In situ characterization of linear-optical networks in randomized boson sampling

Carlton M. Caves; Saleh Rahimi-Keshari; Sima Baghbanzadeh

REVIEW 1 major objections 5 minor 63 references

In situ characterization of linear-optical networks in randomized boson sampling

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Randomized boson sampling can be turned into an efficient in situ characterization of lossy linear-optical networks by switching Alice's measurements to heterodyne detection.

desk verdict A genuinely new in situ characterization protocol for lossy LONs in randomized boson sampling, but the headline resource claim T ~ M/delta^2 is off by a factor of M because the estimator divides by small Haar-random diagonal elements. read the letter →

arxiv 1909.00827 v2 pith:KJYXWMCY submitted 2019-09-02 quant-ph

Saleh Rahimi-Keshari , Sima Baghbanzadeh , Carlton M. Caves This is my paper · ORCID

classification quant-ph

keywords linear-opticalnetworkcharacterizationrandomizedbosonsamplingheterodynemeasurementtwo-modesqueezedvacuumlossytransfermatrixentanglementfidelitytotalvariationdistanceinsitu

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

The paper claims that randomized boson sampling, normally a hard sampling task, can be hijacked into an efficient in situ characterization protocol for lossy linear-optical networks. Alice and Bob share two-mode squeezed-vacuum states; when Alice switches from photon counting to heterodyne measurement on her modes, her outcomes—conditioned on Bob's output modes seeing no counts—form zero-mean Gaussian distributions whose covariances encode one column of the lossy transfer matrix $L$ at a time. Repeating this for every output mode reconstructs $L$, and the total number of characterization runs scales as $T\sim M/\delta^2$. The same entanglement-based fidelity, $F=(1-\chi^2)^M/|\det(I-\chi^2 LU^\dagger)|$, then bounds the total variation distance between the ideal and lossy boson-sampling distributions. If correct, this gives experimenters a way to learn the loss properties of a photonic sampling device while it runs, without altering the device.

What carries the argument

The argument is carried by the two-mode squeezed-vacuum state shared between Alice and Bob, plus Alice's ability to choose, run by run, between photon counting and heterodyne detection. A heterodyne outcome $\alpha$ prepares Bob's input in a coherent state $\chi\alpha^*$, and conditioning on no counts in Bob's output mode $i$ leaves Alice's heterodyne statistics in a zero-mean Gaussian whose covariance ellipse has a special axis determined by the outer product $L_iL_i^\dagger$ of the $i$th column of the lossy transfer matrix; measuring the ellipse's orientation and radii recovers that column. The companion object is the entanglement fidelity $F(\rho_{AB|U},\rho_{AB|L})=(1-\chi^2)^M/|\det(I-\chi^2 LU^\dagger)|$, obtained from a Gaussian coherent-state integral over the thermal input state, which converts the whole network comparison into a scalar and then, via standard fidelity–trace-distance inequalities, into a bound on total variation distance between output distributions.

What would settle it

Construct a LON with known engineered losses and add a controllable dark-count rate to Bob's detectors, then run the heterodyne characterization: if the claim is right, the no-count-conditioned outcomes remain a zero-mean Gaussian with covariance $S_i^{-1}$ and the recovered $L$ matches the engineered network; systematic bias or non-Gaussianity that grows with the dark-count rate would falsify the loss-only assumption and identify the protocol's boundary.

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Extended reading notes

Core claim

The central discovery is that the full lossy transfer matrix $L$ of Bob's network can be recovered from first-order coherence in Alice's conditioned heterodyne statistics. Given heterodyne outcome vector $\alpha$ and no photocount in Bob's output mode $i$, Alice's outcomes are drawn from the zero-mean Gaussian distribution $P(\alpha|0_i,L)\propto e^{-\alpha^* S_i \alpha^T}$ with $S_i=(1-\chi^2)I+\chi^2 L_i L_i^\dagger$, so the covariance matrix $S_i^{-1}$ determines the column $L_i$ of $L$. Estimating these covariances for each output mode yields all of $L$, up to output-phase conventions, with uncertainty $\delta\sim \sqrt{M/T}$. The paper further shows that the quantum fidelity between the joint states after the ideal and lossy networks, $F(\rho_{AB|U},\rho_{AB|L})=(1-\chi^2)^M/|\det(I-\chi^2 LU^\dagger)|$, is a computable process-level comparison, and that the total variation distance between the ideal and lossy RBS photocount distributions is bounded by $\sqrt{1-F^2}$.

