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REVIEW 3 major objections 4 minor 61 references

Quantum teleportation of cat states with binary-outcome measurements

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Pith's one-line read Cat-state qubits can be teleported using only beam splitters, displacements, and binary-outcome measurements, with about 95 percent fidelity in a single round and near-unity fidelity when the final measurement is repeated.

desk verdict Useful cat-state teleportation protocol built from parity and dispersive measurements, but the near-perfect fidelities are postselected and the logical XL correction is probabilistic, so the unconditional claim is not yet established. read the letter →

arxiv 2412.15011 v2 pith:5OLIWEJT submitted 2024-12-19 quant-ph

Jingyan Feng , Mohan Zhang , Matteo Fadel , Tim Byrnes This is my paper · ORCID
classification quant-ph
keywords quantumteleportationSchrödingercatstatesparitymeasurementdispersivebeamsplitteroperationscontinuous-variableinformationcircuitelectrodynamicstrappedions
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

This paper proposes a teleportation protocol for a qubit encoded in Schr\"odinger-cat states, in which Alice holds two bosonic modes, Bob one, and the state is moved from Alice's mode to Bob's using only beam splitters, displacements, and binary-outcome parity or dispersive measurements. The protocol is designed for platforms such as circuit QED, circuit quantum acoustodynamics, and trapped ions, where dispersive readout of a mode through a qubit is natural but homodyne detection is not. Analytically and numerically, the authors find a fidelity around 95 percent for a single measurement round and fidelities that approach one when the final dispersive measurement is repeated roughly ten or more times. They present two variants, one using two single-mode parity measurements and one using a single joint parity measurement, and they map out optimal cat amplitudes and interaction times for both real and imaginary resource-cat amplitudes, including the effect of photon loss. A sympathetic reader would care because this removes the main obstacle to continuous-variable-style teleportation in solid-state and trapped-ion settings.

What carries the argument

The load-bearing element is the displaced single-shot dispersive measurement: apply the displacement $D(\chi)$ to mode $a$, then measure with the dispersive Jaynes-Cummings measurement operators $M_+(\tau)=\sum_n e^{-in\omega t}\cos(n\tau)|n\rangle\langle n|$ and $M_-(\tau)=\sum_n e^{-in\omega t}\sin(n\tau)|n\rangle\langle n|$, with the interaction time set to $\tau=\pi/(2\bar{n})$ where $\bar{n}=|\alpha|^2+|\beta|^2$. This choice puts the zeros of the cosine factor on the $|\pm\bar\chi\rangle$ peaks and the zeros of the sine factor on the $|\pm\chi\rangle$ peaks, turning one bit of readout into an approximate discriminator between the two leftover coherent amplitudes that the parity measurement cannot separate. Repeated application of the same measurement sharpens the discrimination into a Gaussian window in Fock space, which is why the fidelity approaches one as the number of measurements grows; beam-splitter transformations and the parity projections supply the rest of the teleportation circuit, and the logical corrections are realized by $Z_L=e^{-ic^\dagger c\pi}$ and by a further displaced measurement implementing $X_L$.

What would settle it

Compute the exact post-measurement state using the full operators $M_\pm(\tau)$ of Eq. (7) instead of the approximate collapse rules in Eqs. (26)-(27), and average the fidelity over every binary outcome, including the failure branch of the $X_L$ correction; if the fully unconditional average for $\alpha=\beta=4$, $N=1$ falls at or below the classical bound $2/3$, the claimed single-shot performance does not survive. Experimentally, this is a direct test: prepare a known cat-state superposition on mode $a$, run the circuit of Fig. 2, and compare Bob's final mode with the input without postselecting on the correction measurement.

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Extended reading notes

Core claim

The central claim is that an unknown qubit encoded in the even and odd cat states $|0_L\rangle_\alpha$ and $|1_L\rangle_\alpha$ can be teleported with a two-beam-splitter circuit followed by parity projections and a displaced dispersive measurement, and that every measurement in the protocol is binary. After the second beam splitter, the joint state contains two pairs of coherent amplitudes, $\chi=(\alpha+\beta)/\sqrt{2}$ and $\bar\chi=(\alpha-\beta)/\sqrt{2}$; a parity measurement on Alice's two modes collapses the state only partially, and the remaining superposition over $\chi$ and $\bar\chi$ is resolved by displacing mode $a$ by $D(\chi)$ and reading it out with the dispersive measurement operators $M_\pm(\tau)$, with interaction time $\tau=\pi/(2\bar{n})$ chosen so the zeros of $\cos(n\tau)$ and $\sin(n\tau)$ sit at the centers of the two groups of coherent-state peaks. What remains on Bob's mode is the input state up to Pauli corrections $Z_L=e^{-ic^\dagger c\pi}$ and a measurement-induced $X_L$. For cat amplitudes at the few-photon level the single-shot fidelity is about 95 percent, and repeating the final measurement increases the fidelity toward one, so the paper concludes that high-fidelity teleportation is achievable in platforms where quadrature measurements are awkward.

