H}$ (finite volume) or its KMS/GNS extension (infinite volume).

Assumption 3 

(Locality). The Hamiltonian is local (lattice or local QFT), ensuring finite-speed information propagation (Lieb–Robinson type) [3].

Assumption 4 

(Observable class). The label Y is chosen so that $\dot{Y}=i[H,Y]$ lies in the domain of the relevant resolvents/pseudoinverses in the Kubo–Mori operator Hilbert space.

We work in natural units $\hslash =c=k_B=1$.

2. Kubo–Mori Geometry and an Unconditional QICT Definition

2.1. Kubo–Mori Inner Product

For a faithful state $\rho$, the Kubo–Mori (Bogoliubov) inner product on operators is

$${\langle A,B\rangle }_{\mathrm{KM}}\equiv {\int }_0^{1}ds\left({\rho }^s{A}^{†}{\rho }^{1-s}B\right).$$

(1)

This induces the Bogoliubov–Kubo–Mori (BKM) monotone metric on thermal manifolds [1,2].

2.2. Liouvillian Generator and Pseudoinverse

Define the Liouvillian superoperator $\mathcal{L}\left(A\right)=i[H,A]$ and the self-adjoint generator $K\equiv i\mathcal{L}$ on the KM Hilbert space. Because $kerK$ generally contains conserved operators, inverse powers must be understood as pseudoinverses.

Definition 1 

(QICT functional (definition-level)). Let $P_0$ project onto $kerK$. Define the Moore–Penrose pseudoinverse

$$K^{+}\equiv {\left(K{\upharpoonright }_{ker{\left(K\right)}^{\perp }}\right)}^{-1}(1-P_0).$$

(2)

For $Y=Y^{†}$ define $\dot{Y}\equiv \mathcal{L}\left(Y\right)=i[H,Y]$ and set the dimensionless QICT functional

$${\mathfrak{t}}_{\mathrm{copy}}^2\left(Y\right)\equiv {\langle \dot{Y},{\left(K^{+}\right)}^2\dot{Y}\rangle }_{\mathrm{KM}}.$$

(3)

We define the corresponding dimensionful time by

$$t_{\mathrm{copy}}\left(Y\right)\equiv \beta {\mathfrak{t}}_{\mathrm{copy}}\left(Y\right).$$

(4)

3. Exact Spectral Representation and Finiteness

Let $E_K\left(\lambda \right)$ be the spectral measure of the self-adjoint operator K. For any A define the positive measure

$$d{\mu }_A\left(\lambda \right)\equiv {\langle A,d{E}_K\left(\lambda \right)A\rangle }_{\mathrm{KM}}.$$

(5)

Proposition 1 

(Transport-regime invariant spectral formula). If $\dot{Y}\in \mathrm{Dom}\left(K^{+}\right)$, then

$${\mathfrak{t}}_{\mathrm{copy}}^2\left(Y\right)={\int }_{\mathbb{R}\setminus \left\{0\right\}}{\displaystyle \frac{d{\mu }_{\dot{Y}}\left(\lambda \right)}{{\lambda }^2}}.$$

(6)

Proof. 

On $ker{\left(K\right)}^{\perp }$, ${\left(K^{+}\right)}^2={\int }_{\mathbb{R}\setminus \left\{0\right\}}{\lambda }^{-2}d{E}_K\left(\lambda \right)$ as a quadratic form. Insert into (3) and use (5). □

Theorem 1 

(Necessary and sufficient finiteness condition). ${\mathfrak{t}}_{\mathrm{copy}}\left(Y\right)$ is finite iff

$${\int }_{\left|\lambda \right|<{\lambda }_0}{\displaystyle \frac{d{\mu }_{\dot{Y}}\left(\lambda \right)}{{\lambda }^2}}<\infty \mathit{for}\mathit{some}{\lambda }_0>0.$$

(7)

Proof. 

