Experimental Virtual Quantum Broadcasting

Dawei Lu; Hongfeng Liu; Xiangjing Liu; Xinfang Nie; Yutong Luo; Yuxuan Zheng

REVIEW 3 major objections 5 minor 1 cited by

Experimental Virtual Quantum Broadcasting

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Virtual quantum broadcasting, a map that the no-broadcasting theorem forbids for physical processes, is realized on an NMR processor by one linear-combination-of-unitaries circuit and classical post-processing.

desk verdict A credible first experimental realization of virtual quantum broadcasting; the optimality verification is narrower than the abstract implies, but the central claim holds. read the letter →

arxiv 2501.11390 v2 pith:VRER35Y7 submitted 2025-01-20 quant-ph

Yuxuan Zheng , Xinfang Nie , Hongfeng Liu , Yutong Luo , Dawei Lu , Xiangjing Liu This is my paper · ORCID

classification quant-ph

keywords virtualquantumbroadcastingHPTPmapuniversalclonerantisymmetrizerlinearcombinationofunitariesno-broadcastingtheoremNMRprocessorerrormitigation

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

Virtual quantum broadcasting (VQB) is a Hermitian-preserving trace-preserving map that, if realized, would give two perfect local copies of any quantum state while preserving correlations between the copies—something the no-broadcasting theorem says no physical quantum channel can do. This paper reports the first experimental realization of VQB on a nuclear magnetic resonance quantum processor, using one linear-combination-of-unitaries circuit followed by classical post-processing. The virtual output is assembled from two physical channels, the universal cloner $B_+$ and the universal antisymmetrizer $B_-$, via $B(\rho)=d(p_0B_+(\rho)-p_1B_-(\rho))$. The experiments confirm that $B_+$ is the closest physical map to VQB within a parameterized family, and that subtracting the $B_-$ branch suppresses the unavoidable cloning error to near-ideal fidelity. If this works beyond the tested states, it turns a previously theoretical HPTP virtual operation into a concrete recipe available on current hardware, extendable to any finite dimension.

What carries the argument

The load-bearing object is the canonical broadcasting map $B(\rho)=\frac{1}{2}\{\rho\otimes I,S\}$, a Hermitian-preserving trace-preserving (HPTP) map with output equal to a pseudo-density matrix, whose two partial traces both return $\rho$. The argument runs on the decomposition $B=\frac{d+1}{2}B_+-\frac{d-1}{2}B_-$, with $B_+(\rho)=\frac{2}{d+1}\Pi_+(\rho\otimes I)\Pi_+$ (universal cloner) and $B_-(\rho)=\frac{2}{d-1}\Pi_-(\rho\otimes I)\Pi_-$ (universal antisymmetrizer), where $\Pi_\pm=(I\otimes I\pm S)/2$. The circuit implements the linear combination of the unitaries $I\otimes I$ and $S$ through a controlled-SWAP gate on a probe qubit; postselecting the probe on $|0\rangle$ or $|1\rangle$ yields the two channels with probabilities $p_0=(d+1)/(2d)$ and $p_1=(d-1)/(2d)$, and the virtual map is assembled classically as $B(\rho)=d(p_0B_+(\rho)-p_1B_-(\rho))$. The optimality of $B_+$ is carried by Choi-state comparison: because the diamond distance to $B$ is attained on the maximally entangled state, the experiment measures $\|C_B-C_{N_\theta}\|_1$ for the family $N_\theta$ and locates its minimum at $N_{\pi/2}=B_+$.

What would settle it

Prepare a batch of random qubit states $\rho$, run the broadcast circuit for each, reconstruct $p_0B_+(\rho)$ and $p_1B_-(\rho)$, form $B(\rho)=2(p_0B_+(\rho)-p_1B_-(\rho))$, and compute the trace distances between the two partial traces $\mathrm{Tr}_A B(\rho)$, $\mathrm{Tr}_B B(\rho)$ and the input $\rho$; if the average distance significantly exceeds the tomography error bars, the virtual broadcast map is not being realized.

