Encoding Multiple Qubits Boosts Quantum Communication Efficiency

A new method for entanglement-based quantum key distribution protocols sharply improves efficiency, according to Vera Uzunova and Marcin Jarzyna of the University of Warsaw. Encoding multiple qubits per photon pair optimises performance in challenging conditions, such as those expected in satellite quantum communications where signal strength is inherently low and background radiation is high. The findings reveal that optimal efficiency is achieved when a finite production probability of entangled photon pairs exists, a departure from conventional communication models. This multiqubit encoding approach can improve the secret key rate by up to an order of magnitude when compared to traditional single-qubit schemes.

Multiqubit encoding significantly boosts secret key rates in quantum key distribution

Optimising quantum key distribution protocols using multiqubit encoding enhances the secret key rate by up to an order of magnitude compared to traditional single-qubit schemes. This improvement crosses a critical threshold, enabling viable key exchange even with extremely weak signals previously unsuitable for secure communication. The team achieved this by encoding information across multiple qubits within a single photon, effectively increasing the information density of each transmission.

Cryptography underpins modern communication, constantly evolving to meet the requirements of privacy in the storage, processing and transmission of sensitive data, with applications in healthcare, critical infrastructure, telecommunication, financial services, and government agencies. This new approach allows for optimal efficiency at finite entangled photon pair production probability, a departure from previous limitations demanding vanishingly small signal strength for maximum performance. Researchers refined the multiqubit encoding technique by demonstrating its effectiveness across several established quantum key distribution protocols, including the four- and six-state BBM92 and SARG04 protocols.

They carefully modelled the theoretical limits of secret key photon efficiency, considering realistic scenarios like satellite quantum communications where signal strength is inherently low and background radiation is prevalent. A central challenge in cryptography is the secure distribution of this secret key between two legitimate users, Alice and Bob. The team detailed how encoding multiple logical qubits into a single photon utilises temporal slots, mapping qubit basis states onto specific time intervals using interferometric stages. However, these calculations currently assume ideal conditions and do not fully account for real-world detector efficiencies or atmospheric propagation losses affecting signal fidelity. Quantum key distribution (QKD) promises to improve cryptography by incorporating the laws of quantum mechanics into the security model, potentially enhancing secret key rates by up to an order of magnitude compared to single-qubit schemes.

Multiqubit encoding for enhanced cryptographic key distribution

Multiqubit encoding can enhance the secret key rate by up to an order of magnitude compared to single-qubit schemes. The security of QKD originates in the fundamental no-cloning theorem, which states that it is impossible to perfectly copy an unknown quantum state, preventing eavesdroppers from gaining information without altering the transmission. This security is based not on computational difficulty, typical for classical methods, but solely on the underlying laws of physics, allowing for unconditional security in practice, limited by assumptions about the quantum channel and potential side-channels.

Extending quantum communication range to large distances, and ultimately to a global scale, represents one of the main challenges of QKD. Despite recent progress, current fibre optical demonstrations are usually limited to tens or few hundreds of kilometers and require separate fibre networks, making them exceptionally expensive. A potential solution lies in using satellite and free-space QKD, which relies on the much higher optical transmission of the atmospheric channel than fibres. Successful implementation of QKD for free-space optical links has allowed secure key distribution over hundreds and even thousands of kilometers, including entanglement-based key distribution.

Entanglement based QKD extends key distribution range and removes the need for trusted nodes, further increasing security. However, entangled photon sources are typically characterised by low brightness, which, combined with significant losses due to atmospheric propagation, leads to low received signal strength. To obtain optimal performance of such a QKD link, it is vital to extract as much key as possible per single received photon, analogous to classical communication operating in photon starved regimes where the valid figure of merit is photon information efficiency (PIE). The performance of a QKD link should be quantified by an analogous quantity rather than just the key rate.

Most current photonic free-space quantum communication systems use two-dimensional encoding, where each photon can carry at most a single bit of quantum information. However, spatio-temporal or other degrees of freedom of a photon can be used for high-dimensional encoding, realising multiple qubits transmission by a single photon. Employment of high-dimensional quantum states could increase QKD key rate and improve durability with respect to noise and eavesdropping.

This paper investigates the theoretical limits of secret key photon efficiency by implementing high-dimensional encoding for known entanglement-based QKD protocols. The paper organizes itself as follows: Section II provides an introduction to QKD; Section III describes the multiqubit encoding and reception scheme; Section IV details the corresponding QKD link model; Section V presents results on optimising QKD photon efficiency; and Section VI concludes the paper. The goal of QKD is to establish a secret key between two parties, Alice and Bob. In an entanglement based approach, both parties receive one of two photons from an entangled pair emitted by a light source.

