Dimension Witness Violations in Quantum Systems

• Dimension-witness violations are empirical and theoretical results that certify minimum Hilbert-space dimensions through device-independent tests.
• Protocols use linear, nonlinear, and Bell-type inequalities to irreducibly detect high-dimensional quantum correlations and entangled states.
• These techniques underpin practical applications in quantum key distribution, randomness generation, and leakage detection in quantum computing.

Dimension-witness violations are empirical or theoretical results in quantum information that certify, in a device-independent fashion, a lower bound on the Hilbert-space dimension required to reproduce observed data or specific statistical correlations. These violations appear when certain carefully constructed inequalities—linear, nonlinear, or otherwise—are exceeded by experimental results, ruling out any classical or quantum model of lower dimension. The dimension thus witnessed can be classical, quantum, or even irreducible quantum dimension, depending on the protocol and assumptions. Dimension-witness violations underpin certification of high-dimensional entanglement, randomness generation, and quantum computational leakage, with direct impact on quantum cryptography and foundational tests.

1. Classification and Principle of Dimension Witnesses

Dimension witnesses (DW) fall into three broad categories:

• Device-dependent DWs: Require detailed knowledge of the state and measurement operators acting on a system and can only certify dimension with trusted models.
• Device-independent prepare-and-measure DWs: Use only input-output statistics from a black-box scenario; they do not distinguish between classical and quantum carriers if both obey the same dimensionality constraints.
• Device-independent Bell-based DWs: Rely on the violation of Bell-type inequalities, signature of nonlocal correlations. These ensure the underlying dimension is quantum and that it cannot be simulated by any classical no-signaling resource, even of infinite dimension (Cai et al., 2016, Gallego et al., 2010, &&&2&&&).

Formally, if an observed statistical quantity $W$ exceeds a dimension-dependent bound $C_d$, one certifies that the system's dimension must strictly exceed $d$.

2. Theoretical Construction of Dimension-Witness Violations

A. Linear and Nonlinear Witnesses

Most DWs are constructed as linear functionals $W(\vec E)=\sum_{x,y} w_{xy}E_{xy}$ over sets of observed correlators $E_{xy}$. Bounds $C_d$ and $Q_d$ are established for classical and quantum $d$-dimensional models, respectively. Violation of these bounds serves as a certificate.

Nonlinear witnesses (e.g., determinant-based forms, delayed-vector matrix rank) exploit more intricate dependencies, enabling robust detection of higher-dimensional leakage or irreducibility (Strikis et al., 2018, Batle et al., 2023, Białecki et al., 2023):

$W_k = \det[p_{ij}],\quad W_N = \det V_{jk},$

where $V_{jk}$ is a Hankel or Toeplitz matrix of delayed vectors or probabilities.

B. Bell-type Dimension Witnesses

Bell-based DWs utilize multipartite scenarios: for instance, the violation of the CGLMP$_4$ inequality certifies that no less than $d=4$-dimensional quantum systems (ququarts) are necessary to produce the data (Cai et al., 2016). For the CGLMP$_4$ inequality:

$$I_4 = P(a \leq b | 0, 0) + P(a \geq b | 0, 1) + P(a \geq b | 1, 0) - P(a \geq b | 1, 1) - 2 \leq 0\quad\text{(LHV bound)}.$$

Quantum mechanics allows $I_4^{\mathrm{Q}}\approx0.364762$. When experimental results yield $I_4 > 0.315$ (the two-qutrit negativity bound), one certifies $d \geq 4$ in a device-independent manner.

3. Experimental Demonstration and Verification

Photonic Platforms

Dimension-witness violations have been demonstrated using:

• Pairs of photons entangled via spontaneous parametric down-conversion, exploiting both polarization and orbital angular momentum degrees of freedom (Cai et al., 2016, Hendrych et al., 2011).
• Prepare-and-measure tests with variable delays and holographic spatial-light modulators to select high-dimensional subspaces (Aguilar et al., 2017).
• Multi-photon and multi-mode experiments to certify irreducible dimension (i.e., genuine high-dimensionality not simulatible by product systems) (Aguilar et al., 2017).

