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REVIEW 4 major objections 6 minor 53 references

Experimental optimal discrimination of $N$ states of a qubit with fixed rates of inconclusive outcomes

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read An optical experiment demonstrates the quantum-optimal trade-off between error and inconclusive outcomes when discriminating N=2, 3, 5, and 7 symmetric qubit states.

desk verdict Solid experimental demonstration of the two-step FRIO strategy for N symmetric qubit states, with the main caveat being that optimality is benchmarked against ideal pure states rather than the actual mixed ones. read the letter →

arxiv 2411.14537 v1 pith:446PFKT3 submitted 2024-11-21 quant-ph

L. F. Melo , M. A. Solís-Prosser , O. Jiménez , A. Delgado , L. Neves This is my paper · ORCID
classification quant-ph
keywords quantumstatediscriminationfixedrateofinconclusiveoutcomesseparationminimum-errormeasurementsymmetricqubitstatesspatiallightmodulatorpath-encodedphotonicqubitsmaximum-confidence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quantum state discrimination asks how to tell apart nonorthogonal states, and any strategy must trade errors against inconclusive outcomes. This paper reports an optical experiment that realizes the optimal version of this trade-off — the minimal error rate for a fixed rate of inconclusive outcomes — for N=2, 3, 5, and 7 equally likely symmetric states of a qubit. The implementation uses a two-step recipe: a probabilistic state-separation map first increases the distinguishability of the inputs, and a minimum-error measurement is then applied to the successfully transformed outputs. The measured error rates follow the theoretical curve that interpolates between the minimum-error extreme (no inconclusive outcomes) and the maximum-confidence or unambiguous extreme (maximal inconclusive rate), demonstrating the two-step approach in a programmable optical setup that can be extended to higher dimensions.

What carries the argument

The central object is the two-step decomposition of the optimal FRIO measurement: a probabilistic quantum map called state separation, which with maximal success probability $p_s(\theta')=(\sin\theta/\sin\theta')^2$ rotates the symmetric input states $|\alpha_j(\theta)\rangle$ into the more distinguishable states $|\beta_j(\theta')\rangle$, followed by the minimum-error POVM $\hat\Pi_k^{\rm ME}=\frac{2}{N}|u_k\rangle\langle u_k|$ on the successful outputs. The argument is carried by the identities $Q(\theta')=p_f(\theta')=1-p_s(\theta')$ and the closed-form expression $P_{\min}^e(Q)$ of Eq. (13), which together show that fixing the separation angle fixes the inconclusive rate and yields the optimal error rate. Experimentally, the machinery is a path-encoded qubit, a polarization ancilla, a programmable liquid-crystal spatial light modulator that implements the controlled unitary of Eq. (7), and a lens-focal-plane detector array that realizes the projective measurement of Eq. (15).

What would settle it

Compute the true minimum error for the actual mixed states $\hat\rho_j(\theta')$ of Eq. (18) using the measured visibilities $V_j$ and phases $\phi_j$, and check the data at the largest separation angles ($\theta'_6$ and $\theta'_7$), where depolarization is strongest: if the experimental error rates lie above that recomputed optimum by more than the experimental uncertainty, the claim that the implementation closely reaches the minimum error rate for fixed $Q$ is falsified.

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Extended reading notes

Core claim

The central claim is that optimal fixed-rate-of-inconclusive-outcomes (FRIO) discrimination — minimizing the error probability $P_e$ under a fixed rate $Q$ of inconclusive outcomes — can be implemented as a two-step process, and that an optical realization of that process matches the quantum-optimal curve for symmetric qubit states. For $N$ equally likely symmetric pure states, the paper derives and verifies the trade-off $$P_{\min}^e(Q)=\frac{1}{N}\Big[(N-1)(1-Q)-\sqrt{(1-Q)^2-(Q-Q_{\rm MC})^2}\Big],$$ which reduces to the minimum-error bound at $Q=0$ and to the optimal unambiguous ($N=2$) or maximum-confidence ($N>2$) rate at $Q=Q_{\rm MC}=\cos 2\theta$. The experiment prepares path-encoded qubit states with a spatial light modulator, performs the separation by coupling the path modes to a polarization ancilla via a programmable liquid-crystal modulator, and completes the minimum-error step with a lens-focal-plane detector array that implements the quantum Fourier transform measurement. The reported data follow this curve closely for all four $N$, with agreement improving as $N$ grows.

