Basis-Independent Coherence & Dimension Witnesses

• Basis-independent coherence and dimension witnesses are frameworks that certify quantum superposition and minimal Hilbert space dimensions without relying on a fixed measurement basis.
• They employ unitary-invariant overlap measurements and witness inequalities, using techniques like interferometry and HOM interference to assess quantum properties.
• These methods are crucial for scalable benchmarking, device-independent validation, and secure quantum communication in settings with unknown or fluctuating reference frames.

Basis-independent coherence and dimension witnesses constitute a set of theoretical and experimental methodologies that enable certification of quantum coherence and Hilbert space dimensionality without recourse to a predefined reference basis. These approaches are operationally grounded in measurement statistics, overlap relations, and resource-theoretic frameworks that assess genuinely quantum features—such as superposition and the necessity of higher-dimensional state spaces—regardless of the specific coordinate system used. They are indispensable for scalable benchmarking, foundational analysis, and device-independent applications in quantum information science, especially in scenarios where the physical implementation or the proper basis choice is unknown or unreliable.

1. Definitions: Basis-Independence, Coherence, and Dimension Witnesses

Basis-independent coherence is defined as the property of a quantum ensemble, or set of states, that cannot be rendered incoherent (i.e., all simultaneously diagonal) in any orthonormal basis. This differs fundamentally from conventional, basis-dependent coherence measures which rely explicitly on the choice of a computational or measurement basis.

A dimension witness is any operationally constructed function—typically a linear or nonlinear combination of observed probabilities, overlaps, or correlations—which sets a lower bound on the Hilbert space dimension necessary to reproduce the experimentally observed data. Basis-independent dimension witnesses are constructed such that their evaluation and interpretation are invariant under unitary transformations on the underlying Hilbert space. In such witnesses, the observed violation of certain inequalities certifies that the minimal dimension required for the system exceeds a known threshold, and, when quantum correlations or superpositions are identified, that the behavior cannot be simulated classically.

This general framework subsumes:

• Set coherence measures, which quantify coherence properties of a set of quantum states independent of basis by checking simultaneous diagonalizability (Designolle et al., 2020).
• Overlap- and Gram-matrix-based witnesses, which exploit unitary invariance of overlaps $r_{ij} = \mathrm{Tr}[\rho_i \rho_j]$ to define coherence and dimensionality criteria (&&&1&&&, Giordani et al., 2023, Junior et al., 16 Sep 2025).
• Absolute dimension and irreducible dimension concepts, which capture the intrinsic minimal Hilbert space dimension realized or required by a quantum process or ensemble, unaffected by classical postprocessing, sequential composition, or arbitrary basis rotation (Cong et al., 2016, Bernal et al., 3 Sep 2024).

2. Mathematical Formulation and Witness Inequalities

The mathematical structure of basis-independent coherence and dimension witnesses is largely captured by inequalities on measurable invariants, most notably pairwise overlaps. For a set of $n$ states $\{\rho_1, ..., \rho_n\}$, the overlaps $r_{ij} = \mathrm{Tr}[\rho_i \rho_j]$ are invariant under global unitary transformations.

• Classical Polytope Bounds: If all $n$ states are simultaneously diagonalizable (i.e., incoherent), then the vector of overlaps must belong to a convex polytope defined by classical consistency relations. For three states, this yields the inequalities

$$r_{12} + r_{23} - r_{13} \leq 1, \quad r_{12} - r_{23} + r_{13} \leq 1, \quad -r_{12} + r_{23} + r_{13} \leq 1.$$

Violation of any of these conditions certifies basis-independent quantum coherence (Giordani et al., 2021, Giordani et al., 2023, Wagner et al., 2022, Junior et al., 16 Sep 2025).

• Higher-Dimensional and Recursive Witnesses: For larger ensembles, recursively constructed inequalities generalize the basic three-state witness. For example,

$$h_n(\vec{r}) = h_{n-1}(\vec{r}) + r_{0, n-1} - \sum_{i=1}^{n-2} r_{i, n-1} \leq 1$$

for $n \geq 4$. The degree of violation is bounded by the Hilbert space dimension and can be used as a dimension witness (Giordani et al., 2023).