Split (6) into $\left|\lambda \right|<{\lambda }_0$ and $\left|\lambda \right|\ge {\lambda }_0$. The UV part is bounded by ${\lambda }_0^{-2}{\parallel \dot{Y}\parallel }_{\mathrm{KM}}^2$. Hence finiteness reduces to the IR condition (7). □

Remark 1 

(Ballistic/Drude obstruction and the need for an IR regulator). A point mass of $d{\mu }_{\dot{Y}}$ at $\lambda =0$ (Drude weight) violates (7) and implies ${\mathfrak{t}}_{\mathrm{copy}}\left(Y\right)=\infty$ in the ideal limit. This is consistent with ballistic channels where persistent zero-frequency weight obstructs decay of currents [9,10]. Operationally, certification protocols always involve finite resolution, motivating an explicit IR-regularized family.

4. IR-Regularized QICT

Definition 2 

(Regulated QICT). For $ϵ>0$ define

$${\mathfrak{t}}_{\mathrm{copy}}^2(Y;ϵ)\equiv {\int }_{\mathbb{R}}{\displaystyle \frac{d{\mu }_{\dot{Y}}\left(\lambda \right)}{{\lambda }^2+{ϵ}^2}}={\langle \dot{Y},{(K^2+{ϵ}^2)}^{-1}\dot{Y}\rangle }_{\mathrm{KM}}.$$

(8)

The dimensionful regulated time is $t_{\mathrm{copy}}(Y;ϵ)\equiv \beta {\mathfrak{t}}_{\mathrm{copy}}(Y;ϵ)$.

5. Operational Meaning: Canonical Filtered Certification and Hypothesis Testing

We formalize a minimal local certification task and show why the exponential filter is canonical. We also connect the SNR threshold $c\left(\delta \right)$ to standard hypothesis-testing bounds.

5.1. Task: Binary Certification Between Opposite Local Biases

Fix a region $\mathcal{R}$ of linear size ℓ and consider the local Gibbs family

$${\rho }_{\theta }={\displaystyle \frac{e^{-\beta (H-\theta Y_{\mathcal{R}})}}{e^{-\beta (H-\theta Y_{\mathcal{R}})}}},\theta \in \mathbb{R},$$

(9)

where $Y_{\mathcal{R}}$ is the restriction of Y to $\mathcal{R}$. The task is to decide between hypotheses $H_{\pm }:{\rho }_{\pm \theta }$ at target average error probability $\delta \in (0,1/2)$ using measurements localized in $\mathcal{R}$.

5.2. Why the Exponential Filter Is Canonical

We define the exponentially filtered current observable

$$J_{ϵ}\equiv {\int }_0^{\infty }dt{e}^{-ϵt}{\dot{Y}}_{\mathcal{R}}\left(t\right),ϵ>0.$$

(10)

Proposition 2 

(Canonical detector-with-memory realization). Consider a minimal linear Markovian “memory variable” $Q\left(t\right)$ coupled to ${\dot{Y}}_{\mathcal{R}}\left(t\right)$ through the first-order response equation

$$\dot{Q}\left(t\right)=-ϵQ\left(t\right)+{\dot{Y}}_{\mathcal{R}}\left(t\right),Q\left(0\right)=0,$$

(11)

which is the unique causal, time-translation invariant, one-pole (single-timescale) linear filter. Then the long-time readout is

$$\underset{t\to \infty }{lim}Q\left(t\right)=J_{ϵ}.$$

(12)

Proof. 

Solving (11) by variation of constants gives $Q\left(t\right)={\int }_0^{t}ds{e}^{-ϵ(t-s)}{\dot{Y}}_{\mathcal{R}}\left(s\right)$. Taking $t\to \infty$ yields (10). □

Proposition 3 

(Canonical resolvent/Laplace form in linear response). Let K be the self-adjoint Liouvillian generator. In the KM Hilbert space, the Laplace transform of ${\dot{Y}}_{\mathcal{R}}\left(t\right)=e^{tK}{\dot{Y}}_{\mathcal{R}}$ satisfies

$$J_{ϵ}={\int }_0^{\infty }dt{e}^{-ϵt}{e}^{tK}{\dot{Y}}_{\mathcal{R}}={(ϵ-K)}^{-1}{\dot{Y}}_{\mathcal{R}},$$

(13)

as an operator identity on the domain where the resolvent exists.

