Information-Theoretic Impossibility
of Audit-Ready Quantum CPUs
Kernel v3: Auditability Tradeoffs under Information-Theoretic Guarantees

GhostDrift Research Institute
Abstract

This report rigorously investigates whether quantum computing systems can satisfy industrial-grade audit requirements. We define a new standard, the Audit-Ready CPU (ARCPU), characterized by three core requirements: (A) Non-Invasiveness under worst-case inputs (via the Diamond Norm), (B) Responsibility Boundary Closure under Minimax criteria, and (C) Strong Replayability. The main theorem of this paper proves that these requirements are information-theoretically incompatible in systems involving non-commutative state transitions—the essence of quantum computing—based on fundamental principles such as the Helstrom bound, the Data Processing Inequality, and the No-Broadcasting theorem. This impossibility is fundamental and independent of specific algorithms, indicating that satisfying the ARCPU standard necessitates either the abandonment of quantum properties or a compromise on audit specifications. Note that this report primarily targets audit intensities requiring "non-retroactivity" (e.g., in financial, regulatory, or litigation contexts) and does not preclude the validity of statistical verification or computational assumptions (e.g., Mahadev's protocol).

1. Introduction: Game-Theoretic Formulation of Auditing

We formulate the problem of auditing a physical device as a mathematical game played between a Verifier and the World.

Definition 1.1 [World and Responsibility Labels]
Let $\mathcal{W}$ be the parameter space encompassing the system's implementation states and environmental conditions. $\mathcal{W}$ is partitioned into two disjoint sets based on the true locus of responsibility (e.g., Internal Device Fault vs. External Operational Fault): \[ \mathcal{W} = \mathcal{W}_0 \sqcup \mathcal{W}_1 \] We define the responsibility label for a world $w \in \mathcal{W}$ as $\operatorname{Resp}(w) \in \{0, 1\}$, such that $w \in \mathcal{W}_b \iff \operatorname{Resp}(w) = b$.
Definition 1.2 [Audit Strategy and Observation Channels]
The Verifier adopts a "Strategy" $S$. This strategy encompasses (i) input sequence generation and randomization, (ii) preparation of input states potentially entangled with a reference system, (iii) adaptive queries (if necessary), and (iv) final measurement (classicalization) rules. We model the entire process abstractly as a single effective channel plus measurement to maintain generality.

The behavior of the system (CPU) in world $w\in\mathcal{W}$ is modeled as a quantum channel (CPTP map) with an identical output register, conditioned on the audit mode (log flag) $\ell\in\{0,1\}$: $$ \mathcal{N}^{(\ell)}_w:\mathcal{D}(\mathcal{H}_{\mathrm{in}})\to \mathcal{D}(\mathcal{H}_{\mathrm{out}}\otimes \mathcal{H}_{\mathrm{log}}) $$

  • No-Log (Dummy Log) Mode $\ell=0$: The log register is initialized to a fixed state $\tau_{\mathrm{log}}$ (e.g., $\ket{0}\bra{0}$), uncorrelated with the input or internal state. Specifically, there exists an external I/O channel $\mathcal{M}_w:\mathcal{D}(\mathcal{H}_{\mathrm{in}})\to\mathcal{D}(\mathcal{H}_{\mathrm{out}})$ such that: $$ \mathcal{N}^{(0)}_w(\rho)=\mathcal{M}_w(\rho)\otimes \tau_{\mathrm{log}} $$
  • Logged Mode $\ell=1$: The log register is generated dependently on internal processes (subject to subsequent non-invasiveness constraints).

Under strategy $S$, world $w$, and mode $\ell$, the probability distribution of the final audit transcript (classical observation data) is defined as: $$ \Obs^{(\ell)}_w(S)\in\mathcal{P}(\Omega) $$ where $\Omega$ is the sample space of classical strings obtained via the final measurement (comprising I/O logs, audit tags, timestamps, etc.).

Note: Under this definition, Requirement (A♦), "logging does not affect external behavior," is evaluated as the distance between the overall channels induced by log generation relative to the dummy log baseline, rather than the mere presence or absence of a register.