Load-bearing premise

The entire protocol presumes that every imperfection at Bob's end can be absorbed into a subunitary transfer matrix $L$—pure loss at the input, inside the network, or in the detectors—so that dark counts, mode-mismatched photons that still reach the detectors, and excess Gaussian noise lie outside the model and break both the covariance-based reconstruction and the fidelity bound.

Editorial extensions

If this is right

Alice can intersperse characterization runs with RBS sampling runs without changing Bob's apparatus or telling him which is which, so the same experimental session yields both samples and a live estimate of the LON.
The required number of characterization runs grows only linearly in the number of modes, $T\sim M/\delta^2$, for a target per-element uncertainty $\delta$ in the transfer matrix.
From the characterized $L$, the entanglement fidelity $F=(1-\chi^2)^M/|\det(I-\chi^2 LU^\dagger)|$ gives a computable sufficient condition: if $F\ge\sqrt{1-\epsilon^2}$, then the total variation distance between the ideal and actual RBS photocount distributions is at most $\epsilon$.
Bob's unconditional output statistics determine only the output loss matrix $L^\dagger L$, not the lossless part $V$; Alice's conditional heterodyne statistics are the ones that determine the full transfer matrix.
Under uniform loss $L=tU$, the fidelity is $((1-\chi^2)/(1-\chi^2 t))^M$, so the bound degrades exponentially in $M$ as the transmissivity $t$ drops below one.

Reading between the lines

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

Because the whole characterization rests on first-order coherence, the entanglement of the shared squeezed vacuum is doing identifiable work: with a phase-randomized 'classical-classical' input the same moments only recover $L$ up to complex conjugation and require $T\sim M^2/\delta^2$ runs, so the linear-versus-quadratic scaling is a quantitative measure of what the entanglement buys.
The fidelity $F(\rho_{AB|U},\rho_{AB|L})$ is defined for any two linear-optical processes, so a natural extension is to use it as a standard acceptance metric in generalized photonic circuit debugging, not just in boson sampling, whenever both networks are pure-loss.
A direct experimental test of the protocol's boundary would be to add a controlled dark-count rate to Bob's detectors: the no-count-conditioned heterodyne distribution should cease to be the predicted zero-mean Gaussian, and the estimated $L$ should show systematic bias that grows with the dark-count rate.
The paper leaves non-transfer-matrix noise (mode mismatch, excess Gaussian noise) as an open problem; a parameterized extension that adds such noise to the conditional covariance would let one probe whether the model remains identifiable from heterodyne data alone.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things you should know. The core protocol is genuinely new: Alice can characterize Bob's lossy LON in situ by switching from photon counting to heterodyne on her half of the squeezed vacuum, and the conditional second moments extract the full transfer matrix. The distributed Alice–Bob framing is clean, and the entanglement fidelity bound on total variation distance is a real contribution. The paper also deserves credit for being explicit about what the model cannot handle—dark counts, mode mismatch, excess noise—and for not overselling the result as a verification protocol. The appendices are worth reading, especially the analysis of what the classical-classical state can and cannot do.

The problem is the resource scaling in Eq. (3.55). The estimator for off-diagonal elements divides by L_ii, and for a Haar-random LON the typical |L_ii| is O(M^{-1/2}). That amplifies the estimation error from ~1/chi^2 sqrt(T) to ~1/(chi^2 |L_ii| sqrt(T)) ~ M/sqrt(T), so the number of runs needed for a given per-element error scales as T ~ M^2/delta^2, not M/delta^2. The stress-test note is correct. This is not a minor technicality: the protocol is still polynomial in M, so the main idea survives, but the practical efficiency claim is off by a factor of M, which matters for the near-term experiments the paper targets.

