Utility of Quantumness (UQ)

• Utility of Quantumness (UQ) is a framework that quantifies quantum resources using generalized robustness, coherence, discord, and entanglement measures.
• It establishes both qualitative and quantitative operational advantages through channel discrimination tasks and precise performance benchmarks.
• By unifying resource theories with practical metrics such as SWaP-C and application readiness levels, UQ guides the design and evaluation of quantum devices in diverse fields.

The Utility of Quantumness (UQ) formalizes the operational value of quantum resources, encompassing foundational resource theory, practical device-level benchmarks, and unified frameworks connecting coherence, discord, and entanglement. UQ quantifies the advantage provided by nonclassical states—whether as measured by generalized robustness in discrimination tasks, speedup considering system overheads, or unified operational distance to classicality—enabling rigorous evaluation and comparison of quantum versus classical strategies across theory and implementation. This article details the definitions, mathematical structures, experimental criteria, and cross-domain applications that jointly constitute UQ as established in contemporary literature.

1. Foundational Definitions: Generalized Robustness and UQ

The Utility of Quantumness is rigorously identified with the generalized robustness measure $R_{\mathcal F}(\rho)$ in arbitrary resource theories, where $\mathcal F(\mathcal H)$ is a closed, possibly nonconvex set of free states on $d$-dimensional Hilbert space $\mathcal H$ (Kuroiwa et al., 2023). The primal form is

$$R_{\mathcal F}(\rho) = \min_{s\ge 0,\; \tau\in\mathcal D(\mathcal H)} \left\{ s\;:\;\frac{\rho + s\,\tau}{1+s} \in \mathcal F(\mathcal H) \right\}$$

Intuitively, this quantifies the minimal mixing with an arbitrary state $\tau$ required to "wash out" the quantumness of $\rho$. In convex resource theories, the dual form yields a linear witness $W$ such that

$$1 + R_{\mathcal F}(\rho) = \max_{W \ge 0} \left\{ \operatorname{Tr}[W\,\rho] \;:\; \operatorname{Tr}[W\,\sigma] \le 1 \;\forall\,\sigma \in \mathcal F(\mathcal H) \right\}$$

For nonconvex $\mathcal F$, multi-copy polynomial witnesses $\widetilde W_m(\rho, s)$ on $\mathcal H^{\otimes m}$ (for $m=2,\ldots,d$) are required, jointly satisfying

$$\max_{m}\operatorname{Tr}[\widetilde W_m(\rho, s)\,\rho^{\otimes m}] < 0,\quad \max_{m}\operatorname{Tr}[\widetilde W_m(\rho, s)\,\sigma^{\otimes m}] \ge 0\;\forall\sigma\in\mathcal F$$

for $s < R_{\mathcal F}(\rho)$. Thus, $R_{\mathcal F}(\rho)$ admits a geometric/primal definition and a multi-copy nonlinear witness dual interpretation.

2. Qualitative and Quantitative Operational Advantages

The generalized robustness $R_{\mathcal F}(\rho)$ underpins both qualitative and quantitative advantages in channel discrimination:

• Qualitative (Multi-Copy): For any non-free $\rho\notin\mathcal F$, there exists a multicopy channel discrimination task (using ensembles $\{\Lambda_i^{(m)}\}$ and measurements on $\rho^{\otimes m}$ for some $m$) in which the optimal success probability for $\rho$ strictly exceeds that for any $\sigma\in\mathcal F$ (Kuroiwa et al., 2023). Formally,

$$\min_{\sigma \in \mathcal F} \max_{m} \frac{p_{\rm succ}(\rho^{\otimes m})}{p_{\rm succ}(\sigma^{\otimes m})}> 1$$

• Quantitative (Single-Copy, Multiple Constraints): If $\mathcal F = \bigcup_{k} \mathcal F_k$ with convex $\mathcal F_k$, then the worst-case advantage in single-copy discrimination is exactly $1 + R_{\mathcal F}(\rho)$:

$$\inf_k \max_{\text{tasks}} \frac{p_{\rm succ}(\rho)}{\max_{\sigma\in\mathcal F_k}p_{\rm succ}(\sigma)} = 1 + R_{\mathcal F}(\rho)$$

Thus, $R_{\mathcal F}(\rho)$ is robust—every non-free state provides a genuine advantage and this advantage is exactly quantified in the most adversarial scenario.

