Generalized State Discrimination for Tunable Quantum Key Distribution: The phiQKD Protocol

TL;DR

This work introduces Generalized State Discrimination (GSD), a tunable discrimination framework controlled by the tilting angle $φ$, which smoothly interpolates between unambiguous state discrimination and minimum-error discrimination. By embedding GSD in a two-state QKD protocol (phiQKD), the authors replace the fixed IDP measurement with a tunable POVM, enabling adaptivity to channel noise and imperfections while maintaining security under asymptotic, finite-key, and composable models. For the two-state pair $|0 angle$ and $|+ angle$, the protocol achieves a composable secure key rate of $R_{secure} o 0.181958$ bits per signal, about 16% higher than standard B92, with improved sifting and lower QBER. Beyond QKD, the GSD framework offers a general approach to designable quantum measurements, potentially benefiting quantum sensing, metrology, and hypothesis testing through adaptive measurement strategies that balance correctness and conclusiveness.

Abstract

We introduce a tunable framework for generalized quantum state discrimination (GSD) and apply it to quantum key distribution (QKD) through a protocol we call phiQKD. Building upon the two-state B92 protocol, phiQKD replaces the traditional unambiguous state discrimination (IDP) measurement with a one-parameter family of hybrid POVMs characterized by a tilting angle $φ$. This allows for continuous control over the trade-off among correct, incorrect, and inconclusive outcomes. While the asymptotic key rate improvement over B92 is modest, phiQKD offers a practical advantage by enabling adaptability to noise and channel imperfections via measurement tunability. By evaluating the protocol under asymptotic, finite-key, and composable security models, we show that, treating quantum measurement as a tunable design parameter, rather than a fixed operation, enables flexible protocol optimization and improved performance under realistic constraints.

Figures (13)

• Figure 1: Bloch Sphere representation of the POVM basis. When trying to discriminate between two states $\lvert\psi_1\rangle$ and $\lvert\psi_2\rangle$, the directions of the $\Pi_o, \Pi_1, \Pi_2$ show the detector configuration for inconclusive, correct and incorrect detection.
• Figure 2: Bloch Sphere representation of the basis for Helstrom method of state discrimination: Detector configuration for the optimum minimum-error discrimination of two pure states with equal a priori probabilities. A von Neumann measurement with two orthogonal detectors placed symmetrically around $\lvert\psi_1\rangle$ and $\lvert\psi_2\rangle$ will achieve the optimum.
• Figure 3: Bloch Sphere representation of the generalized approach to state discrimination: Here, $\lvert\psi_1\rangle,\lvert\psi_2\rangle$ are two quantum states with inner angle $\theta$. By tilting these quantum states in an angle $\phi$, the primed quantum states i.e. $\lvert\psi_1^\prime\rangle,\lvert\psi_2^\prime\rangle$ are found. Again, by taking the orthogonal states of the primed states the new basis states $\lvert\psi_1^{\prime \perp}\rangle,\lvert\psi_2^{\prime \perp}\rangle$ are constructed which will be used in the formulation of the POVM elements $\Pi_1^\prime, \Pi_2^\prime$ respectively. $\Pi_0^\prime$ represents the basis for inconclusive outcomes. Note on the Bloch sphere, the angle between state vectors corresponds to twice the angle used in their quantum state representation.
• Figure 4: Probabilities vs. Tilting Angle ($\phi$): Correct detection probability ($P_s$) [Green], incorrect discrimination probability ($P_e$) [Red], and probability of inconclusive result ($P_q$) [Blue] vary smoothly and continuously in the discrimination range [IDP, MED].
• Figure 5: Quantum circuit in Qiskit for implementing generalised state discrimination (GSD). The state to be sent, in this case $\lvert\psi_2\rangle=\lvert+\rangle$, is prepared or initialized in qubit $q_0$ and the unitary operation $U$ and the measurements on the qubits $q_0, q_1$(an ancilla qubit) represent the measurement in the GSD proposed basis.