Watch

Extended reading notes

Core claim

The paper's central claim is that virtual quantum broadcasting, although not a physically allowed process, can be realized as a virtual operation on real hardware. The canonical broadcasting map $B(\rho)=\frac{1}{2}\{\rho\otimes I,S\}$ is decomposed as $B=\frac{d+1}{2}B_+-\frac{d-1}{2}B_-$, where $B_+$ is the universal cloner and $B_-$ is the universal antisymmetrizer. A controlled-SWAP circuit on three qubits, based on the linear combination of unitaries, realizes $B_+$ when the probe qubit is postselected on $|0\rangle$ and $B_-$ when postselected on $|1\rangle$; the exact HPTP map is then recovered by classical subtraction, $B(\rho)=d(p_0B_+(\rho)-p_1B_-(\rho))$. Using a four-qubit NMR processor, the paper reconstructs the outputs for pure and mixed one-qubit inputs, reconstructs the Choi states of $B$, $B_+$, $B_-$, and a one-parameter family $\{N_\theta\}$, and finds the minimal trace distance to $B$ at $\theta=\pi/2$, where $N_{\pi/2}=B_+$. It further demonstrates that the $B_-$ branch cancels the no-cloning error, raising the broadcast fidelity to near $1$.

Load-bearing premise

The load-bearing premise is that the ensemble-NMR reconstruction of the post-selected branches faithfully represents the true conditional states, because the experiment cannot projectively measure the probe qubit and instead infers each $p_i\rho_i$ from full state tomography, so any reconstruction error becomes a direct error in the claimed $B(\rho)$.

Editorial extensions

If this is right

Virtual quantum broadcasting of arbitrary qubit states is experimentally achievable on current ensemble hardware with a single linear-combination-of-unitaries circuit and classical post-processing.
Within the tested family of physical maps, $B_+$ is the closest to the virtual broadcast map, so no physical broadcasting channel can beat the universal cloner; improvement has to come from classical post-processing, exactly as the protocol does.
The antisymmetrizer branch is an error-mitigation resource: subtracting it from the cloner branch suppresses the no-cloning fundamental error and brings the output fidelity near $1$.
Because the procedure generalizes to any finite dimension $d$ and the success probability $p_0=(d+1)/(2d)$ is bounded away from zero, the same circuit pattern can be scaled to higher-dimensional virtual broadcasting without discarding data.

Reading between the lines

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

If the reconstruction is as faithful as reported, the same circuit pattern should transfer to platforms with projective measurement, where the two branches $B_+$ and $B_-$ can be sampled directly instead of being inferred from full tomography, reducing overhead.
The measured trace-distance curve as a function of $\theta$ suggests a general diagnostic: any experimental broadcast attempt can be scored by its distance to the virtual map, making VQB a benchmark for cloner hardware.
Because the qubit that carries the input state is left untouched by the circuit, the scheme could in principle be iterated to spawn multiple copies of the same state for multi-recipient tasks, a step the paper only gestures toward.
The same two-branch subtraction pattern may apply to other Hermitian-preserving trace-preserving virtual operations whenever the unphysical map can be split into two physical channels, extending the error-mitigation reading beyond cloning.

Share X Bluesky LinkedIn Reddit HN

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

The paper does what it says on the tin: it implements virtual quantum broadcasting on a four-qubit NMR processor using an LCU circuit and classical post-processing. The central claim, that B(ρ)=d(p0B+−p1B−) can be realized and that the reconstructed map matches theory for several input states, holds up. The Choi-state eigenvalues showing B± are completely positive while B is not is a clean, direct check. This is the first experimental realization of VQB as far as I can tell, and the circuit is simple enough that the result should be reproducible on other platforms.

What is not new is the map and its decomposition—those come from Parzygnat et al. [12]. The experimental contribution is the circuit, the validation protocol, and the error-mitigation framing. The framing is the weakest part. The claim that the universal cloner is "the closest physical map to VQB" is only verified within a one-parameter family Nθ that is constructed to contain B+ at θ=π/2. The trace-distance scan over that family is a consistency check, not a proof of global optimality. The global statement is a theorem imported from [12]. The abstract overstates this, and the paper should be revised to say "within our chosen family" rather than implying a complete experimental verification.

The error-mitigation section is also softer than it looks. The suppression of the "fundamental error" is essentially a restatement of the defining identity B = d(p0B+ − p1B−). Since the reduced states of B(ρ) are ρ by construction, the post-processed marginals are guaranteed to be near the input once B± are reconstructed correctly. It is fair to present this as a demonstration of virtual broadcasting, but calling it an "error mitigation protocol" without noting that it is definitional is a bit generous.