2. Rigorous Definition of ARCPU Standards

We now define the mathematical requirements for a system to be certified as an "Audit-Ready CPU" (ARCPU). Here, we adopt Worst-case guarantees to preclude any potential loopholes.

We adopt worst-case / minimax criteria not to model "average" audits, but to ensure that the closure of the responsibility boundary itself serves as irrefutable evidence in scenarios where post-hoc shifting of specifications or operations might occur. Consequently, $\varepsilon$ and $\delta$ are treated as design parameters representing audit intensity, rather than fixed constants.

Definition 2.0 [ARCPU Tiers (Stratification of Audit Intensity)]
To clarify distinctions in audit intensity, we stratify ARCPU requirements into three tiers:
  • Tier-1: Non-Invasiveness of External I/O (A♦)
  • Tier-2: Responsibility Boundary Closure under Worst-Case (B★)
  • Tier-3 (Strongest): Preservation and Double Supply of Evidence/Boundary State (C#)
Important: The impossibility proven via No-Broadcasting (Theorem 4.2) targets Tier-3 and does not negate relaxed audit frameworks (e.g., statistical/computational guarantees) that satisfy only Tier-1/2.
Definition 2.1 [Requirement A♦: Diamond Non-Invasiveness]
Log acquisition must not perturb the system's external behavior. We define this using the Diamond Norm (Completely Bounded Trace Norm). For a given tolerance $\varepsilon \in [0, 1]$, the following must hold for any world $w \in \mathcal{W}$: \[ \dnorm{\mathcal{N}_w^{(1)} - \mathcal{N}_w^{(0)}} \le 2\varepsilon \] Here, $\|\cdot\|_\diamond$ denotes the optimal distinguishability of quantum channels [1].
Lemma 2.2 [Operational Meaning of Non-Invasiveness]
When Requirement (A♦) is satisfied, the Total Variation Distance (TV) between observation distributions with and without logging is bounded as follows for any audit strategy $S$ (including those utilizing reference systems) and any post-processing measurement: \[ \TV(\Obs_w^{(1)}(S), \Obs_w^{(0)}(S)) \le \varepsilon \] This follows from the standard result that $\frac{1}{2}\|\mathcal{N}_w^{(1)} - \mathcal{N}_w^{(0)}\|_\diamond$ quantifies the bias (advantage) in optimal success probability for distinguishing unknown channels in a single shot. This guarantees that detection of invasiveness, even using correlated inputs, is suppressed to $\varepsilon$.
Definition 2.3 [Requirement B★: Minimax Responsibility Closure]
We require that the responsibility label $b\in\{0,1\}$ can be uniquely estimated from the audit transcript. The Verifier designs a Strategy $S$ and a Decision Function $\psi:\Omega\to\{0,1\}$, while the World selects the worst-case $w$ within a responsibility class.

First, define the worst-case risk for any pair $(S,\psi)$: $$ \mathrm{Risk}(S,\psi) := \sup_{b\in\{0,1\}}\ \sup_{w\in\mathcal{W}_b}\ \Pr_{\omega\sim \Obs^{(1)}_w(S)}\big[\psi(\omega)\neq b\big] $$

Define the Minimax risk as: $$ \mathrm{Risk}_\star := \inf_{S,\psi}\ \mathrm{Risk}(S,\psi) $$ Requirement (B★) is satisfied if, for a permissible error rate $\delta\in[0,1/2)$: $$ \mathrm{Risk}_\star \le \delta $$

Note: This definition implies a Minimax criterion where the "Verifier optimizes the design, while the World exploits the worst conditions," avoiding the excessive requirement that "any arbitrary strategy $S$ must succeed" ($\sup_S$).

3. Information-Theoretic Impossibility of Responsibility Closure

In this section, we prove that (A♦) and (B★) are fundamentally incompatible in quantum systems. First, we derive a mathematical reduction from system engineering premises.