Other soft spots are minor by comparison. The step from estimating outer products LiL_i† to uniqueness of the full L is asserted rather than fully proven, though Appendix B gives supporting structure. There is no error propagation analysis for the estimated L, only the unsupported 1/sqrt(T) statement. The fidelity bound derivation in Appendix A is solid, and the uniform-loss example is helpful.

Who should read this? Anyone working on characterization or validation of linear-optical networks, especially for boson sampling. It deserves a serious referee, but the referee should insist on a corrected resource analysis, ideally with numerical checks on Haar-random LONs. I would not cite the T ~ M/delta^2 claim as is, but the protocol and fidelity bound are citable after revision.

Referee Report

1 major / 5 minor

Summary. This paper proposes and analyzes a protocol for characterizing a lossy linear-optical network (LON) in the setting of randomized boson sampling (RBS). Alice and Bob share M two-mode squeezed-vacuum states. In RBS runs Alice uses photon counting; in characterization runs she performs heterodyne detection on her modes. Conditioned on Bob reporting no counts in a given output mode, Alice's heterodyne outcomes are Gaussian, with a covariance matrix that encodes the corresponding column of the lossy transfer matrix L. The paper derives estimators for diagonal and off-diagonal elements of L (Eqs. (3.53)–(3.54)), states a sample-complexity bound T∼M/δ^2 (Eq. (3.55)), and derives a fidelity between the ideal and lossy joint states that bounds the total variation distance between the ideal and lossy RBS probability distributions (Eqs. (3.69)–(3.70)). It also discusses limitations (dark counts, mode mismatch, excess noise) and provides appendices with a Kraus-operator description of lossy LONs and an analysis of characterization with classical-classical correlations.

Significance. The basic idea is attractive and the core derivations are explicit: the conditioned heterodyne distribution is Gaussian, the covariance is a rank-one perturbation of the identity, and the fidelity bound is obtained by a clean Gaussian integral. The paper is honest about the scope of the pure-loss model. The fidelity measure (entanglement fidelity) and the bound on total variation distance are useful contributions in their own right. The derivations are explicit and checkable, which is a strength. However, the advertised efficiency scaling T∼M/δ^2 is the central quantitative claim, and as written it does not follow from the estimators; this requires a nontrivial revision of either the estimation procedure or the scaling claim.

major comments (1)

[Sec. III B 3, Eqs. (3.53)–(3.55)] The key scaling claim T∼M/δ^2 is not supported by the estimators presented. From Eq. (3.54), L_ji = −⟨α_j α_i^*⟩_i/(χ^2 L_ii), so the standard error in L_ji is ∼1/(χ^2 |L_ii| sqrt(T)), not the stated ∼1/(χ^2 sqrt(T)). For a Haar-random U, |U_ii|^2 has a Beta(1, M−1) distribution, so |L_ii| is typically O(M^{−1/2}); with χ^2 ≃ 1/√M this gives per-element uncertainty ∼M/√T and hence T ∼ M^2/δ^2. The paper explicitly uses only the M moments with k = i (Eq. (3.50)) and does not exploit the full covariance matrix, so the linear scaling does not follow. Note that this quadratic scaling is the same as the paper finds for the classical-classical-state protocol in Appendix B (end of B.3), which undercuts the stated advantage of the TMSV-based method. The claim can likely be repaired by estimating the special eigenvector of the full covariance matrix S_i^{-1} (the eigenvalue gap is O(χ^2), so the eigenvector error is O(1/(χ^2√T))), but as written the protocol and its complexity claim are internally inconsistent.

minor comments (5)

