Researchers have established a definitive link between the maximum achievable key rate in quantum key distribution and the tolerable error rates of practical protocols. The work demonstrates a necessary and sufficient condition for strictly positive two-way assisted capacities of the qubit Pauli channel, revealing a fundamental limit on reliable information transmission.

This analysis identifies the possibility of undiscovered discrete-variable QKD protocols capable of tolerating higher quantum bit error rates than currently known, suggesting existing designs may be unnecessarily conservative. The Pirandola-Laurenza-Ottaviani-Banchi bound establishes that -log_2(1-η) is the maximum possible number of key bits per channel use achievable in repeaterless QKD.

PLOB Bound & QBER Limits for Discrete-Variable QKD

This connection between the PLOB bound and practical key rate limits had remained largely unaddressed until now, with new analysis translating QBER thresholds into bounds on maximum transmission distance for discrete-variable QKD protocols over both fiber and free-space links. Researchers have extended this analysis to repeater-assisted protocols, revealing fundamental limits on noise robustness and identifying gaps in existing literature.

This finding challenges existing assumptions about noise tolerance and opens avenues for designing more robust QKD systems. The analysis establishes a specific QBER, denoted as E_μ, which is a lower bound for the overall protocol QBER, detailed in Eq. (10) of the study, and provides a framework for modeling QBERs with greater precision.

By combining noise models with the established QBER limits, the research team calculated maximum achievable distances for QKD protocols under various conditions. Assuming a symmetric scenario where QBERs across multiple mutually unbiased bases (MUBs) are equal, the study presents a best-case estimate for protocol performance. However, the researchers caution that achieving a positive key rate when the total QBER approaches 1/2 does not guarantee the existence of a corresponding repeater-based protocol.

The team found that a similar limitation applies to 3-MUB protocols. Understanding these limits is essential for designing protocols that approach optimal performance. “While this provides the maximum key rate achievable in repeaterless QKD, it does not directly impose a bound on the maximum quantum bit error rate (QBER) that is tolerable by discrete-variable (DV) protocols,” the researchers write, highlighting the nuanced relationship between theoretical bounds and practical implementation.

Qubit Pauli Channels Define QKD Error & Transmission

Diffraction-limited free-space implementations can achieve exceptionally long transmission distances, supporting the feasibility of deep-space quantum communications, according to new analysis of qubit Pauli channels and their impact on quantum key distribution. The work establishes a direct link between the characteristics of these channels, specifically, the interplay between photon loss and qubit errors, and the ultimate reach of secure quantum communication systems.

Quantum channels in discrete-variable QKD involve both a lossy channel transmitting photons with a probability denoted as η, and a qubit Pauli channel affecting the encoding degree of freedom. The lossy channel governs detection rates, while the Pauli channel dictates the quantum bit error rate, together determining the achievable secret key rate.

This framework allows for a precise connection between Pauli error probabilities and the bit and phase error rates observed in QKD protocols utilizing multiple mutually unbiased bases. For protocols relying on two MUBs, like the BB84 protocol, the bit error rate E_X and phase error rate E_Z satisfy specific relationships; similarly, protocols employing three MUBs, such as the six-state protocol, adhere to additional constraints.

The analysis highlights a particularly simple Pauli channel, the bit-and-phase-flip (BPF) channel, with errors limited to the X and Z bases, making it compatible with two-MUB QKD systems. The team demonstrated that for a qubit Pauli channel, and for a depolarizing channel, which introduces equal errors across all bases, the two-way assisted capacity C_2, encompassing entanglement distribution, quantum information transmission, and QKD, satisfies a critical condition.

This condition, derived from previous work establishing ultimate rates in the absence of relays, provides equivalent criteria for a quantum channel to achieve a strictly positive capacity. The proof relies on the nonpositive partial transposition of the two-qubit Choi state of a qubit Pauli channel, a characteristic not generally extendable to higher-dimensional systems like qutrits where bound entanglement exists. Each link in a quantum repeater chain introduces both a lossy channel and a Pauli channel, with the overall capacity of the chain determined by the bottleneck value across all links.

Specifically, if each link implements a 2-MUB protocol with QBERs E_Z^i and E_X^i, the condition for successful key extraction is. Violating this threshold prevents any key rate from being generated. The paper states the mathematical relationship governing the chain’s performance. This finding has implications for designing more robust and efficient quantum communication networks capable of spanning greater distances.

Two-Way Assisted Capacity Criteria for Quantum Channels

This NPT property, detailed in a related publication, underpins the derivation of threshold QBER values for discrete-variable QKD (DV-QKD) protocols utilizing either two or three mutually unbiased bases (MUBs). Specifically, the researchers found that for a depolarizing channel, the condition for positive capacity is met when certain error probabilities are satisfied, offering a quantifiable benchmark for protocol viability.

