Source-linked AI summary
Quantum discord as a resource for quantum cryptography
Stefano Pirandola
TL;DR
The paper asks when quantum discord, rather than entanglement, is sufficient for secure QKD. It analyzes trusted-noise device-dependent protocols and contrasts them with untrusted-noise and ideal scenarios. It finds that non-zero discord can support secure, entanglement-free device-dependent QKD, whereas entanglement is necessary in device-independent and ideal QKD, with discord remaining an upper bound on key rates.
Problem
The paper addresses whether secure QKD can be achieved without entanglement and identifies the operational role of quantum discord across trusted-noise and untrusted-noise settings.
Method
The paper models device-dependent QKD with an inaccessible trusted-noise purification system P, recasts prepare-and-measure protocols as classical-quantum discord-based schemes, and compares them with device-independent and ideal protocols.
Results
Non-zero discord supports secure device-dependent QKD even without entanglement, while entanglement remains necessary for device-independent and ideal QKD; discord upper-bounds their optimal key rates.
Takeaways & Limitations
Quantum discord is necessary for secure QKD and can replace entanglement as the cryptographic resource only when trusted noise remains inaccessible to Eve.
Abstract
from arXiv · showhide
Quantum discord is the minimal bipartite resource which is needed for a secure quantum key distribution, being a cryptographic primitive equivalent to non-orthogonality. Its role becomes crucial in device-dependent quantum cryptography, where the presence of preparation and detection noise (inaccessible to all parties) may be so strong to prevent the distribution and distillation of entanglement. The necessity of entanglement is re-affirmed in the stronger scenario of device-independent quantum cryptography, where all sources of noise are ascribed to the eavesdropper.
A Bob Alice
Device-dependent QKD can remain secure with trusted noise inaccessible to Eve, even when entanglement is absent, provided input discord is non-zero. When noise is untrusted or absent, entanglement again governs security, while discord bounds achievable key rates.
- Device-dependent QKD: Device-dependent QKD is the only scenario where secure key distribution can occur without distillable or bound entanglement, when trusted noise is present.Trusted noise is represented by an inaccessible purification system P, yielding K ≥ Ic.
- Device-dependent QKD: Non-zero input discord D(A|a) > 0 enables secure device-dependent protocols even when entanglement is completely absent.The paper explicitly gives separable Gaussian states as an example.
- Device-dependent QKD: Prepare-and-measure protocols based on non-orthogonal states can be recast as entanglement-free discord-based protocols using classical-quantum states.Security is preserved when the purification of the classical-quantum state remains inaccessible to Eve.
- Device-independent QKD: If all extra noise is untrusted and controlled through side channels, QKD becomes equivalent to entanglement distillation.In this device-independent setting, discord remains necessary but is reduced to an upper bound on coherent information and key rates.
- Secret-key rates: The optimal forward rate equals output discord minus Eve’s entanglement of formation, with the discord orientation linked to the reconciliation direction.The reverse rate has the analogous expression with D(B|A) and Ef(B, E).
- Ideal QKD: For ideal QKD with no extra-noise system P, discord still upper-bounds optimal direct and reverse key rates, and the bound can be tight in reverse reconciliation.A continuous-variable protocol over a pure-loss channel can achieve K(◭) = D(B|A).
Supplementary Material
The protocol constructs secret-key rates for separable Gaussian states under collective Gaussian attacks and finds broad parameter regions where both reconciliation rates are positive despite absent entanglement. In pure-loss channels, reverse reconciliation achieves a rate equal to output discord and remains positive for every nonzero transmissivity.
- Protocol and state: A separable Gaussian input state with nonzero discord is sent through an entangling-cloner attack, with Bob receiving one output and Eve storing the other modes.The input correlation parameter satisfies |g| ≤ µ − 1; except at g = 0, the state has nonzero discord.
- Protocol and state: Alice and Bob heterodyne their modes to obtain correlated variables X and Y, while conditional Gaussian states and covariance matrices determine Eve’s information.The construction computes Eve’s average and measurement-conditioned covariance matrices, spectra, and entropies.
- Rate calculation: The key rates K(Y |X) and K(X|Y ) are obtained by subtracting Eve’s Holevo information from Alice and Bob’s mutual information.The two Holevo quantities are I(E, X) = S(E)−S(E|X) and I(E, Y ) = S(E)−S(E|Y ).
- Rate calculation: Wide parameter regions yield strictly positive direct- and reverse-reconciliation rates even though the input and output states are not entangled.The result is exhibited using maximum separable-state correlation g = µ−1 and the large-modulation limit µ →+∞.
- Rate regions: For a pure-loss channel with ω = 1, the asymptotic rate is positive for any τ > 0.532.The figure maps positivity as a function of transmissivity τ and thermal variance ω; reverse reconciliation has a wider positive region at low ω.
- Discord bound: In the ideal reverse-reconciliation scenario, the optimal backward rate equals the output discord, and under pure loss this rate is positive for every 0 < τ < 1.The rate can be achieved with heterodyne detection at Bob and coherent detection at Alice.