A more technical concern: the ensemble NMR implementation never performs a projective measurement on the probe. The conditional states p0ρ0 and p1ρ1 are inferred from full tomography of the three- or four-qubit state. That is a standard and legitimate procedure in NMR, but it means the "post-selection" is classical post-processing on tomographic data. The paper should state this clearly in the main text, not just in the SM. It also means the fidelity of the reconstructed B(ρ) is directly tied to the tomography accuracy, which the SM does discuss.

Raw data are not provided beyond the averaged plots, but the error analysis in the SM is reasonable, with fidelities above 0.96 for the four-qubit states and error bars from five repetitions. That is enough for a proof of principle.

Overall: this is a solid, honest experimental paper with one overstated conclusion. It deserves serious peer review. I would recommend acceptance after the authors tone down the optimality claim and reposition the error-mitigation section. The citation pattern is fine, and the connection to the pseudo-density-matrix experiments is acknowledged.

Would I bring it to reading group? Yes, if only to discuss how virtual operations differ from physical ones. I would cite it as the experimental realization of VQB.

Referee Report

3 major / 5 minor

Summary. The paper reports an NMR experimental realization of virtual quantum broadcasting (VQB), an HPTP map that satisfies the broadcasting condition through post-processing rather than as a physical CPTP channel. The authors implement an LCU circuit with a probe qubit to realize the universal cloner B+ and the universal antisymmetrizer B-, reconstruct the conditional states via full state tomography, and combine them as B = d(p0 B+ - p1 B-) to obtain the VQB output. They also prepare Choi states, verify the CP property of B± and the non-CP property of B by eigenvalue measurements, scan a one-parameter family of CPTP maps Nθ to argue that B+ is the closest physical map to B, and present the post-processed results as an error-mitigation demonstration with near-unity fidelity.

Significance. If the claims are properly qualified, this is a valuable first experimental demonstration of a virtual HPTP operation: the LCU circuit design is explicit, the post-processing algebra in SM Sec. B is correct, and the Choi-state eigenvalue measurements directly confirm the expected CP/non-CP structure. The experimental implementation is careful, with five repeated trials, error bars, and a documented error model in SM Sec. E. The central realization claim---that VQB can be implemented by a single circuit followed by classical post-processing---is credible and significant for the ongoing program of virtual quantum operations. The main advertised secondary claims, however, go beyond what the data support.

major comments (3)

[Section "Validation of optimality of the universal cloner B+", Fig. 3(d), and abstract] The experimental data support only the statement that B+ minimizes the Choi trace distance ||CB - CNθ||1 within the specially constructed one-parameter family Nθ, where Nθ is deliberately chosen so that Nπ/2 = B+. The abstract and conclusion claim that the experiment "demonstrates that the universal cloner is the closest physical map to VQB" among all CPTP maps. That global statement is a theorem imported from Ref. [12]; for Nθ ≠ B+, the Choi trace distance is only a lower bound on the diamond distance, and no equality is established in the manuscript for those points. The data therefore do not experimentally verify global optimality. The claims should be revised to "closest within the chosen family" or "consistent with the theoretical optimality of B+".
[Section "VQB as an error-mitigation protocol", Eq. (3), Fig. 4(c)] The near-unity fidelity after post-processing is a direct consequence of the defining identity B(ρ) = d(p0 B+(ρ) - p1 B-(ρ)) and the broadcasting condition TrA B(ρ) = TrB B(ρ) = ρ, provided B± are correctly reconstructed from the tomography. Thus Fig. 4(c) is primarily a consistency check of the reconstruction rather than an independent demonstration that the "fundamental error" imposed by quantum mechanics is suppressed. To make the error-mitigation claim nontrivial, the authors should either reframe this section as a direct demonstration of the VQB identity or benchmark the post-processed results against a physical cloner run under the same experimental noise.
[SM Sec. E and Fig. 3(d)] The experimental procedure for scanning θ is not fully specified: the number of θ values, their spacing, and the number of independent repetitions per θ are not stated in the main text or clearly in the SM. Since the optimality claim is based on a curve of trace distances, the sampling density matters for the strength of the claim. Please specify the full θ grid and state clearly that the minimum is found only on that grid.

minor comments (5)

[Fig. 1 caption] The phrase "The size of the qubit system represents the cloning fidelity" is unclear; please explain what is meant by "size" in this schematic.
[Main text, Experimental setup paragraph] The sentence "The circuit applies for the input state ρ with an any finite dimension d, we take d = 2 in our experiment" is grammatically awkward and should be rewritten.
[SM Eq. (12)] Equation (12) contains garbled rendering ("/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne...") that obscures the mathematical derivation; this must be fixed.
[SM Sec. E] The reported fidelity bounds "above 0.98" and "above 0.96" for the reconstructed states are not accompanied by the actual values or the procedure used to compute fidelity; please provide these details.
[Conclusions] The statement "no data is wasted given the probabilistic nature of our method" would benefit from clarification that all probe outcomes, including both P0 and P1 branches, are used in the classical linear combination.