The analysis models quantum channels in DV-QKD as involving both a system carrier and a degree of freedom used for encoding and decoding, acknowledging the interplay between these elements. The researchers examined specific Pauli channels, including the bit-and-phase-flip (BPF) channel, the simplest model compatible with two-MUB QKD protocols, and the depolarizing channel, parametrized by probabilities and acting on qubit states.

Building on prior work that established ultimate rates for entanglement distribution and QKD in the absence of relays, the team derived analytical expressions for two-way assisted entanglement distribution, quantum, and secret-key capacities for specific channels like the bosonic lossy channel and dephasing channels. These capacities, denoted as D₂, Q₂, and K₂ respectively, coincide for the channels studied, providing a unified framework for understanding information transfer limits.

For a sequence of Pauli channels P_i with probabilities, they defined a bottleneck value to represent the overall channel quality. If this threshold is violated, no key rate can be extracted, highlighting a critical design constraint for quantum networks. Our investigation not only establishes the fundamental limits of noise robustness for DV-QKD, but also reveals gaps in the existing literature. In particular, we identify the possibility of undiscovered DV-QKD protocols capable of tolerating higher QBERs than those currently known.

Maximum Tolerable QBER with Two Mutually Unbiased Bases

The Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound defines the maximum number of key bits achievable per channel use in quantum key distribution, specifically where η represents channel transmissivity; this limit, while known for repeaterless QKD, had not previously been directly linked to tolerable quantum bit error rates in discrete-variable protocols. Researchers determined that QKD remains viable as long as a specific inequality holds:. This threshold arises from the interplay between QBERs and the probability of identity Pauli errors, which diminish as error rates increase.

In the most challenging scenario, where an eavesdropper introduces no Y errors, the maximum QBER is bounded, ensuring a positive key rate if this limit is not exceeded. “If this inequality is satisfied, then there certainly exists a QKD protocol with a positive key rate,” the paper states, highlighting a clear criterion for successful quantum communication. Extending this analysis to quantum repeaters, the team considered chains of links, each introducing both loss and Pauli noise. A violation of this threshold definitively prevents key extraction, emphasizing a critical design constraint for practical quantum networks.

The implications of this work extend beyond current protocols, suggesting the possibility of undiscovered BB84-like protocols and advanced data processing techniques. “Our results imply the existence of undiscovered BB84-like protocols and/or data processing methods that are able to overcome the best performance known so far,” the researchers conclude. This suggests that existing protocols may be unnecessarily conservative in their noise tolerance, and that further exploration of protocol design and signal processing could unlock significantly improved performance.

Three MUB QKD Protocols & Symmetric QBER Thresholds

The bit-and-phase-flip channel, a specific type of Pauli channel inducing only X and Z errors, presents the simplest QKD model compatible with protocols utilizing two mutually unbiased bases (MUBs). Researchers established a direct link between observed quantum bit error rates (QBERs) and the possibility of secure key exchange within these two-MUB systems, finding QKD remains viable as long as the QBER satisfies a specific condition.

According to Eq. In scenarios where an eavesdropper avoids introducing Y errors, the analysis reveals a maximum QBER bound for the BPF channel. This threshold of 1/4 coincides with the QBER induced by an intercept-resend attack against the BB84 protocol, suggesting a fundamental connection between these two approaches to quantum communication security.

The team’s findings indicate a QKD protocol can achieve a positive key rate up to this limit with appropriate data processing, prompting investigation into whether existing protocols are unnecessarily conservative in their noise tolerance. Previous studies have shown BB84 can be secured up to with two-way data processing, and more recently up to, raising the question of whether even more powerful mechanisms can approach the 1/4 upper bound.

Extending the analysis to protocols based on three MUBs, incorporating observed QBERs for X, Z, and Y errors, the researchers derived a similar threshold. They found that QKD is possible as long as the combined QBERs satisfy a specific relationship, ultimately leading to a symmetric condition where E<1/3. This limit also corresponds to the threshold for an intercept-resend attack, reinforcing the pattern observed with two-MUB protocols.

To assess the maximum distance achievable by QKD protocols, the researchers combined their noise models with the established QBER thresholds, assuming a symmetric scenario where all MUBs share a common QBER value. This assumption represents a best-case scenario, allowing for an upper bound on the maximum distance.

For a 2-MUB protocol, the bound is determined by a formula incorporating the 1/4 threshold, while a 3-MUB protocol yields a slightly modified equation. The work provides distances beyond which QKD protocols are certainly not secure, offering a rigorous framework for evaluating performance and designing more robust quantum systems.

👉 More information
🗞 Fundamental Limits on Quantum Bit Error Rate and Distance in Quantum Key Distribution
✍️ Stefano Pirandola
🧠 DOI: http://link.aps.org/doi/10.1103/3bgy-tpy9

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