Assumption 3.1 [BM: Boundary Mediation]
Let $\mathcal{H}_{\text{core}}$ be the internal state space of the system (CPU) and $\mathcal{H}_B$ be the boundary state space serving as the interface with the outside. All information flowing to the outside (Output/Log) is accessible only via the boundary $\mathcal{H}_B$; there is no direct access to $\mathcal{H}_{\text{core}}$. That is, the entire observation process can be described as the composition of a CPTP map on $\mathcal{H}_B$ (after tracing out $\mathcal{H}_{\text{core}}$) and classical post-processing. Note: This assumption is not specific to quantum mechanics but abstracts observation constraints inherent in general hardware audits (I/O boundaries).
Lemma 3.2 [IL: Interface Locality / Reduction]
Under Assumption (BM), for any strategy $S$, there exists a channel $\Gamma_S: \mathcal{D}(\mathcal{H}_B) \to \mathcal{P}(\Omega)$ such that the observation distribution depends solely on the boundary state $\rho_w^B = \Tr_{\text{core}}(\rho_w^{\text{total}})$: \[ \Obs_w^{(\ell)}(S) = \Gamma_S(\rho_w^B) \] (This implies that for either mode $\ell \in \{0, 1\}$, the final distribution is representable as a map from the boundary state.) Thus, information regarding the difference in world $w$ reduces to the difference in the boundary state $\rho_w^B$.

Next, we introduce fundamental theorems of Quantum Information Theory.

Definition 3.2’ [Normalized Trace Distance]
In this paper, we utilize the following definition for the distance between quantum states: $$ \Delta(\rho,\sigma):=\frac12\|\rho-\sigma\|_1 $$ ($\|\cdot\|_1$ denotes the trace norm). With this normalization, for any measurement (classicalization) $\mathsf{M}$, the following holds: $$ \TV(\mathsf{M}(\rho),\mathsf{M}(\sigma)) \le \Delta(\rho,\sigma) $$
Lemma 3.3 [Fundamental Inequalities]
  1. Helstrom Bound and TV Distance: The minimum error probability $p_e^*$ for distinguishing two distributions $P, Q$ is given by $p_e^*(P, Q) = \frac{1 - \TV(P, Q)}{2}$ [2].
  2. Data Processing Inequality (Including Measurement): For any CPTP map $\Lambda$, $\Delta(\Lambda(\rho),\Lambda(\sigma))\le \Delta(\rho,\sigma)$. In particular, for any measurement (classicalization) $\mathsf{M}$, $\TV(\mathsf{M}(\rho),\mathsf{M}(\sigma))\le \Delta(\rho,\sigma)$ holds [3].
Theorem 3.4 [ARCPU Impossibility Theorem: Minimax Closure vs Diamond Non-Invasiveness]
Let Assumption (BM) hold, and assume there exist two worlds $w_0 \in \mathcal{W}_0,\, w_1 \in \mathcal{W}_1$ with different responsibility labels. Let their respective boundary states be $\rho := \rho_{w_0}^B,\ \sigma := \rho_{w_1}^B$.

Furthermore, suppose Requirement (A♦) holds with tolerance $\varepsilon$ and Requirement (B★) holds with error rate $\delta$. Then, the following necessary condition applies: $$ \Delta(\rho,\sigma) + 2\varepsilon \ \ge\ 1-2\delta. $$ Consequently, the simultaneous satisfaction of (A♦) and (B★) is impossible whenever: $$ \Delta(\rho,\sigma) + 2\varepsilon \ <\ 1-2\delta $$

Note 3.4’ [This Theorem establishes a "Performance Bound", not merely "Impossibility"]
Theorem 3.4 provides a lower bound (tradeoff) that the distinguishability of boundary states $\Delta(\rho,\sigma)$ must satisfy against invasiveness $\varepsilon$ and error rate $\delta$. Thus, the conclusion is not that "Quantum is inherently unauditable," but rather that enforcing audit specifications solely through information-theoretic guarantees reveals an unavoidable lower bound between parameters.

