DEFESA DE TESE DE DOUTORADO #443 – LUCAS FELIPE BEZERRA DE MELO OLIVEIRA – 21/08/2025

Quantum mechanics forbids deterministic and error-free discrimination of nonorthogonal quantum states in a single-shot measurement. Due to this fundamental limitation, many strategies were devised to optimize the task and later became fundamental tools for quantum information processing. Most of these strategies involve outcomes that can be correct, incorrect, or inconclusive in identifying the state. Our work focuses on a general optimized measurement scheme that minimizes the error probability under the constraint of a fixed rate of inconclusive outcomes (FRIO). The optimal FRIO measurement is general in the sense that it encompasses other fundamental strategies as particular cases: at zero inconclusive rate, it reduces to the minimum-error (ME) measurement; at a critical inconclusive rate, it corresponds to optimal unambiguous discrimination (UD) for linearly independent states or to optimal maximum-confidence (MC) measurement for certain linearly dependent states!. Between these two extremes, FRIO balances error and inconclusive rates, outperforming ME and UD/MC in their respective limitations.

In this thesis, we explore the theoretical, experimental, and fundamental aspects of the optimal FRIO measurement applied to the discrimination of N equiprobable symmetric pure states in any dimension n≤N, a family of states of particular interest in quantum communication. On the theoretical side, we first derive analytical solutions for the strategy in this scenario. We then introduce a concatenated FRIO scheme, in which the remaining information from inconclusive outcomes is recovered via ME measurement (for n>2), thereby increasing the overall probability of correct identifications compared to the standard scheme. On the experimental side, we present an optical implementation of the optimal FRIO measurement for N=2,3,5, and 7 states of a qubit (n=2) encoded in photonic spatial modes. In addition to successfully demonstrating the FRIO strategy and its extreme cases (ME and UD/MC) for a qubit, we show that our setup can be straightforwardly extended to qudits (n>2). Final!ly, on the fundamental side, we investigate two distinct aspects. First, we characterize quantum coherence as a resource underlying both standard and concatenated FRIO measurements, and show that this approach has practical applications, particularly in secure quantum communication. Next, we derive entropic wave–particle duality relations for uniform N-path interferometers with symmetric which-way detector states, showing that the ME measurement yields the tightest bound, followed by concatenated and standard FRIO. We further show that the relation is saturated at nontrivial points only when the number of paths is non-prime, and identify the corresponding detector states. Our results bring theoretical and experimental advances to the field of quantum state discrimination and emphasize its importance from both practical and foundational standpoints.

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