Table: Experimental dimension-witness results for $I_4$ (from (Cai et al., 2016)).

State	Measured $I_4$	Certification
MES	$0.333 \pm 0.007$	$d \geq 4$
MVS	$0.354 \pm 0.009$	$d \geq 4$ ($\sim 4\sigma$)

Quantum Computing Hardware

The method-of-delays witness, implemented on IBM Quantum devices, detects leakage into higher dimensions (e.g., transmon qubits evolving beyond the nominal two levels). A violation is declared when the rank of the Hankel matrix formed from repeated measurements exceeds the theoretical dimension-bound ($d^2$) (Strikis et al., 2018, Białecki et al., 2023).

4. Scalability, Robustness, and Variants

Detection Efficiency and Noise Robustness: Dimension-witness protocols require calibration against detector inefficiency and shared randomness. For linear witnesses, the observed value scales as $\eta W_{\max}$ under loss; certification holds as long as $\eta > \frac{W_C(d)}{W_Q(d)}$ (Dall'Arno et al., 2012).

Binary Outcome Protocols: Recent protocols achieve dimension witnessing with only binary outcome measurements, circumventing the need for high-$d$ projective measurements and enabling practical certification of arbitrarily large dimensions (Czechlewski et al., 2018).

Irreducibility Certification: Advanced witnesses distinguish truly irreducible $d$-dimensional systems from ensembles of lower-dimensional subsystems with adaptive classical control. This requires entangled measurements and self-testing protocols coupled to swap-fidelity functionals and SDP-based bounds (Cong et al., 2016, Aguilar et al., 2017).

5. Applications and Impact

Dimension-witness violations underpin:

• Quantum Key Distribution (QKD): Secure randomness generation and key extraction are certified via dimension witnesses in semi-device-independent settings; double classical violation in three-observer protocols enables parallel randomness extraction (Li et al., 2017).
• High-dimensional Quantum Communication: Larger Hilbert-space dimension supports enhanced information bandwidth, reduced vulnerability to eavesdropping, and resource-efficient cryptography (Aguilar et al., 2017).
• Certification of Quantum Resources: Experimental dimension-witness violations are the only device-independent means to assure generation, transmission, and manipulation of high-dimensional entangled states without full system characterization (Cai et al., 2016, Aguilar et al., 2017).
• Leakage Detection in Quantum Computers: Model-independent dimension witnesses constitute effective tools for identifying unadvertised system dimensions and gate calibration drift (Strikis et al., 2018, Białecki et al., 2023).

6. Limitations, Open Problems, and Outlook

Dimension-witness violations are subject to several caveats:

• Robustness to Loss: Statistical confidence may degrade with detection inefficiency, requiring careful analysis of threshold values $\eta_{\text{th}}$ (Dall'Arno et al., 2012).
• Irreducibility Distinction: While some witnesses certify the presence of high-dimensionality, only refined protocols can exclude simulation by sequential or product lower-dimensional systems (Cong et al., 2016).
• Generalizations and Tightness: For many witness constructions, analytic tightness of the bounds below the certified dimension is not guaranteed; numerical approaches dominate for high $d$ (Pal et al., 2017). The scaling of quantum-classical gaps (e.g., for Bell violations) with dimension remains an open question in some settings (Buhrman et al., 2010).
• Practical Feasibility: Implementing genuinely multi-outcome measurements, required by some DWs, poses a technical challenge; recent protocols using binary outcomes or null-tests with single measurements alleviate this issue (Czechlewski et al., 2018, Batle et al., 2023).

Future work focuses on extending dimension-witness protocols to even larger systems, improving robustness to noise and loss, and integrating dimension witnessing into large-scale quantum communication and computation platforms. Applications in device-independent randomness certification and foundational tests of quantum nonlocality continue to motivate the development of scalable, resource-efficient witness constructions.