3. Unified Frameworks: UQ as a Resource-Theoretic Distance

A general operational framework further unifies quantumness measures through interferometric protocols (Yuan et al., 2017). The UQ of $\rho_{AB}$ is defined via mutual information extracted about phase-encoded inputs under local, weak reference frame-free, or strong adversarial settings: $${\rm UQ}_0(\rho_{AB}) = \max_M I(X:\Phi),\quad {\rm UQ}_{\rm w}(\rho_{AB}) = \min_{J'_A}\,\max_M\,I(X:\Phi),\quad {\rm UQ}_{\rm s}(\rho_{AB}) = \min_{\text{purifications}}\min_{J_A^e}\max_M I(X:\Phi)$$ Yielding specialized resource measures as minimal distances:

• Coherence: $$C_{J_A}(\rho_A) = \min_{\sigma\in\mathcal I_{J_A}} D(\rho_A \|\sigma)$$
• Discord: $D(\rho_{AB}) = \min_{J_A} Q_{J_A}(\rho_{AB})$
• Entanglement: $$E(\rho_{AB}) = \min_{\sigma\in\mathcal S} D(\rho_{AB} \|\sigma)$$

The resource-conversion theorems establish that coherence can be activated into discord, and discord into entanglement by suitable ancilla addition and local operations, preserving the quantitative resources throughout the conversion process.

4. Utility Metrics and Practical Benchmarks

Quantum utility, as an application-centric instantiation of UQ, is operationalized via performance metrics comparing quantum and classical devices of matched SWaP-C (size, weight, power, cost) (Herrmann et al., 2023). Quantum utility is claimed when one of the following holds:

• Less computing time
• Less power consumption
• Greater accuracy

Composite benchmark scores for UQ: $$\mathrm{Score}_1 = \frac{\mathrm{Performance}}{\mathrm{Runtime}\times \mathrm{Power}},\quad \mathrm{Score}_2 = \frac{\mathrm{Performance}}{\mathrm{Volume}\times \mathrm{Runtime}\times \mathrm{Power}}$$ Application Readiness Levels (ARLs), from conceptualization (ARL 1) to fully demonstrated quantum utility (ARL 5), structure implementation maturity. Case studies in quantum chemistry, simulation, machine learning, and data analysis instantiate these scores and levels, with variational quantum eigensolvers (VQE) reaching ARL 3 where projected quantum runtimes and power consumption surpass classical HPC baselines at realistic system sizes.

5. Experimental Validation and Milestone Frameworks

Empirical UQ assessments involve milestone-based benchmarking with annealing quantum processors (McGeoch et al., 2023). Three principal milestones adjust the total quantum time to include progressively more realistic overheads:

• $UQ_{M0}$: speedup using pure anneal time only,
• $UQ_{M1}$: incorporates programming and readout overheads,
• $UQ_{M2}$: further adds minor-embedding overheads.

On a D-Wave Advantage QPU, $UQ_{M1}$ achieved wins in 99% of evaluated tasks, and $UQ_{M2}$ in 19%, the latter precisely where classical solvers failed most (large, time-constrained instances). Contrasting convergence profiles between quantum annealing (weak size dependence, distributional sampling) and classical heuristics (strong NP-hard slowdown) further rationalize these results.

Milestone	Quantum Time Definition	Fraction of Tasks with Quantum Utility
$\mathrm{M}_0$	$T_\mathrm{anneal}$	Demonstrated in prior work
$\mathrm{M}_1$	$$T_\mathrm{program} + T_\mathrm{anneal} + T_\mathrm{readout}$$	$99\%$
$\mathrm{M}_2$	$$T_\mathrm{embed} + T_\mathrm{program} + T_\mathrm{anneal} + T_\mathrm{readout}$$	$19\%$

6. Applications, Extensions, and Unifying Perspectives

UQ is established as a universal resource measure governing core aspects of random number generation, self-testing, and discrimination tasks (Yuan, 2017, Yuan et al., 2017, Kuroiwa et al., 2023). Coherence quantifies intrinsic randomness and is distillable into random bits at rates set by its relative-entropy measure. Entanglement underpins device-independent randomness expansion and measurement-device-independent entanglement witnessing, essential for robust cryptographic protocols. The framework admits extensions to alternate metric spaces (trace norm, Bures), multipartite and continuous-variable settings, steering, and nonlocality resource axes.

Empirical and theoretical guidelines recommend explicit reporting of performance/SWaP-C ratios, system-level power and volume, algorithmic readiness classification (ARL), and strict comparison to equally spec’d classical baselines to substantiate utility claims.

7. Synthesis and Outlook

Utility of Quantumness (UQ), whether formalized as generalized robustness, mutual-information extractability, or practical speedup metrics, synthesizes the operational and foundational advantage of quantum states and devices. UQ connects discrimination-type tasks, resource-theoretic axioms, and transformative application criteria, admitting both qualitative and quantitative advantages in nonconvex scenarios and real-world implementations. The measurement and reporting standards emerging from UQ theory and milestones directly shape future benchmarks, device development, and cross-domain deployment of quantum technologies. The current literature mandates rigorous, scenario-aware comparison and provides methods for quantifying, verifying, and realizing UQ in both abstract resource theory and applied engineering contexts (Kuroiwa et al., 2023, Yuan et al., 2017, McGeoch et al., 2023, Yuan, 2017, Herrmann et al., 2023).