Example: If we require audit intensity $\varepsilon=0.05$ (log influence $\le 5\%$ in TV) and $\delta=0.1$ (worst-case error $\le 10\%$), Theorem 3.4 implies $\Delta(\rho,\sigma)\ge 1-2\delta-2\varepsilon=0.7$ is required. In other words, to close the responsibility boundary while maintaining non-invasiveness, the boundary states must be strongly separated.

Proof

(1) From (B★), there exists a strategy $S$ and decision function $\psi$ such that the worst-case risk is $\le \delta$. Specifically, this pair $(S,\psi)$ succeeds in distinguishing the two worlds $w_0, w_1$ with error $\le \delta$. Invoking the Helstrom bound (Lemma 3.3(1)), the Total Variation distance between the corresponding observation distributions (Logged Mode $\ell=1$) must satisfy: $$ \TV\big(\Obs^{(1)}_{w_0}(S),\Obs^{(1)}_{w_1}(S)\big)\ \ge\ 1-2\delta \quad\text{--- (Eq 1)} $$

(2) By the triangle inequality: $$ \begin{aligned} \TV\big(\Obs^{(1)}_{w_0}(S),\Obs^{(1)}_{w_1}(S)\big) &\le \TV\big(\Obs^{(1)}_{w_0}(S),\Obs^{(0)}_{w_0}(S)\big) \\ &\quad +\TV\big(\Obs^{(0)}_{w_0}(S),\Obs^{(0)}_{w_1}(S)\big) \\ &\quad +\TV\big(\Obs^{(0)}_{w_1}(S),\Obs^{(1)}_{w_1}(S)\big). \end{aligned} \quad\text{--- (Eq 2)} $$

(3) From (A♦) and Lemma 2.2, for any $w$ and strategy $S$: $$ \TV\big(\Obs^{(1)}_{w}(S),\Obs^{(0)}_{w}(S)\big)\le \varepsilon \quad\text{--- (Eq 3)} $$ holds. Thus, the 1st and 3rd terms of (Eq 2) are each bounded by $\varepsilon$.

(4) From (BM) and Lemma 3.2 (IL), the observation distribution in Mode $\ell=0$ depends solely on the boundary state. That is, there exists a map $\Gamma^{(0)}_{S}$ (dependent on $S$) such that: $$ \Obs^{(0)}_{w_i}(S)=\Gamma^{(0)}_{S}(\rho_{w_i}^B)\quad(i=0,1). $$ Applying the Data Processing Inequality (Lemma 3.3(2)): $$ \TV\big(\Obs^{(0)}_{w_0}(S),\Obs^{(0)}_{w_1}(S)\big) = \TV\big(\Gamma^{(0)}_{S}(\rho),\Gamma^{(0)}_{S}(\sigma)\big) \le \Delta(\rho,\sigma). \quad\text{--- (Eq 4)} $$

(5) Combining (Eq 2), (Eq 3), and (Eq 4) yields: $$ \TV\big(\Obs^{(1)}_{w_0}(S),\Obs^{(1)}_{w_1}(S)\big)\ \le\ \Delta(\rho,\sigma)+2\varepsilon. \quad\text{--- (Eq 5)} $$ Combining this result with (Eq 1) provides the necessary condition: $$ 1-2\delta\ \le\ \Delta(\rho,\sigma)+2\varepsilon $$ The proof is complete.

Lemma 3.5 [Conditional: Non-Commutativity Implies Imperfect Distinguishability]
Assume there exist two worlds $w_0\in\mathcal{W}_0,\ w_1\in\mathcal{W}_1$ with different responsibility labels, and their boundary states $\rho:=\rho_{w_0}^B,\ \sigma:=\rho_{w_1}^B$ satisfy: $$ [\rho,\sigma]\neq 0 $$ Then, it follows that: $$ \Delta(\rho,\sigma) < 1 $$ That is, as long as boundary states are non-commutative, they cannot be perfectly distinguished (i.e., they are non-orthogonal).
(Proof Sketch) If $\Delta(\rho,\sigma)=1$, then $\rho$ and $\sigma$ must have orthogonal supports (disjoint supports). Orthogonal supports imply $\rho\sigma=0$, which further implies $\rho\sigma=\sigma\rho=0$, making them commutative. Therefore, if $[\rho,\sigma]\neq 0$, then $\Delta(\rho,\sigma)\neq 1$, which implies $\Delta(\rho,\sigma)<1$.