Load-bearing premise

The theoretical curve used as the benchmark assumes the separated states are exactly the ideal pure symmetric states, but the liquid-crystal modulator partially depolarizes the ancilla — an effect the paper's own footnote 5 acknowledges and leaves for future work — and introduces phase errors, especially at large separation angles, so the comparison against the ideal curve does not account for the true mixedness of the prepared states.

Editorial extensions

If this is right

  • The two-step recipe provides a constructive way to reach any point on the optimal error–inconclusive trade-off curve: set the separation angle $\theta'$ to fix $Q$, then run the minimum-error measurement.
  • For quantum communication protocols that need to keep both the error rate and the inconclusive rate below the levels of the standard extreme strategies, the demonstrated scheme offers a tunable intermediate operating point controlled by a single programmable element.
  • Because the separation stage uses only a two-dimensional ancilla and the measurement stage works for any $N$, the same experimental platform can be extended to symmetric qudit states encoded in $d$ path modes.
  • The improvement of the agreement as $N$ increases supports the interpretation that experimental imperfections largely independent of $N$ are the main source of deviation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the depolarization of the transmissive liquid-crystal modulator at high gray levels is the dominant imperfection, then replacing that device with a phase-only modulator or adding active polarization correction should pull the high-$Q$ data points closer to the ideal curve; this is a testable upgrade of the present setup.
  • The two-step structure suggests the same decomposition could be used for optimal fixed-trade-off discrimination in non-symmetric state sets, provided a suitable probabilistic separation map is available, rather than only for the symmetric qubit case demonstrated here.
  • Because the paper compares its data against the ideal pure-state curve rather than the true optimum for the actual mixed states, the claim that the implementation closely reaches the minimum error rate is a conservative statement; recomputing the optimum for the measured visibilities $V_j$ and phases $\phi_j$ would provide a stricter test.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper presents an experimental demonstration of optimal fixed-rate-of-inconclusive-outcomes (FRIO) discrimination for N=2, 3, 5, and 7 equally likely symmetric qubit states encoded in photonic path modes. The implementation follows a two-step strategy: first, a probabilistic optimal state separation stage based on a programmable spatial light modulator, which fixes the inconclusive rate Q; second, a minimum-error (ME) measurement on the successfully separated states, implemented via an optical Fourier transform and detector arrays. The theoretical part derives the error probability Pe(θ') and inconclusive rate Q(θ') as functions of the separation angle and obtains the FRIO benchmark P_min^e(Q) of Eq. (13), interpolating between minimum-error (Q=0) and maximum-confidence/unambiguous (Q=Q_MC) discrimination. The authors compare experimental success probabilities, correct-discrimination probabilities, and the resulting error-versus-Q tradeoff with theoretical curves and conclude that the scheme closely reaches the minimum error rate for fixed Q.

Significance. If fully established, the paper would be a valuable addition to the experimental quantum state discrimination literature: it goes beyond the previously demonstrated two-state FRIO case to N=2,3,5,7 symmetric qubit states, and it explicitly demonstrates the two-step decomposition (probabilistic state separation followed by a minimum-error measurement) as a practical route to optimal FRIO discrimination. The theoretical formulas are standard and correctly cited, and the paper provides a careful characterization of the separated states (Eq. (18), Fig. 3), an explicit depolarization model for the SLM (Eq. (B2)), and a plausible path toward higher-dimensional extensions. The experimental data in Figs. 4 and 5 show reasonable agreement with the theoretical curves, with the caveat discussed below that the benchmark curve assumes ideal pure output states. The paper is clearly written and the experimental methodology is reproducible in its main elements.