Proof. 

The Bochner integral ${\int }_0^{\infty }{e}^{-ϵt}{e}^{tK}dt$ is the standard Laplace representation of the resolvent ${(ϵ-K)}^{-1}$ for $ϵ>0$. □

Remark 2 

(Why “canonical” is appropriate). Proposition 2 shows that (10) is the unique readout of the simplest causal Markovian detector with a single memory timescale ${ϵ}^{-1}$. Proposition 3 shows that the same exponential filter is the natural object produced by Laplace-transforming linear-response dynamics, and therefore appears directly in resolvent-based correlation bounds (Sec. Section 5.3).

5.3. SNR Bound Controlled by Regulated QICT

Define the SNR for the statistic $J_{ϵ}$ as

$$\mathrm{SNR}\left(ϵ\right)\equiv {\displaystyle \frac{|{\langle J_{ϵ}\rangle }_{+\theta }-{\langle J_{ϵ}\rangle }_{-\theta }|}{\sqrt{{\mathrm{Var}}_{\mathrm{KM}}\left(J_{ϵ}\right)}}}.$$

(14)

Theorem 2 

(Operational bound: regulated QICT controls filtered certification). In linear response around $\theta =0$,

$$\mathrm{SNR}\left(ϵ\right)\le 2\left|\theta \right|t_{\mathrm{copy}}(Y_{\mathcal{R}};ϵ).$$

(15)

Proof. 

Using (13), write $J_{ϵ}={(ϵ-K)}^{-1}{\dot{Y}}_{\mathcal{R}}$. Linear response gives ${\langle J_{ϵ}\rangle }_{\theta }-{\langle J_{ϵ}\rangle }_0=\beta \theta {\langle Y_{\mathcal{R}},J_{ϵ}\rangle }_{\mathrm{KM}}+O\left({\theta }^2\right)$ (Kubo formula [4]). Applying Cauchy–Schwarz in the KM inner product, $|{\langle Y_{\mathcal{R}},J_{ϵ}\rangle }_{\mathrm{KM}}|\le \parallel Y_{\mathcal{R}}{\parallel }_{\mathrm{KM}}{\parallel J_{ϵ}\parallel }_{\mathrm{KM}}$. A sharper bound uses the identity $\parallel J_{ϵ}{\parallel }_{\mathrm{KM}}^2={\langle {\dot{Y}}_{\mathcal{R}},{(ϵ-K)}^{-1}{(ϵ-K)}^{-1}{\dot{Y}}_{\mathcal{R}}\rangle }_{\mathrm{KM}}={\langle {\dot{Y}}_{\mathcal{R}},{(K^2+{ϵ}^2)}^{-1}{\dot{Y}}_{\mathcal{R}}\rangle }_{\mathrm{KM}}={\mathfrak{t}}_{\mathrm{copy}}^2(Y_{\mathcal{R}};ϵ)$, which follows from the spectral theorem. Substituting yields (15) after multiplying by $\beta$ and using the definition of SNR with ${\mathrm{Var}}_{\mathrm{KM}}\left(J_{ϵ}\right)={\parallel J_{ϵ}\parallel }_{\mathrm{KM}}^2$. □

5.4. From SNR to Target Error Probability: A Standard Hypothesis-Testing Bound

Let $P_{\pm }$ be the classical distributions of the measurement outcome X produced by a given protocol under ${\rho }_{\pm \theta }$. For equal priors, the optimal average error (Bayes risk) is

$$P_{\mathrm{err}}^{★}={\displaystyle \frac{1}{2}}\left(1-\mathrm{TV}(P_{+},P_{-})\right),$$

(16)

where $\mathrm{TV}$ is total-variation distance [12]. Thus any sufficient lower bound on $\mathrm{TV}(P_{+},P_{-})$ yields an error guarantee.