[After Eq. (3.36)] The word "Husismi" should be "Husimi".
[Sec. III B 3] The step from the outer products L_i L_i† to a complete reconstruction of L is asserted with "A little thought shows that..." but is not demonstrated; a short explicit reconstruction algorithm would help the reader verify the uniqueness claim.
[Sec. IV] The Nobel Prize narrative is stylistically unusual for a research paper; it could be shortened or moved to a more informal venue.
[Abstract] The phrase "without Bob's knowing" is colloquial; "without Bob's knowledge" would be more standard.
[Eq. (3.75)] The approximation for the fidelity in the uniform-loss case is written with "≃" but the two exponents are of different order in χ^2; clarifying the regime of validity would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the covariance-to-L inversion and fidelity bound are explicit and self-contained.

full rationale

The paper's central derivation is self-contained. The conditional heterodyne distribution given no counts in output mode i is derived directly from the two-mode squeezed-vacuum state and the coherent-state action of the lossy LON (Eqs. 3.15-3.44). The covariance S_i^{-1} is then inverted in Eqs. (3.50)-(3.54) to express the column L_i in terms of measured second moments; this is an explicit parameter inversion from a well-defined likelihood, not a fitted parameter relabeled as a prediction. The fidelity bound in Eq. (3.70) uses the characterized L and the requested U as arguments; the inequality D ≤ sqrt(1-F^2) is a standard result, and F(ρ_AB|U, ρ_AB|L) is a definitional function of the same L used in the characterization, so no circularity arises. Self-citations to prior coherent-state characterization [34,35], randomized boson sampling [15], and SU(1,1) interferometry [32] are motivational or contextual; the main inversion and fidelity calculation do not depend on these citations as load-bearing premises. The admitted limitations in Sec. IV (dark counts, mode-mismatched photons, and excess Gaussian noise) delimit the model but do not make the derivation circular. The possible T-scaling issue raised by the skeptic is an accuracy concern about the estimator variance and Haar-random typical column magnitudes, not a reduction of the result to its own inputs.

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

No free parameters are fitted to data; the only tunable physical input is the squeezing parameter χ, which is an experimental resource, not a fitted constant. The protocol rests on the loss-only transfer-matrix model and on standard Gaussian quantum optics. No new particles, forces, or unobserved entities are postulated.

assumptions (5)

domain assumption The lossy LON is fully characterized by a subunitary M×M transfer matrix L acting on coherent states as |β⟩ → |βL⟩.
Introduced in Sec. III A and used throughout; excludes dark counts, mode-mismatched photons, and excess Gaussian noise (Sec. IV).
domain assumption The M SPDC sources produce identical two-mode squeezed-vacuum states with known squeezing parameter χ, with relative phases set so χ is real.
Eq. (2.1) and Fig. 1; Alice uses χ in the inversion formulas (3.53)-(3.54).
domain assumption Bob is honest and implements the requested LON; Alice can interleave characterization runs without changing anything at Bob's end.
Protocol steps in Sec. II; the 'in situ' and 'without Bob's knowing' claims depend on Bob's actions being independent of Alice's measurement choice.
domain assumption Weak squeezing, χ^2 ≲ 1/√M, holds for the efficiency estimates and simplified reconstructions.
Stated in Sec. II and used for T ~ M/δ^2 (Eq. 3.55) and Eqs. (3.53)-(3.54); exact formulas exist but the scaling is derived in this regime.
standard math Standard properties of Gaussian states, coherent-state transformations, and fidelity inequalities used in derivations.
Used for the conditional Gaussian distribution (3.44), the Gaussian integral (A31), and the fidelity and trace-distance inequality array (3.61).

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Cite this review

Pith. "Pith review of In situ characterization of linear-optical networks in randomized boson sampling." pith.science (2026). https://pith.science/paper/KJYXWMCY

