Entanglement verification is a cryptographic functionality that allows one or more parties to verify the presence and quality of quantum entanglement between distant or untrusted devices or parties[1,2]. This functionality is crucial in quantum networks and cryptographic protocols where trust in devices or remote quantum systems cannot be assumed. This is a fundamentally “quantum” functionality. 

At a high level, the goal is to certify that two or more systems share a quantum state exhibiting genuine entanglement, often under adversarial conditions, using only limited trusted operations (e.g., classical control, limited quantum capabilities, or statistical tests). In cryptographic contexts, this functionality often ensures that remote devices are correlated in ways that cannot be explained by classical strategies, even if the devices are maliciously controlled.

Entanglement verification protocols are typically formulated in settings where:

• The verifier has limited or no quantum capabilities (e.g., classical or semi-quantum),
• The provers (or devices) are untrusted, and
• The security guarantees are statistical or device-independent, based on observed violations of Bell inequalities.

There is no direct classical analogue to entanglement verification, since this is a uniquely quantum functionality. 
The device-independent nature of quantum entanglement verification distinguishes it even more sharply from classical analogues: it can be realised without trusting the internal workings of the devices involved.

• Quantum Key Distribution (QKD): E.g., Ekert protocol (E91) uses entanglement-based correlations to distribute secret keys.
• Device-Independent Cryptography: Entanglement verification is foundational for protocols that guarantee security from the violation of Bell inequalities.
• Delegated/Blind Quantum Computation: Verifying entanglement between server devices ensures they are quantum and non-colluding[3].
• Quantum Network Benchmarking: Certification of entanglement distribution across quantum communication links.

The cryptographic properties of entanglement verification depend on the setting (device-dependent vs. device-independent), but several key features apply generally:

Soundness:  If the devices are not genuinely entangled, they will fail the verification test with high probability. Typically analyzed statistically, using the CHSH inequality or other Bell-type tests.

Completeness: Honest devices sharing a maximally entangled state (e.g., a Bell pair) will pass the test with high probability.

There are additional properties such as:

• Device Independence: In device-independent settings, verification is based purely on observed statistics: no trust is required in the internal operations of the devices. Security is guaranteed even when the devices are built or controlled by an adversary, as long as they cannot communicate during the test.
• Self-Testing:  Some entanglement verification protocols also perform self-testing, certifying that the shared state and measurements must (up to local isometries) be close to a specific entangled state (e.g., a Bell state or GHZ state).
• Rigidity: In certain protocols like [1], there is a property called rigidity that ensures that any strategy achieving close to optimal violation must essentially use the correct entangled state and measurements.

Multipartite Generalizations: Verification protocols exist for GHZ states, W-states, and graph states, allowing certification of entanglement in multi-party settings.

Limitations: Verification assumes space-like separation (no communication during tests) and often needs statistical confidence over many runs. Finite statistics and noise are practical challenges.

1. Reichardt, Ben W., Falk Unger, and Umesh Vazirani. “Classical command of quantum systems.” Nature 496, no. 7446 (2013): 456-460.
2. Šupić, Ivan, and Joseph Bowles. “Self-testing of quantum systems: a review.” Quantum 4 (2020): 337.
3. Broadbent, Anne, Joseph Fitzsimons, and Elham Kashefi. “Universal blind quantum computation.” In 2009 50th annual IEEE symposium on foundations of computer science, pp. 517-526. IEEE, 2009.