• Quantum Bounds and Convex Constraints: The quantum mechanically allowed region for overlap vectors is strictly larger than the classical polytope, but for a set that spans at most a $d$-dimensional Hilbert space, the region is still constrained. For three pure states, the Gram matrix positivity condition leads to

$$1 - r_{12} - r_{13} - r_{23} + 2 \mathrm{Re}(\Delta_{123}) \geq 0,$$

where $\Delta_{123}$ is the Bargmann (triple product) invariant (Fernandes et al., 22 Mar 2024). For more states, the set of simulatably accessible overlap data is further limited.

• Analytical and SDP-Based Dimension Witnesses: For an ensemble $\{\rho_x\}$, analytical witnesses and semidefinite programming (SDP) approaches test whether the measurement statistics $\mathrm{Tr}[\rho_x M_{b|y}]$ can be reproduced with convex mixtures of $r$-dimensional preparations (possibly with noise). Violation of the analytical bound

$$W(\mathcal{E}) \leq \sum_{i=1}^r \lambda_i\left( \sum_x O_x \right)$$

(with $O_x = \sum_{b, y} c_{b x y} M_{b|y}$, $\lambda_i$ the leading eigenvalues) witnesses absolute dimension $> r$ (Bernal et al., 3 Sep 2024).

3. Experimental Implementation

Several experiments have implemented basis-independent coherence and dimension certification, most notably via interferometric and overlap-based protocols.

• Interferometric Overlap Measurement: For photonic or optical spatial modes (e.g., integer and fractional OAM), the overlap $r_{ij}$ is measured by interfering two modes and recording the intensity pattern at a detector. The modulus of the cross-term in the Fourier transform of the interference pattern directly yields the overlap:

$$r_{AB} = |\langle \psi_A | \psi_B \rangle|^2 \simeq |\tilde{I}(\vec{q})|^2$$

where $\tilde{I}(\vec{q})$ is the spatial Fourier transform at the appropriate spatial frequency (Junior et al., 16 Sep 2025). A crucial operational aspect is that a single intensity image per pair suffices and active phase stabilization is unnecessary.

• Hong-Ou-Mandel (HOM) Interference: For quantum states of light, SWAP tests implemented via HOM interference determine overlaps by mapping coincidence rates to $r_{ij}$ (Giordani et al., 2021, Giordani et al., 2023). This approach generalizes to higher-dimensional quantum states and is robust against reference frame misalignment.
• Programmable Multi-Port Interferometry: Coherence and dimension witnesses for qudits have also been certified in programmable universal photonic processors capable of generating arbitrary high-dimensional quantum state superpositions and measuring all required pairwise overlaps (Giordani et al., 2023).
• Fractional OAM State Certification: For fractional OAM light modes, the witness framework demonstrates that measured sets of overlaps cannot be explained by integer OAM (or lower-dimension) bases, certifying both coherence and dimensionality in a basis-independent manner. The overlap between two fractional OAM states of charges $\ell$ and $\ell'$ with phase discontinuities $\alpha$ and $\alpha'$ is given analytically as

$$|\langle \ell', \alpha' | \ell, \alpha \rangle|^2 = \operatorname{sinc}^2[(\ell-\ell')\pi] - \frac{\beta(2\pi - \beta)}{\pi^2} \sin(\ell \pi) \sin(\ell' \pi) \cdot \operatorname{sinc}\left(\frac{(\ell-\ell')\beta}{2}\right) \operatorname{sinc}\left((\ell-\ell')\left(\pi - \frac{\beta}{2}\right)\right)$$

with $\beta = \alpha - \alpha'$ (Junior et al., 16 Sep 2025).

4. Resource Theoretic and Operational Significance

Basis-independent witnesses are tightly integrated into quantum resource theories.

