Fundamental Limits on QBER and Distance in Quantum Key Distribution

The gist This work establishes fundamental limits on quantum bit error rate (QBER) and achievable communication distance in quantum key distribution by deriving upper bounds based on qubit Pauli channels and applying them to realistic physical noise models.

How it works

The paper establishes a necessary and sufficient condition under which all two-way assisted capacities of the qubit Pauli channel are strictly positive Building on this result, they derive the threshold QBER values required to guarantee the security of a DV-QKD protocol employing either two or three mutually unbiased bases (MUBs) Next, they adopt a standard noise model that relates the QBER to the channel transmissivity and other relevant system parameters Exploiting this relationship, they translate the QBER threshold conditions into bounds on the maximum transmission distance achievable by DV-QKD protocols over fiber-based and free-space links Finally, they extend their analysis from the repeaterless setting to repeater-assisted protocols.

Quantum Channels in DV-QKD

In DV-QKD protocols, there are two quantum channels involved, one affecting the system carrier and the other affecting the degree of freedom used for encoding and decoding The first quantum channel is a bosonic lossy channel that transmits photons with probability η, referred to as the channel transmissivity The second channel is a qubit Pauli channel. Together, they determine the final secret key rate.

The action of a qubit Pauli channel P on a state ρ is defined by P(ρ) = X T k=0 pkPkρP† k, (1) The Pauli error probabilities can be connected with the values of the QBERs in the encoding bases. If security is based on two MUBs, such as X and Z of the BB84 protocol, then EX = p2 + p3 and EZ = p1 + p2 If security is based on three MUBs, X, Z and Y, then EY = p1 + p3.

Capacity Thresholds

The paper establishes criteria for a quantum channel to satisfy C2 = 0 For a qubit Pauli channel, they show that C2(P) = 0 if pmax:= max pk ≤ 1/2 For a depolarizing channel, the condition is C2(Pdepol) = 0 if p ≥ 2/3.

Maximum Tolerable QBER

For protocols based on two MUBs, QKD is possible as long as EX + EZ < 1/2 + p2 Assuming the symmetric condition EX = EZ:= E, this becomes E < 1/4, which coincides with the QBER induced by an intercept-resend attack against the BB84 protocol. For protocols based on three MUBs, they obtain EX + EZ + EY < 1 Assuming symmetry EX = EZ = EY:= E, this leads to E < 1/3, which corresponds to the threshold for an intercept-resend attack.

Noise Model for QBER in DV-QKD

The QBER of a QKD protocol can be related to the total transmissivity η of the lossy channel connecting Alice and Bob The irreducible QBER has the form E = γedet + (1 − γ)Y0/2γ + (1 − γ)Y0, where γ depends on Alice’s source. For a decoy-state QKD protocol with intensity µ, the overall QBER is lower-bounded by Eµ given by Eq. (B7).

Maximum Distance

The maximum distance achievable in fiber is bounded by d < -10 10/α log10 Ω, where Ω depends on the source type and intensity. For a 3-MUB protocol with a single-photon source, the bound is d < 7.73 × 10 7 km. In free space, the diffraction-limited upper bound is d < πw0aR/λr squared Ω', where Ω' takes expressions depending on the source type.

Repeater-Based Protocols

For a repeater chain with N+1 links, the end-to-end two-way assisted capacity C2 is bounded by p min max ≤ 1/2 This translates to the condition that max i of E iX + E iZ < 1/2 for a 2-MUB protocol, or maxi of E iX + E iZ + E iY < 1 for a 3-MUB protocol. The distance constraint identified for the repeaterless configuration also governs the longest individual segment within a repeater chain.

Conclusions

The work establishes maximum tolerable QBER and bounds their maximum distance in both fiber-based and free-space implementations The results imply the existence of undiscovered QKD protocols and/or data processing methods that are able to overcome the best performance known so far. In the free-space scenario, geometric diffraction imposes a fundamental upper bound on the maximum transmission distance

Acknowledgments

This work was supported by the Integrated Quantum Networks (IQN) Research Hub (EPSRC, Grant No. EP/Z533208/1).

Appendix A: Zero-capacity thresholds for repeater chains

The upper bound Φ(P) in Eq. (A1) comes from the relative entropy of entanglement ER (REE) computed over its Choi state σP It is immediate to check that the two-qubit Choi state ρP in Eq. (A2) is an NPT state for pmax > 1/2 Because, in dimensions 2 × 2, all NPT states have nonzero distillable entanglement ED(ρP) > 0 Thus, this basic strategy for entanglement distribution guarantees that entanglement can be distilled from the output state of the Pauli channel if we have pmax > 1/2.

Appendix B: QBER noise modelling

The minimum QBER reads Ek ≃ edetηk + Y0/2ηk + Y0, where ηk is the transmissivity of a k-photon signal. In the case of an ideal single-photon source, then the QBER is given by Eq. (B4) for k = 1.

Appendix C: Zero-capacity thresholds for repeater chains

The end-to-end capacity between Alice and Bob satisfies C2(chain) ≤ min i ER(σPi) = min i Φ(Pi) This means p min max ≤ 1/2 certainly implies C2(chain) = 0. Unfortunately, the opposite cannot be proven using the simple approach of the repeaterless case.

References

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[2] D. Bruss, “Optimal Eavesdropping in Quantum Cryptography with Six States,” Phys. Rev. Lett. 81, 3018 (1998)

[3] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, et al.

1. This system can calculate fundamental limits on QBER and distance by establishing the maximum quantum bit error rate (QBER) compatible with secure QKD and deriving corresponding upper bounds on communication distance based on a fundamental capacity threshold for qubit Pauli channels.

2. The improved AI system can assess the security robustness of existing or proposed QKD protocols by checking if observed error rates satisfy the derived thresholds, such as determining if "EX + EZ < 1/2 + p2 for two MUBs or EX + EZ + EY < 1" for three MUBs.

3. The system can predict achievable transmission distances in fiber and free-space links by translating QBER thresholds into bounds, such as using the derived formula "d < − 10/α log10 Ω for fiber or the diffraction-limited bound d < πw0aR λ / r2 Ω′" for free space.

4. The AI can evaluate the performance of different QKD implementations by considering source types, such as comparing bounds derived from single-photon implementation (γ = η) versus attenuated source with intensity µ, using the relationship "η > − 1/µ ln (1 − Γ)."

5. The system can perform a preliminary analysis for repeater-based networks by checking if the bottleneck value satisfies the capacity condition, specifically ensuring that for a chain of N links, "max i of E iX + E iZ < 1/2 for 2-MUB protocols or max i of E iX + E iZ + E iY < 1" for 3-MUB protocols.

6. The AI can provide a rigorous upper bound on the maximum transmission distance in free space by incorporating the effects of atmospheric extinction and beam spreading, using the decomposition ηch = ηfsηatm and applying constraints like ηatm(d) ≤ 1 and ηfs(d) ≤ ηdiff(d).