Note: We do not claim here that non-commutative pairs always exist. Existence depends on the design of the world class $\mathcal{W}$ (e.g., implementation variances involving continuous parameters, noise, temperature, clock jitter). The claim is a conditional information-theoretic proposition: If the responsibility boundary involves non-commutative overlaps, the impossibility region of Theorem 3.4 is activated.

4. Strong Replay Closure and No-Broadcasting

We independently prove quantum mechanical impossibility for Requirement (C).

Definition 4.1 [Requirement C#: Strong Replay (Double Supply of Boundary Evidence)]
To guarantee reproducibility in an audit, one must not only recalculate outputs from logs but also be able to reconstruct the "boundary state" itself, which serves as the foundation of correctness. Specifically, there exists a CPTP map (Broadcasting Map) $\mathcal{B}: \mathcal{D}(\mathcal{H}_B) \to \mathcal{D}(\mathcal{H}_B \otimes \mathcal{H}_B)$ such that for any state $\rho \in \mathcal{K} \subset \mathcal{D}(\mathcal{H}_B)$, the following holds: \[ \Tr_1 \mathcal{B}(\rho) = \Tr_2 \mathcal{B}(\rho) = \rho \]

Note: Requirement (C#) represents the Tier-3 (Strongest) formulation, maximizing "preservation of evidence (boundary state itself)" in audits. In practice, weaker forms of reproducibility—such as (i) preservation of classically hashed measurement results, (ii) statistical verification, or (iii) verification based on computational assumptions—may suffice. These forms are not the target of negation in this paper.

Theorem 4.2 [Impossibility via No-Broadcasting Theorem]
If the boundary state set $\mathcal{K}$ contains non-commutative state pairs, a map $\mathcal{B}$ satisfying (C#) does not exist [5]. Therefore, in quantum computing, Strong Replayability ("continuing calculation while preserving evidence (non-invasively)") is principally impossible.

4.1 Relation to Existing Quantum Verification Research

Proposition 4.3 [Non-Contradiction and Specification Difference]
"Classical Verification of Quantum Computations" by Mahadev et al. [6] performs verification under interactive protocols and assumptions of computational hardness (e.g., LWE). Its requirements differ from the ARCPU standards defined herein (simultaneous satisfaction of (A♦) Non-Invasiveness, (B★) Minimax Responsibility Boundary, and (C#) Strong Replay). The impossibility theorem in this paper demonstrates that ARCPU standards impose excessive (physically impossible) demands on quantum systems, and does not negate the effectiveness of existing computational verification methods.

5. Conclusion: Implications for Specification Compromise

We have demonstrated that satisfying the ARCPU standards defined in this paper (especially the Tier-2/3 worst-case closure under information-theoretic guarantees) is impossible in certain parameter regions, or at least cannot avoid the performance tradeoff bound given by Theorem 3.4. Any counterargument to this conclusion does not constitute a mathematical refutation, but rather a declaration of "Specification Compromise" in one of the following forms:

References

[1]
J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018. (Diamond Norm & Channel Distance)
[2]
C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, 1976.
[3]
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Ed., Cambridge University Press, 2010.
[4]
C. A. Fuchs and A. Peres, "Quantum-State Disturbance versus Information Gain," Physical Review A, vol. 53, 1996.
[5]
H. Barnum et al., "Noncommuting Mixed States Cannot Be Broadcast," Phys. Rev. Lett., 76, 2818, 1996.
[6]
U. Mahadev, "Classical Verification of Quantum Computations," SIAM J. Comput., 51(1), 2022.

【Note on the Nature of this Report】

This document is an Engineering Audit Report evaluating standards based on "System Accountability" and "Auditability under Finite Resources" for industrial implementation, rather than a pursuit of pure mathematical truth. The mathematical models are constructed to maximally visualize operational risks in reality.

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