@misc{pith2026190900827,
  author       = {Pith},
  title        = {Pith review of: In situ characterization of linear-optical networks in randomized boson sampling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KJYXWMCY}},
  note         = {Machine review of arXiv:1909.00827}
}

read the original abstract

We introduce a method for efficient, in situ characterization of linear-optical networks (LONs) in randomized boson-sampling (RBS) experiments. We formulate RBS as a distributed task between two parties, Alice and Bob, who share two-mode squeezed-vacuum states. In this protocol, Alice performs local measurements on her modes, either photon counting or heterodyne. Bob implements and applies to his modes the LON requested by Alice; at the output of the LON, Bob performs photon counting, the results of which he sends to Alice via classical channels. In the ideal situation, when Alice does photon counting, she obtains from Bob samples from the probability distribution of the RBS problem, a task that is believed to be classically hard to simulate. When Alice performs heterodyne measurements, she converts the experiment to a problem that is classically efficiently simulable, but more importantly, enables her to characterize a lossy LON on the fly, without Bob's knowing and without changing anything at Bob's end (this is what we mean by in situ). We introduce and calculate the fidelity between the joint states shared by Alice and Bob after the ideal and lossy LONs as a measure of distance between the two LONs. Using this measure, we obtain an upper bound on the total variation distance between the ideal probability distribution for the RBS problem and the probability distribution achieved by a lossy LON. Our method displays the power of the entanglement of the two-mode squeezed-vacuum states: the entanglement allows Alice to choose for each run of the experiment between RBS and a simple characterization protocol based on first-order coherence between complex amplitudes.

Figures

Figures reproduced from arXiv: 1909.00827 by the authors.

**Figure 2.** plots how this fidelity scales with the transmissivity t, for different values of M. IV. DISCUSSION AND CONCLUSION We present in this paper a protocol for characterizing randomized boson sampling that is implemented jointly by two parties, Alice and Bob, who share weakly squeezed, quantum-entangled, two-mode squeezed-vacuum states produced by a set of spontaneous parametric down-converters. Alice instructs Bob to c… view at source ↗

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Works this paper leans on

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This larger network is characterized by a unitary operator˜U and a corresponding unitary transfer matrix ˜U = (L R S ˜L ) , (A1) which describes, as in Eq

Lossy networks using auxiliary loss modes To make progress, we need to do some preliminary work by developing the description of the lossy network as part of a larger, lossless network that has ˜M auxiliary modes; the auxiliary modes are initialized in vacuum and receive the photons lost from Bob’s original network. This larger network is characterized by...

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(A23) The factor ⏐⏐⟨nB|U| nA⟩ ⏐⏐ is zero unless|nB| =|nA|, so it gives the indicated restriction on the sum

Fidelity bound (3.69) We turn now to the ﬁdelity bound (3.69) and start with the ﬁdelity (3.63) between the ideal and lossy RBS distributions, F ( PBS|nA,U,P BS|nA,L ) = ∑ nB √ PBS(nB|nA,U)PBS(nB|nA,L) = ∑ nB |nB|=|nA| |⟨nA|U†|nB⟩| √ ⟨nB|EL (|nA⟩⟨nA|)|nB⟩. (A23) The factor ⏐⏐⟨nB|U| nA⟩ ⏐⏐ is zero unless|nB| =|nA|, so it gives the indicated restriction on ...

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A 2 we formulate bounds on the classical ﬁdelity between the joint distributionsPQ|U andPQ|L of Eqs

Fidelity bounds in ﬁxed photon-number sectors In App. A 2 we formulate bounds on the classical ﬁdelity between the joint distributionsPQ|U andPQ|L of Eqs. (2.6) and (3.7). At Eq. (A27), this involved averaging the ﬁdelity between the ideal and lossy boson-sampling distributions over all input photo- count recordsnA. In this section, we formulate ﬁdelity b...

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Conditional heterodyne statistics Starting with a joint probability distribution P (α,nB|L), we want to condition the heterodyne outcomes on photocount records at Bob’s end. The only conditioning we use below is on a ﬁxed set of modes having no counts, but for generality here, we introduce a set ∆ of photocount records and the projector onto the subset sp...

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Conditional heterodyne moments Our characterization procedure runs on conditional heterodyne moments. The moment-generating characteristic function, Φ(ξ,ξ†|∆,L) = ⟨ eξα†−αξ†⟩ = ∫ d2MαP (α|∆,L)eξα†−αξ† , (B19) is the Fourier transform of the relevant distribution and corresponds to F (α,α†) = eξα†−αξ† . The characteristic function generates moments via its...