Load-bearing premise

The protocol depends on two approximations: that the qubit-based detector's two answers sit at the centers of the two leftover wave-packets and separate them cleanly, and that the bit-flip correction, itself a probabilistic measurement, is counted only when it succeeds; if either approximation gives way, the teleported state keeps extra errors.

Editorial extensions

If this is right

  • Teleportation of cat-state qubits becomes available in circuit QED, quantum acoustodynamics, and trapped-ion systems that already have dispersive readouts, with no homodyne detection required.
  • Even a single measurement round exceeds the classical teleportation bound of $2/3$, so genuine quantum teleportation can be certified with binary measurements alone.
  • Repeating the final dispersive measurement about ten times raises the fidelity above 99 percent, putting near-unity teleportation within reach of current cat-state experiments.
  • Using one joint parity measurement instead of two single-mode parity measurements reduces the number of binary outcomes from three to two with essentially unchanged fidelity.
  • The protocol tolerates a few percent loss with little degradation, and with repeated measurements and purely imaginary $\beta$ it maintains near-perfect fidelity for loss probabilities up to about 35 percent.

Reading between the lines

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

  • The same displaced-binary-measurement step could be reused in other continuous-variable tasks where two coherent-state amplitudes must be told apart without homodyne detection, such as entanglement swapping, quantum repeaters, or error correction of cat codes.
  • Because the logical $X_L$ correction is itself implemented by a probabilistic measurement, an unconditional version of the protocol should include the failure branch of that measurement; the reported fidelities condition on the correction succeeding, so the fully unconditional average may be somewhat lower and is worth computing.
  • The fidelity maps are symmetric under exchange of $\alpha$ and $\beta$ along the diagonal, which suggests a resource duality between the input cat amplitude and the entanglement cat amplitude; allocating a fixed total coherent amplitude between the two could be a way to optimize single-shot fidelity.
  • An adaptive version of the final measurement, choosing the interaction time based on the observed Fock-space distribution instead of fixing $\tau=\pi/(2\bar{n})$, could push single-shot fidelity higher in the imaginary-$\beta$ case where interference shifts the optimal time.
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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

3 major / 4 minor

Summary. The paper proposes a teleportation protocol for a qubit encoded in Schrödinger-cat states, using only beam-splitter operations, displacements, and binary-outcome measurements (parity and dispersive measurements in the Jaynes-Cummings regime). The protocol works with three bosonic modes: Alice holds two modes, Bob one; entanglement is generated by a beam splitter acting on a cat state, a second beam splitter correlates Alice's modes, and binary measurements followed by conditional corrections teleport the state. Two variants are presented, using single-mode or joint parity measurements, and additional dispersive measurements distinguish the residual coherent-state amplitudes. Numerical fidelities around 95% are reported for a single final measurement, approaching unity for many repeated measurements, and the protocol is analyzed under photon loss. The central claim is that high-fidelity, near-perfect teleportation is possible in platforms where homodyne detection is unnatural, such as circuit-QED, QAD, and trapped ions.

Significance. If the reported performance can be substantiated as an unconditional teleportation fidelity, the protocol would be a useful alternative to standard CV teleportation for bosonic platforms with Jaynes-Cummings-type interactions. The manuscript has clear strengths: the state evolution after the beam splitters is derived analytically from first principles, the measurement operators are explicit POVM elements, the optimal interaction time in Eq. (24) is derived rather than fitted, and the numerical simulations cover both variants and decoherence. The protocol's reliance only on binary-outcome measurements is a genuinely useful feature for the target platforms. However, as written, the headline fidelities are conditional on postselected measurement branches, and the logical X_L correction is itself a probabilistic measurement. The claim of near-perfect teleportation therefore currently exceeds what the computed quantities establish.