major comments (4)
  1. [Section V A, Eq. (18), Fig. 5] The headline claim that the implementation 'closely reaches the minimum error rate for a fixed value of Q' is tested against Eq. (13), which is the FRIO optimum for ideal symmetric pure states |β_j(θ')>. However, the experimentally separated states are the mixed states ρ_j(θ'_t) of Eq. (18), with measured visibilities V_j that are in some cases significantly below sin 2θ'_t (notably for θ'_6 and θ'_7, as acknowledged in Section V A). The ME POVM of Eq. (9) is optimal for the ideal pure states, not necessarily for these mixed states, and the paper does not compute the true FRIO-optimal error rate for the measured states. Therefore, agreement with Eq. (13) does not by itself establish that the achieved error rate is the minimum possible for the states actually produced. This is a fixable but load-bearing gap: the authors already have the measured V_j and φ_j, so they could compute the true optimum (or a rigorous bound) for the measured mixed states and compare their data against it. Without that comparison, the central optimality claim is not fully supported.
  2. [Section V, Fig. 5, endpoint Q = Q_MC] The benchmark mismatch is most acute at the endpoint Q = Q_MC = 0.7771. For N=2, Eq. (13) gives P_min^e(Q_MC)=0, corresponding to optimal unambiguous discrimination. But unambiguous discrimination of two nonorthogonal mixed states is impossible unless the states are orthogonal, and the measured N=2 states at θ'_7 have visibilities below the ideal value. The data point near Q=0.7771 must therefore lie strictly above the true minimum error for the actual mixed states if V_j < sin 2θ'_7. The paper does not quantify this gap; the authors should compute the true mixed-state optimum for the measured visibilities and phases, or provide a quantitative argument that the deviations are negligible within their reported error bars.
  3. [Section V B and Eqs. (11)] The experimental FRIO probabilities are constructed as [Pe]_expt = [ps]_expt (1 - [pβ_c]_expt) and Q_expt = 1 - [ps]_expt. This is consistent with Eq. (11) only if the two averages can be multiplied without introducing bias. Since [ps]_expt and [pβ_c]_expt are averaged separately over the input states and the measured quantities come from the same intensity distributions, the paper should explicitly justify that no conditioning or post-selection bias enters this product. In particular, the authors should clarify that the same set of input states and the same detected events underlie both averages, and that the camera detection efficiency compensation described in Section V B does not distort the joint statistics.
  4. [Section III D and Eq. (13)] The derivation leading to Eq. (13) via Eqs. (6), (10), and (11) is algebraically clean, but the paper does not prove that the two-step POVM of Section III D is globally optimal for the FRIO problem; it relies for that on prior results (Refs. [20,21,34]). This is acceptable for an experimental demonstration, but the claim in Section VI that the paper demonstrates 'both theoretically and experimentally' the optimal FRIO discrimination is slightly overstated. The theoretical optimality is imported from the literature; the experimental contribution is the implementation and test of the two-step scheme against that known benchmark.
minor comments (6)
  1. [Eq. (7)] Please define the ordering of the ancilla basis used in the 2×2 block of Eq. (7), and state explicitly that ξ(θ') = tan θ cot θ' lies in [0,1] for θ ≤ θ' ≤ π/4, which is used later.
  2. [Section IV C and Eq. (17)] The sign convention and branch choice in the Fourier transform leading to x_k = -λf m_k/(NΔ) would benefit from an explicit statement connecting Eq. (16) to the projective measurement of Eq. (15); currently the reader must reconstruct the signs from context.
  3. [Table I and Fig. 3] In Fig. 3, the experimental markers for θ'_6 and θ'_7 lie noticeably inside the ideal parallels. Please provide the numerical values of the measured visibilities V_j (or at least their range) either in the caption or in a small table, so the purity loss can be assessed quantitatively without reading the figure by eye.
  4. [Section V A, footnote 7] The attribution of V_j > sin 2θ'_t to 'inaccuracies in the preparation of the input states' is plausible but is not supported by a quantitative model; a sentence explaining which preparation imperfection (e.g., imperfect phase or amplitude in the grating masks) produces this effect would be helpful.
  5. [Appendix B, Eq. (B2)] The depolarizing model of Eq. (B2) describes the ancilla polarization, but the relation between the ancilla depolarization and the qubit visibility V_j in Eq. (18) is not derived explicitly. A short derivation or a reference to one would strengthen the connection between the SLM characterization and the measured states.
  6. [Abstract and text] There are several small typographical issues: 'th second column' in the Table I caption; 'a useful platform' in the introduction; the word 'minium' in the arXiv metadata. Also, the acronym FRIO is used in the abstract without being spelled out there; please define it at first use in the abstract or tolerate the fact that the introduction defines it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the theoretical benchmark is an independent prior result and the experimental data are measured directly rather than fitted to it.