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Characterization using CC state The classical-classical (CC) state (4.1) has perfect, but purely classical photon-number correlations between theA modes and theB modes; thus it can be used for randomized boson sampling in exactly the same way as the two-mode squeezed-vacuum 27 input|ΨAB⟩ of Eq. (2.2). For both these states, the marginal state of either th...

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RBS-only characterization A reward—really quite a considerable reward—for having done the analysis of heterodyne statistics and the CC state is that we can now address and answer the question of what kind of characterization can be done with Alice’s conditional photostatistics in the RBS runs of our protocol. If we are interested only in photocounting at ...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

This larger network is characterized by a unitary operator˜U and a corresponding unitary transfer matrix ˜U = (L R S ˜L ) , (A1) which describes, as in Eq

Lossy networks using auxiliary loss modes To make progress, we need to do some preliminary work by developing the description of the lossy network as part of a larger, lossless network that has ˜M auxiliary modes; the auxiliary modes are initialized in vacuum and receive the photons lost from Bob’s original network. This larger network is characterized by...

work page

[2] [2]

(A23) The factor ⏐⏐⟨nB|U| nA⟩ ⏐⏐ is zero unless|nB| =|nA|, so it gives the indicated restriction on the sum

Fidelity bound (3.69) We turn now to the ﬁdelity bound (3.69) and start with the ﬁdelity (3.63) between the ideal and lossy RBS distributions, F ( PBS|nA,U,P BS|nA,L ) = ∑ nB √ PBS(nB|nA,U)PBS(nB|nA,L) = ∑ nB |nB|=|nA| |⟨nA|U†|nB⟩| √ ⟨nB|EL (|nA⟩⟨nA|)|nB⟩. (A23) The factor ⏐⏐⟨nB|U| nA⟩ ⏐⏐ is zero unless|nB| =|nA|, so it gives the indicated restriction on ...

work page

[3] [3]

A 2 we formulate bounds on the classical ﬁdelity between the joint distributionsPQ|U andPQ|L of Eqs

Fidelity bounds in ﬁxed photon-number sectors In App. A 2 we formulate bounds on the classical ﬁdelity between the joint distributionsPQ|U andPQ|L of Eqs. (2.6) and (3.7). At Eq. (A27), this involved averaging the ﬁdelity between the ideal and lossy boson-sampling distributions over all input photo- count recordsnA. In this section, we formulate ﬁdelity b...

work page

[4] [4]

Conditional heterodyne statistics Starting with a joint probability distribution P (α,nB|L), we want to condition the heterodyne outcomes on photocount records at Bob’s end. The only conditioning we use below is on a ﬁxed set of modes having no counts, but for generality here, we introduce a set ∆ of photocount records and the projector onto the subset sp...

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[5] [5]

Conditional heterodyne moments Our characterization procedure runs on conditional heterodyne moments. The moment-generating characteristic function, Φ(ξ,ξ†|∆,L) = ⟨ eξα†−αξ†⟩ = ∫ d2MαP (α|∆,L)eξα†−αξ† , (B19) is the Fourier transform of the relevant distribution and corresponds to F (α,α†) = eξα†−αξ† . The characteristic function generates moments via its...

work page

[6] [6]

diagonal

Characterization using CC state The classical-classical (CC) state (4.1) has perfect, but purely classical photon-number correlations between theA modes and theB modes; thus it can be used for randomized boson sampling in exactly the same way as the two-mode squeezed-vacuum 27 input|ΨAB⟩ of Eq. (2.2). For both these states, the marginal state of either th...

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[7] [7]

RBS-only characterization A reward—really quite a considerable reward—for having done the analysis of heterodyne statistics and the CC state is that we can now address and answer the question of what kind of characterization can be done with Alice’s conditional photostatistics in the RBS runs of our protocol. If we are interested only in photocounting at ...

work page

[8] [8]

Preskill, Quantum computing and the entanglement frontier, 2012, arXiv:1203.5813

J. Preskill, Quantum computing and the entanglement frontier, 2012, arXiv:1203.5813

arXiv 2012