major comments (3)
  1. [Sec. IV B, IV E, Figs. 4-6, 8, 10] The near-perfect fidelities are computed on postselected branches only. Sec. IV B states that for N=1000 the results use the outcome k+ = N, k- = 0, and Sec. IV E acknowledges that earlier plots consider only sigma'_a = + and calls this performing a postselection. Eq. (40) averages only over the two dispersive outcomes for the fixed parity outcome sigma_a = sigma_b = +. The paper never reports an unconditional teleportation fidelity that includes all parity outcomes, all k sequences of repeated measurements, and all dispersive outcomes, nor does it report the total success probability of the protocol. The abstract's statement that 'near-perfect fidelity can be obtained' is therefore only a statement about a selected branch unless an unconditional number is provided.
  2. [Sec. III E, Eqs. (34)-(35)] The logical X_L correction is implemented as D(-beta) M+(pi/(2 nbar)) D(beta) or D(-beta) M+(pi/(4 nbar)) D(beta), which is a probabilistic measurement rather than a deterministic unitary. The text concedes that 'ordinarily we would need to consider the M- outcome as well' and dismisses it as low probability without quantifying it. The fidelities in Eqs. (36) and (41) apply the correction C as if it were the ideal Pauli operation, so neither the failure probability of the X_L branch nor the state produced on the M- outcome enters any reported number. Since the complete protocol requires X_L for several measurement outcomes, the unconditional fidelity and success probability including the probabilistic X_L step must be computed, or the protocol must be restricted to a heralded-success description with the success probability explicitly stated.
  3. [Sec. VII, Sec. IV C] The comparison to the 66% classical bound for qubit teleportation is made for the fixed input state mu = 1/2, nu = sqrt(3)/2 used throughout the simulations. The Massar-Popescu bound in Ref. [51] applies to the fidelity averaged over all input qubit states (or at least over a suitable Haar measure), and a state-dependent fidelity can exceed 2/3 for a non-teleporting channel. To support the claim that the protocol exceeds the classical teleportation limit, the authors should report the Bloch-sphere-averaged fidelity, or explicitly state and justify a worst-case fidelity benchmark.
minor comments (4)
  1. [Sec. IV E] The heading 'Average over dispersive measurement outcomes' and Eq. (40) describe an average only over sigma'_a for the fixed parity outcome sigma_a = sigma_b = +, not an average over all teleportation branches; the wording should be clarified to avoid implying a full unconditional average.
  2. [Sec. IV C] The text states that Eq. (24) predicts the optimal interaction time tau = pi/(2 nbar), but Fig. 5 shows the numerical optimum near xi = 2.05-2.06 for several parameter sets. The paper should state explicitly that Eq. (24) is an approximation and quantify the fidelity difference between xi = 2 and the true optimum.
  3. [Fig. 11(b)] The inset mentions 'the averaged curve' without defining what is averaged; specifying the averaging procedure (for example, a local average over Fock index n) would make the figure self-contained.
  4. [Sec. III D 1] The sentence removing superscripts for modes after Eq. (22) is slightly confusing because Eq. (20) and the table retain mode labels; a consistent notation for mode ordering would help readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the teleportation fidelity is computed from a stated dispersive-measurement model, and the self-citations to measurement operators and Gaussian approximations are independent published results rather than load-bearing premises.

full rationale

The paper's derivation is self-contained. The logical-state decomposition (10)-(15), the Bell-state creation (17)-(19), and the post-beam-splitter state (20) follow algebraically from the beam-splitter transformations and the definitions of the cat states; no result is defined in terms of the fidelity it is meant to predict. The measurement operators (7) are taken from the published POVM model of Ref. [42]; this is a real external input, and the numerical fidelities are computed by applying these operators to the derived states rather than by assuming the answer. The interaction time tau = pi/(2 n-bar) in Eq. (24) is obtained from the average boson number n-bar = |alpha|^2 + |beta|^2 of the displaced branches and is then checked numerically by scanning xi in Fig. 5, with the optimum (xi about 2.05) close to the predicted value (xi = 2); this is an optimization consistency check, not a fit of a parameter to the claimed fidelity. The Gaussian approximation (29)-(30) cited to Refs. [46,47] is used only to interpret the repeated-measurement outcomes; the actual simulation uses the exact binomial measurement operator (28), so this citation is not load-bearing. The paper's high fidelities are conditional on particular measurement branches (e.g., sigma'_a = +, k_+ = N, k_- = 0), and the physical X_L correction is itself a probabilistic measurement whose failure branch is not included; these are legitimate concerns about the unconditional success probability and the strength of the teleportation claim, but they are completeness/correctness issues rather than circularity, because the conditional fidelity is not used to define the input parameters or the measurement model. Accordingly, no circular step is present; at most there are minor self-citations that do not carry the argument.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The protocol relies on standard and experimentally demonstrated operations for bosonic modes coupled to a qubit. No free parameters are fitted to data; the interaction time is derived from the state amplitudes. No new entities are postulated.