full rationale

The paper's central comparison is between independently measured experimental probabilities and the theoretical FRIO curve P_min^e(Q) of Eq. (13). The experimental points in Fig. 5 are computed as [Pe]_expt = [ps]_expt (1 - [pBeta_c]_expt) and Q_expt = 1 - [ps]_expt, where [ps]_expt is obtained from directly measured intensity ratios at the two PBS outputs and [pBeta_c]_expt from counts at detector positions implementing the ME measurement. These quantities are not fitted parameters and are not adjusted to match Eq. (13). The theoretical curve itself is derived by combining Eq. (6) for the optimal separation success probability, Eq. (10) for the ME error of the separated states, and Eq. (12) for Q_MC, and the paper explicitly notes that Eq. (13) agrees with prior results from Bagan et al., Herzog, and Chefles and Barnett. The same-group citations used for the input formulas are antecedent theoretical results used as building blocks, not outputs of this experiment, and no conclusion is forced by a self-citation chain or by definitional equivalence. The observed depolarization of the transmissive LCD, which makes the characterized states in Eq. (18) mixed rather than the ideal pure states of Eq. (4), is a genuine benchmark-validity concern that could affect how strongly the data corroborate the optimality claim, but it is a correctness/robustness issue, not a circularity. Accordingly, no circular step is present.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on prior theoretical results (state separation optimality, FRIO formula, ME POVM) and on the equivalence between coherent-light statistics and single-photon statistics. No new free parameters are fitted to the final data; the only free choices are the input angle and the target separation angles, which are experimental controls. No invented particles, forces, or dimensions are introduced.

free parameters (2)
  • Input state polar angle theta = 19.5 degrees
    Chosen as the input state parameter to allow access to all target separation angles up to 45 degrees with the available SLM (Appendix B). It is not fitted to the result but is a free experimental choice that affects the theoretical curves.
  • Target separation angles theta'_t (t=1..7) = 19.5, 22.6, 25.5, 29.5, 34.2, 40.0, 45.0 degrees
    Chosen to sample the FRIO curve from Q=0 to Q_MC. These are experimental control settings selected from the gray-level calibration, not fit parameters.
assumptions (5)
  • domain assumption The optimal state separation success probability is ps(theta')=(sin theta / sin theta')^2
    Taken from Solis-Prosser et al. (2016), which includes authors of this paper. Used in Eq. (6) and Eq. (11). Not rederived here.
  • domain assumption The minimum-error POVM for N symmetric qubit states is Pi_k^ME=(2/N)|u_k><u_k|
    Standard result from Ban et al. and Jimenez et al. Used in Eq. (9) and in the derivation of Eq. (10).
  • standard math The optimal FRIO error rate formula P_min^e(Q) in Eq. (13) is correct
    Taken from Bagan et al. and Herzog. Used as the theoretical benchmark for the experimental data.
  • standard math Naimark extension allows an N-outcome POVM on a qubit to be realized as a projective measurement in an N-dimensional Hilbert space
    Standard theorem, cited as Peres. Used in Section IV C to justify the optical Fourier transform implementation.
  • domain assumption A classical coherent laser field with cameras reproduces the measurement statistics of single-photon states for this discrimination task
    The experiment uses bright coherent light to simulate quantum states. The equivalence is argued in footnote 4 and Section VI, based on prior work [29,31]. This is a physical assumption, not formally proven in this paper.

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Pith. "Pith review of Experimental optimal discrimination of $N$ states of a qubit with fixed rates of inconclusive outcomes." pith.science (2026). https://pith.science/paper/446PFKT3

@misc{pith2026241114537,
  author       = {Pith},
  title        = {Pith review of: Experimental optimal discrimination of $N$ states of a qubit with fixed rates of inconclusive outcomes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/446PFKT3}},
  note         = {Machine review of arXiv:2411.14537}
}
abstract

In a general optimized measurement scheme for discriminating between nonorthogonal quantum states, the error rate is minimized under the constraint of a fixed rate of inconclusive outcomes (FRIO). This so-called optimal FRIO measurement encompasses the standard and well known minimum-error and optimal unambiguous (or maximum-confidence) discrimination strategies as particular cases. Here, we experimentally demonstrate the optimal FRIO discrimination between $N=2,3,5,$ and $7$ equally likely symmetric states of a qubit encoded in photonic path modes. Our implementation consists of applying a probabilistic quantum map which increases the distinguishability between the inputs in a controlled way, followed by a minimum-error measurement on the successfully transformed outputs. The results obtained corroborate this two-step approach and, in our experimental scheme, it can be straightforwardly extended to higher dimensions. The optimized measurement demonstrated here will be useful for quantum communication scenarios where the error rate and the inconclusive rate must be kept below the levels provided by the respective standard strategies.

Figures

Figures reproduced from arXiv: 2411.14537 by the authors.

Figure 1
Figure 1. FIG. 1. Symmetric states of a qubit on the Bloch sphere for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Experimental setup (see text for details). HWP: half [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Characterization of the state separation process for [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Experimental results: average probabilities of successful state separation (black squares) and correct discrimination of the separated [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Average error probability, [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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