A convenient route is to relate $\mathrm{TV}$ to the KL divergence via Pinsker’s inequality, $\mathrm{TV}(P_{+},P_{-})\le \sqrt{\frac{1}{2}{D}_{\mathrm{KL}}(P_{+}\parallel P_{-})}$ [11], and then bound $D_{\mathrm{KL}}$ in terms of the (sub-)Gaussian concentration of the chosen statistic X. For protocols based on sums or filtered integrals of local currents, the outcome is often well-approximated by a sub-Gaussian variable; we therefore state a conservative lemma:

Lemma 1 

(Sub-Gaussian SNR ⇒ error bound). Suppose under each hypothesis $H_{\pm }$ the statistic X is sub-Gaussian with the same variance proxy ${\sigma }^2$, and means ${\mathbb{E}}_{\pm }\left[X\right]=\pm \mu$. Then the threshold test $\mathrm{sign}\left(X\right)$ obeys

$$P_{\mathrm{err}}\le exp\left(-{\displaystyle \frac{{\mu }^2}{2{\sigma }^2}}\right).$$

(17)

Equivalently, achieving $P_{\mathrm{err}}\le \delta$ is guaranteed if

$$\mathrm{SNR}\equiv {\displaystyle \frac{|{\mathbb{E}}_{+}\left[X\right]-{\mathbb{E}}_{-}\left[X\right]|}{\sqrt{\mathrm{Var}\left(X\right)}}}\ge c\left(\delta \right),c\left(\delta \right)=\sqrt{2ln{\displaystyle \frac{1}{\delta }}}.$$

(18)

Proof. 

For sub-Gaussian X with mean $\mu$ and variance proxy ${\sigma }^2$, the Chernoff bound gives $\mathbb{P}(X\le 0)\le exp(-{\mu }^2/2{\sigma }^2)$ [13]. Under $H_{+}$, the error is ${\mathbb{P}}_{+}(X\le 0)$ and under $H_{-}$ it is ${\mathbb{P}}_{-}(X\ge 0)$, which are equal by symmetry. This yields (17). Rewriting in terms of SNR gives (18). □

Remark 3. 

Lemma 1 provides a standard, conservative translation from SNR to target error level, and fixes a concrete choice of $c\left(\delta \right)$. More refined (and typically stronger) bounds can be obtained from exact Gaussian formulas or from Chernoff information if $P_{\pm }$ are known. In a quantum setting, Helstrom’s theorem gives the minimum error in terms of trace distance [14]; the present approach is intentionally “operational” in the sense that it bounds the performance of explicit local measurement protocols.

6. Transport-Agnostic Master Golden Relation

A regime-independent matching statement may be written as

$$m_S\left(T_{*}\right)=\sqrt{{\kappa }_{\mathrm{eff}}\left(T_{*}\right)}{k}_I\left(T_{*}\right),k_I\left(T_{*}\right)\equiv t_{\mathrm{copy}}^{-1}(Y;{ϵ}_{*}),$$

(19)

where ${ϵ}_{*}$ is the operational IR scale used in matching.