assumptions (7)
  • domain assumption The dispersive JC measurement operators M±(tau) = sum_n e^{-i n omega t} cos(n tau)|n><n| and sin(n tau)|n><n| form a valid POVM.
    Used throughout Sec. III D; taken from Ref. [42], a published result. If the actual measurement differs, the protocol's collapse analysis fails.
  • domain assumption Ideal beam splitter transformations can be implemented between any two modes.
    Invoked in Sec. III B and III C; experimental references [39,40,31] demonstrate such operations in trapped ions, cQED, and QAD.
  • domain assumption Cat states of the form |±alpha> can be prepared and manipulated.
    The protocol requires preparing cat states on modes a and b; references [32-37] provide methods. If cat state preparation is imperfect, the resource state quality degrades.
  • standard math For |alpha| large, |±alpha> ≈ (1/sqrt(2))(|0L>_alpha ± |1L>_alpha), with corrections O(e^{-2|alpha|^2}).
    Eq. (14) is the key approximation; the paper notes corrections are at the 0.01% level for |alpha|=2 and uses |alpha|>=3.
  • standard math Repeated dispersive measurements are described by Eq. (28) with the binomial distribution, and the Gaussian approximation (29)-(30) holds.
    Used in Sec. III D and IV to argue that N measurements sharpen the collapse; cited from Refs. [46,47].
  • domain assumption The input is an arbitrary logical qubit mu|0L>_alpha + nu|1L>_alpha with alpha real, and the resource cat state has amplitude beta (real or imaginary).
    This defines the teleportation task. The restriction to real alpha is stated at the start of Sec. III A.
  • domain assumption Modes a, b, c are independent and a measurement on mode a does not affect mode c.
    The fidelity traces out modes a,b; this assumes decoupled modes, standard in the dispersive regime.

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Pith. "Pith review of Quantum teleportation of cat states with binary-outcome measurements." pith.science (2026). https://pith.science/paper/5OLIWEJT

@misc{pith2026241215011,
  author       = {Pith},
  title        = {Pith review of: Quantum teleportation of cat states with binary-outcome measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5OLIWEJT}},
  note         = {Machine review of arXiv:2412.15011}
}
read the original abstract

We propose a teleportation protocol involving beam splitting operations and binary-outcome measurements, such as parity measurements. These operations have a straightforward implementation using the dispersive regime of the Jaynes-Cummings Hamiltonian, making our protocol suitable for a broad class of platforms, including trapped ions, circuit quantum electrodynamics and acoustodynamics systems. In these platforms homodyne measurements of the bosonic modes are less natural than dispersive measurements, making standard continuous variable teleportation unsuitable. In our protocol, Alice is in possession of two bosonic modes and Bob a single mode. An entangled mode pair between Alice and Bob is created by performing a beam splitter operation on a cat state. An unknown qubit state encoded by cat states is then teleported from Alice to Bob after a beamsplitting operation, measurement sequence, and a conditional correction. In the case of multiple measurements, near-perfect fidelity can be obtained. We discuss the optimal parameters in order to maximize the fidelity under a variety of scenarios.

Figures

Figures reproduced from arXiv: 2412.15011 by the authors.

Figure 1
Figure 1. FIG. 1: Schematic illustration of the physical system that [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Our proposed quantum teleportation protocol. The [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The various coherent states that appear in our quan [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: FIG. 5: Fidelity (36) as a function of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Average fidelity (40) as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: (b) shows the results for the same parameters as [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: compared to the single-mode parity measurement results of [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Fidelity (52) as a function of 1 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The probability distributions in the Fock basis [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13: The probability distributions in the Fock basis [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Fidelity (41) in the space of [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]

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Pith tools

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