Fidelity Determines Optimal State Discrimination Probability for Single and Multi-Qubit Systems

Distinguishing between quantum states is a core challenge in quantum information science, with applications ranging from secure communication to advanced computation. Hyunho Cha and Jungwoo Lee, from NextQuantum and Seoul National University, alongside their colleagues, investigate how understanding the underlying structure of these states and the measurements used to identify them impacts the accuracy of state discrimination. Their work reveals that while the relationships between states fully define discrimination ability for single quantum bits, this connection breaks down with more complex systems. By analytically determining the optimal discrimination probability for three single-qubit states, and exploring the properties of the measurements themselves, the researchers demonstrate how partial knowledge of these measurements can significantly improve existing methods for bounding discrimination accuracy, potentially simplifying complex calculations and advancing the field of quantum state identification.

Researchers are continually striving to improve our ability to distinguish between different quantum states – the fundamental building blocks of quantum information. This work focuses on establishing tighter limits on the probability of correctly identifying an unknown quantum state from a given set, which is crucial for optimising quantum technologies such as communication and computation. The core challenge lies in the inherent uncertainty of quantum mechanics; unlike classical bits, quantum states exist in a superposition of possibilities.

Determining the best strategy for ‘decoding’ which state has been presented requires sophisticated mathematical tools, and this research explores various mathematical bounds – inequalities that provide upper limits on the probability of successful discrimination. A key concept is identifying a crucial subset of states, denoted as I+(E), which are most relevant for discrimination. By focusing calculations on this smaller set, researchers aim to simplify the problem and achieve tighter bounds.

The Löwner-Heinz inequality, a mathematical principle governing positive semi-definite matrices, plays a vital role in refining these bounds, and assuming that all states are equally likely further contributes to improved accuracy. Researchers explore methods for finding both subsets and supersets of I+(E) to enhance the precision of the bounds, ultimately seeking a more complete understanding of the limits of quantum state discrimination. Researchers have also developed a method for constructing valid measurement strategies, known as Positive Operator-Valued Measures (POVMs), in quantum state discrimination.

While the method doesn’t uniquely pinpoint the set of essential states I+(E), it successfully identifies a portion of it, and a larger set containing I+(E) can also be found by applying specific conditions. These conditions involve finding a set of non-negative coefficients that satisfy certain mathematical relationships, defining a valid POVM and ensuring that the probabilities of all possible measurement outcomes sum to one. The success of this method is measured by calculating the probability of correctly identifying the quantum state, which is always greater than or equal to the optimal probability achievable with any measurement strategy.

This research provides a partial solution to identifying the set of essential states I+(E), offering a way to determine whether a given state belongs to this set. The method relies on a lower bound for the optimal probability of discrimination, closely related to the sum of the squares of the quantum states. By combining this inequality with another mathematical expression, researchers can establish a criterion for determining membership in I+(E).

If replacing the set I+(E) with a different set – excluding a specific state – violates the inequality, it conclusively demonstrates that the excluded state belongs to I+(E). For single-qubit systems, the pairwise fidelities – a measure of the similarity between states – completely determine the optimal discrimination probability. However, this simple relationship does not hold in higher-dimensional systems.

Researchers derived a closed-form expression for three equiprobable single-qubit states possessing equal pairwise fidelities, and importantly, knowing which measurement operators must be zero leads to tighter bounds on the optimal discrimination probability. In certain instances, subsets of non-vanishing operators can be inferred without employing computationally intensive techniques like semidefinite programming (SDP). Determining all non-vanishing operators without SDP – either analytically or through more efficient numerical methods – remains a significant challenge, and a deeper understanding of how to quickly identify this set could further enhance the ability to find the optimal measurement strategy, reducing the computational time required.