6.1. Generalized Golden Relation with Free Dynamical Exponent z

If, around $T_{*}$, the relevant slow mode is relaxational with rate

$${\mathsf{\Gamma }}_k\sim {\mathfrak{D}}_z\left(T\right)k^z,z>0,$$

(20)

then at local resolution $k\sim 1/a$ one obtains the generalized form

$$m_S\left(T_{*}\right)=C_{\mathsf{\Lambda }}^{\left(z\right)}\left(T_{*}\right)\sqrt{{\kappa }_{\mathrm{eff}}\left(T_{*}\right){\chi }_Y\left(T_{*}\right)},C_{\mathsf{\Lambda }}^{\left(z\right)}\left(T_{*}\right)\propto {\displaystyle \frac{a{\left(T_{*}\right)}^z}{\beta {\mathfrak{D}}_z\left(T_{*}\right)}}.$$

(21)

Diffusion corresponds to $z=2$ and ${\mathfrak{D}}_2=D_Y$. In sub-diffusive regimes ($z>2$) the same structure holds with the appropriate z and ${\mathfrak{D}}_z$; in ballistic limits, the unregulated $ϵ\to 0$ object may diverge and the operational ${ϵ}_{*}$ must be kept finite.

7. Benchmark Estimates for ${\mathit{C}}_{\mathbf{\Lambda }}$ from Independent Chaos and Transport

We cite primary sources and provide numerical brackets for canonical transport inputs.

7.1. Strongly Coupled Benchmark: $\mathcal{N}=4$ SYM Diffusion

For $\mathcal{N}=4$ SYM at strong coupling, the R-charge diffusion constant is [6]

$$D={\displaystyle \frac{1}{2\pi T}}.$$

(22)

Taking a thermal microscopic resolution $a={\left(2\pi T\right)}^{-1}$ gives $a/D=1$. Assuming near-saturation of the chaos bound (${\lambda }_L\approx 2\pi T$) [5], the diffusive ($z=2$) specialization of (21) yields an order-one benchmark for $C_{\mathsf{\Lambda }}^{\left(2\right)}$.

7.2. Weakly Coupled Benchmark: Kinetic Theory Scaling

In weakly coupled gauge plasmas, kinetic theory gives the parametric scaling [7,8]

$$D\left(T\right)\sim {\displaystyle \frac{\xi }{g^4\left(T\right)Tlog(1/g\left(T\right))}},$$

(23)

with $\xi =\mathcal{O}\left(1\right)$ depending on the channel. With $a={\left(2\pi T\right)}^{-1}$ and $\xi \simeq 1$, the ratio

$${\displaystyle \frac{a}{D}}\sim {\displaystyle \frac{g^4\left(T\right)log(1/g\left(T\right))}{2\pi }}$$

(24)

is numerically small for perturbative g. For instance, $g=0.5$ gives $g^{4}log(1/g)\approx 0.043$ and hence $a/D\approx 6.8\times {10}^{-3}$.

8. Structural Risk: Uncertainty Propagation into the Mass Band

Assuming (21) in a window where a relaxational description applies, relative uncertainties propagate as

$${\displaystyle \frac{\delta m_S}{m_S}}={\displaystyle \frac{\delta C_{\mathsf{\Lambda }}^{\left(z\right)}}{C_{\mathsf{\Lambda }}^{\left(z\right)}}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta {\kappa }_{\mathrm{eff}}}{{\kappa }_{\mathrm{eff}}}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\delta {\chi }_Y}{{\chi }_Y}}.$$

(25)

If $C_{\mathsf{\Lambda }}^{\left(z\right)}\propto a^z/\left(\beta {\mathfrak{D}}_z\right)$ then

$${\displaystyle \frac{\delta C_{\mathsf{\Lambda }}^{\left(z\right)}}{C_{\mathsf{\Lambda }}^{\left(z\right)}}}=z{\displaystyle \frac{\delta a}{a}}-{\displaystyle \frac{\delta {\mathfrak{D}}_z}{{\mathfrak{D}}_z}}-{\displaystyle \frac{\delta \beta }{\beta }}.$$

(26)

If ${\mathfrak{D}}_z\left(T\right)\propto T^p$ locally, a matching-scale uncertainty $\delta T_{*}$ injects

$${\left.{\displaystyle \frac{\delta m_S}{m_S}}\right|}_{\mathfrak{D}}\simeq -p{\displaystyle \frac{\delta T_{*}}{T_{*}}}.$$

(27)