They then perform measurements independently in preselected random bases, rejecting events with differing bases and performing error correction and privacy amplification on the remaining bit sequences. Any noise, errors or unwanted effects result from an action of a possible eavesdropper, Eve. This paper considers s = 4 four- and s = 6 six-state BBM92 protocols, as well as entanglement-based versions of the four- and six-state SARG04 protocols, detailed in App. A. The theoretical asymptotic secret key rate K of any QKD protocol is given by K = I(A, B) −χ(B, E), where I(A, B) is the mutual information between Alice and Bob, χ(B, E) is the Holevo information between Bob and Eve, and we assumed classical data in the postprocessing. Measurements are performed in eigenbases of Pauli X, Y and Z operators, with respective quantum bit error rates (QBER) denoted by eX, eY, eZ. The value of QBER in different bases can be associated with projections of Alice’s and Bob’s shared bipartite entangled state ρAB onto different Bell states through a set of equations.

Specifically, pI denotes probability of no error, pX represents probability of a bit flip, pZ the probability of a phase flip and pY is the probability of simultaneous bit and phase flip. In some protocols, such as four-state BBM92, measurements are not performed in all three X, Y, Z bases, requiring minimisation of the key formula over QBER corresponding to the unobserved basis. The total probability of a bit error is then given by ebit = pX + pY and for phase error one obtains eph = pZ + pY. For a protocol specified by given values of bit and phase errors probabilities, the key rate in Eq. is given by K = 1 −h(ebit) −h(eph|ebit), where h(x) = −x logo(1 −x) log2(1 −x) is the binary entropy function.

In practice, Alice and Bob can use their measurement bases with unequal probabilities. For example, assuming the fraction of measurements in the Z basis is denoted by q for the four-state BBM92 protocol using X and Z bases, the average QBER is qeZ + (1 −q)eX. However, in such instances, instead of the standard approach which uses both bases for key generation, one may use the Z basis to produce the key while measurements in the X basis serve for QBER estimation, particularly beneficial when the effective quantum channel between Alice and Bob affects the bases in an asymmetric way. For the aforementioned BBM92 protocol one obtains the key rate KBBM92s=4 = q [1 −h(eZ) −h(eX)]. Note that q only affects the statistical accuracy of the QBER estimates.

Satellite-based optical communication is associated with inevitable significant losses. This, combined with the fact that typical entanglement sources are characterised by a small pair of production probability of ppair ≪1 leads to low signal strength at the receiver. In such a regime, one wants to maximise the amount of secret key bits that can be extracted per single received photon since the latter is a precious resource. One way to do this is to encode multiple qubits in each photon to optimally utilise a weak signal.

Consider a single photon occupying one of M = 2m temporal slots composing a frame of duration ∆t. The index of the photon-carrying slot, k, can be written in binary representation as k = k m−1 k 1 k 0, where k i = 0 or 1. In this representation, the state of a photon occupying the k-th slot naturally corresponds to a combination of computational basis states of m qubits |k⟩∼|k m−1 ⟩|k 1 ⟩|k 0 ⟩. This encoding resembles pulse position modulation, which is standard in classical communication. To encode an arbitrary state of m qubits, one needs to represent all possible superpositions of basis states similarly. The multiqubit encoding can enhance the secret key rate by up to an order of magnitude compared to single-qubit schemes.

Each photon pair can be used to encode multiple qubits to optimally utilise a weak signal. By optimising source intensity and the number of encoded qubits, the theoretical information limit for quantum key distribution (QKD) efficiency is studied. Optimal efficiency is attained for finite entangled photon pair production probability, unlike conventional communication efficiency which is maximised with vanishing signal strength. Multiqubit encoding can enhance the secret key rate by up to an order of magnitude compared to single-qubit schemes.

The majority of current free-space quantum communication systems employ two-dimensional encoding, allowing each photon to carry at most one bit of quantum information. However, degrees of freedom such as the spatio-temporal properties of a photon can be used for high-dimensional encoding, effectively transmitting multiple qubits with a single photon. Employment of high-dimensional quantum states may increase the QKD key rate and improve strength with respect to noise and eavesdropping.

Satellite quantum key distribution benefits from stronger signals

Securing communications via satellite presents unique hurdles; the sheer distance weakens signals, demanding increasingly sensitive detectors and sophisticated error correction. Amplifying the signal isn’t a solution, as this simultaneously boosts noise and the potential for eavesdropping, a problem addressed by quantum key distribution. John Rarity and Andrew Shields, along with colleagues, have shown a tension between optimising for signal strength and maximising the information carried per photon, a trade-off previously understood through the perspective of vanishing signal strength. This work demonstrates a key point about satellite-based quantum key distribution (QKD); it reveals that maximising efficiency doesn’t necessarily require vanishingly weak signals, a departure from traditional communication approaches.

Researchers demonstrated that optimising quantum key distribution efficiency does not require signals to be as weak as previously thought. This is important because it improves the potential performance of satellite quantum communications, where signal strength is naturally limited by distance. By encoding multiple qubits per photon pair, the secret key rate can be enhanced by up to ten times compared to current single-qubit methods. The study focused on optimising both the source intensity and the number of encoded qubits to achieve this improved efficiency.