• Set Coherence and Operational Games: Set coherence quantifies the role of a collection of states in tasks such as quantum state or subchannel discrimination, providing a maximal or mean robustness measure $R(\vec{\rho})$ reflecting the advantage of the set over any simultaneously diagonalizable set (Designolle et al., 2020). The operational meaning is precise: $R_1(\vec{\rho})$ is the increase in average success probability in discrimination tasks relative to any incoherent ensemble.
• Dimension Witnesses as Resource Certifiers: Absolute dimension witnesses link the accessible information or discrimination success probability to the minimal necessary Hilbert space dimension. In high-dimensional communication protocols, these witnesses provide device-independent guarantees that the system genuinely utilizes the claimed Hilbert space dimension and is not functionally classical or lower-dimensional due to noise, postprocessing, or hardware imperfections (Bernal et al., 3 Sep 2024).
• Irreducible Dimension: The irreducible dimension framework excludes strategies that sequentially combine lower-dimensional subsystems to simulate apparently high-dimensional behavior, instead certifying the necessity of genuinely $d$-dimensional resources for the observed correlations (Cong et al., 2016).

5. Graph-Based and Unitary-Invariant Witness Methodologies

The general framework for certifying basis-independent coherence and dimension expands through:

• Graph-Theoretic Polytopes: Overlaps, or more generally, pairwise “confusabilities” for an event graph $G$ (with overlaps or correlations assigned to edges) must lie within a classical polytope defined by transitivity and equivalence class connectivity. Violation of cycle and facet inequalities derived from this polytope witnesses not only basis-independent coherence but also more general nonclassical properties such as nonlocality and contextuality (Wagner et al., 2022).
• Unitary-Invariant Bargmann Invariants and Imaginarity: Basis-independent witnesses extend to imaginarity (resourcefulness of complex amplitudes). For tuples of pure states, characterization of allowable (e.g., third-order) Bargmann invariants, subject to positive semidefiniteness of the associated Gram matrix, provides a route to witness quantum features that cannot be captured by real quantum theory. For four-state sets, pairwise overlaps suffice; for three-state sets, global invariants are needed (Fernandes et al., 22 Mar 2024).
• Simulation and SDP Tools: Analytical and semidefinite programming approaches implement the above ideas in practical settings, determining the minimal dimension required to simulate given ensembles (absolute dimension) or constructing witnesses for ensemble robustness to noise and device imperfections (Bernal et al., 3 Sep 2024).

6. Applications, Limitations, and Future Directions

Basis-independent coherence and dimension witnesses are applicable across quantum information processing:

• Quantum Communication and Key Distribution: Secure quantum protocols can be certified to employ genuinely high-dimensional and coherent states even when the basis is fluctuating or unknown, enhancing security and capacity (Cai et al., 2016, Giordani et al., 2023).
• Benchmarking and Device Validation: Programmable photonic platforms and multi-mode quantum processors can be systematically validated for dimensionality and superposition resource, enabling scalable verification in regimes where full tomography is intractable (Giordani et al., 2023, Junior et al., 16 Sep 2025).
• Robustness Under Noise: Witnesses remain meaningful and practically certifying even under loss, dark counts, phase noise, and in the presence of arbitrary basis rotations (Bowles et al., 2013, Junior et al., 16 Sep 2025).
• Metrology and Sensing: Direct certification that quantum coherence and high-dimensionality participate in phase estimation and sensing tasks without basis assumptions ensures that observed quantum advantages reflect genuine resources, not classical or basis-aligned artifacts (Giordani et al., 2021).
• Open Problems: While current techniques can exclude lower-dimensional (or incoherent) models, boundaries for higher $n$ and for mixed or nonorthogonal states are less well-characterized mathematically. Further, the reach of these methods in multipartite, correlated systems, and their integration with device-independent randomness or security analysis in cryptographic protocols, remain active areas of research.

A plausible implication is that as technological platforms for high-dimensional quantum information mature, basis-independent witnesses and absolute dimensionality assessments will become foundational to both certification standards and the practical exploitation of quantum advantage in communication, computation, and sensing.