Circularity Check

1 steps flagged · score 4.0 of 10

Error-suppression result reduces to the defining identity B=d(p0B+−p1B−); the core LCU realization is self-contained, while optimality claims exceed the one-parameter scan.

self definitional [Main text, Eq. (2), Eq. (3), and section 'VQB as an error-mitigation protocol', Fig. 4(c)]
"B(ρ)= d(p0B+(ρ)− p1B−(ρ)). (3) ... We classically process the data according to Eq. (3), and present the corresponding results in Fig. 4(c). We observe that the fundamental errors are significantly suppressed, and the fidelities are near 1, demonstrating the effectiveness of VQB as an error mitigation protocol for broadcasting arbitrary quantum states."

The decomposition (2) is the definition of B from B± and the projection probabilities, and Eq. (3) is exactly that decomposition rewritten with p0=(d+1)/2d and p1=(d−1)/2d. Therefore the post-processed combination is B(ρ) by construction, and the broadcasting condition Tr_A B(ρ)=Tr_B B(ρ)=ρ is an algebraic identity following from B(ρ)=1/2{ρ⊗I,S}. Once the conditional states p0B+(ρ) and p1B−(ρ) are correctly reconstructed, the near-unit fidelity in Fig. 4(c) is guaranteed by Eq. (3); it is a consistency check of the tomography, not an experimentally discovered suppression of a fundamental error.

full rationale

The central claim—that VQB can be realized by one LCU circuit plus classical post-processing—is not circular: the circuit is a concrete construction whose post-selected branches are measured and tomographically reconstructed, and the comparison of the reconstructed B(ρ) with the theoretical map is an honest experimental consistency check. The optimality statement is also not circular in the strict sense, because the global result that B+ minimizes diamond distance is imported from Ref. [12] (an independent theoretical source), and the experiment only scans a one-parameter family Nθ with Nπ/2=B+; this limits the evidential weight but does not make the claim definitionally forced. The one genuine circular step is the error-mitigation demonstration in Fig. 4(c), whose 'near-ideal fidelity' follows by definition from Eq. (3) once B+ and B− are reconstructed. This is a mild, explicit circularity in an advertised application, not in the core implementation claim, so the score is 4 rather than higher.

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

No free parameter is fitted to data: the family parameter θ is scanned, and p0,p1 are derived theoretically and checked experimentally. The paper introduces no new physical entities; the virtual map B and its decomposition are imported from Ref. [12]. The main axioms are the uniqueness and operational interpretation of VQB, the LCU circuit algebra, and the NMR ensemble-to-pure-state modeling.

assumptions (5)

domain assumption The canonical VQB map B(ρ)=1/2{ρ⊗I,S} is uniquely selected by covariance, permutation invariance, and consistency with classical broadcasting.
Adopted from Ref. [12]; all experiment targets this specific HPTP map without re-deriving uniqueness.
domain assumption HPTP maps can be operationally realized through classical post-processing of measurement statistics from physical channels (virtual operations).
This is the conceptual foundation of the experiment, taken from Refs. [12,31-33]; if this interpretation fails, the claim of realizing VQB fails.
standard math The LCU circuit with control in |+>, system A in I/d, system B in ρ, followed by probe measurement, produces B± with probabilities p0=(d+1)/2d and p1=(d-1)/2d.
Derived in SM Sec. B, Eqs. (13)-(16); relies on standard algebra of swap projectors.
domain assumption NMR pseudo-pure state preparation, shaped-pulse control, and quantum state tomography accurately emulate ideal pure-state circuit operations and allow extraction of post-selected states.
SM Secs. D-E; no direct projective measurement is possible in ensemble NMR, so all conditional states come from tomographic reconstruction.
domain assumption Diamond distance between two channels equals trace distance between their Choi states, and B+ is globally the closest CPTP map to B.
The Choi-state comparison is standard (Refs. [63-65]), but the global optimality statement is a theorem from Ref. [12], not established by the family scan in Fig. 3(d).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Experimental Virtual Quantum Broadcasting." pith.science (2026). https://pith.science/paper/VRER35Y7

@misc{pith2026250111390,
  author       = {Pith},
  title        = {Pith review of: Experimental Virtual Quantum Broadcasting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VRER35Y7}},
  note         = {Machine review of arXiv:2501.11390}
}

read the original abstract

The quantum no-broadcasting theorem states that it is fundamentally impossible to perfectly replicate an arbitrary quantum state, even if correlations between the copies are allowed. While quantum broadcasting cannot occur through any physical process, it can be achieved via postprocessing of experimental data using a process called virtual quantum broadcasting (VQB). In this work, we report the experimental implementation of a quantum circuit based on the linear combination of unitaries, integrated with a post-processing protocol, to realize VQB in a nuclear magnetic resonance system. VQB can be expressed as a linear combination of two channels: the universal cloner, which broadcasts the target quantum state, and the universal antisymmetrizer, which reduces broadcasting error. We implement both channels within the same circuit and demonstrate that the universal cloner is the closest physical map to VQB. In addition, we show how the universal antisymmetrizer can be utilized to mitigate imperfections in the cloner, enabling near-ideal fidelity. Our method is applicable to broadcasting quantum systems of any dimension.

Figures

Figures reproduced from arXiv: 2501.11390 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: (a). We implement the quantum circuit depicted in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 7.** Figure 7: FIG. 7. (a) The NMR parameters and molecular structure of [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8. (a) Initialization circuit for the universal cloner experiment. (b-d) Initialization circuits for preparing: (b) pure states of the form [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Virtual Cloning of Quantum States
quant-ph 2025-07 accept novelty 7.0 of 10

A set of quantum states can be virtually cloned if and only if their density matrices are linearly independent, and the optimal 1-to-2 cloning cost for any pair of pure states equals sqrt(1+|⟨ψ1|ψ2⟩|²).

Reference graph

Works this paper leans on

76 extracted references · 57 canonical work pages · cited by 1 Pith paper

[12]

A. W. Harrow, The church of the symmetric subspace, arXiv:1308.6595 (2013)

arXiv 2013
[1]

Wehner, D

S. Wehner, D. Elkouss, and R. Hanson, Quantum internet: A vision for the road ahead, Science 362, eaam9288 (2018)

2018
[2]

The experimental results align closely with the theoretical predictions, demon- strating the success and reliability of our circuit and the post- processing procedure. V alidation of optimality of the universal cloner B+.—We design and implement an experiment to validate the univer- sal cloner B+ as the closest CPTP map from a specially de- signed set of ...

work page
[3]

The Choi state CB is constructed by d(p0CB+− p1CB−)= CB

Finally, the Choi state of B can be constructed as follows: set θ = π/2 and perform the pro- jective measurement {P0, P1} on the probe, which yields the collapsed states of the remaining three qubits, CB+ and CB−, with probabilities p0 and p1, respectively (see SM [47]). The Choi state CB is constructed by d(p0CB+− p1CB−)= CB. We experimentally construct ...

work page
[4]

Caleffi, M

M. Caleffi, M. Amoretti, D. Ferrari, J. Illiano, A. Manzalini, and A. S. Cacciapuoti, Distributed quantum computing: a survey, Computer Networks 254, 110672 (2024)

work page 2024
[5]

A. S. Cacciapuoti, M. Caleffi, F. Tafuri, F. S. Cataliotti, S. Gher- ardini, and G. Bianchi, Quantum internet: Networking chal- lenges in distributed quantum computing, IEEE Network 34, 137 (2019)

work page 2019
[6]

Zhang and Q

Z. Zhang and Q. Zhuang, Distributed quantum sensing, Quan- tum Sci. Technol. 6, 043001 (2021)

work page 2021
[7]

C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017)

2017

Show all 76 references

[8]

W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature 299, 802 (1982)

1982
[9]

Barnum, C

H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schu- macher, Noncommuting mixed states cannot be broadcast, Phys. Rev. Lett. 76, 2818 (1996)

1996
[10]

D. Bruß, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchi- avello, and J. A. Smolin, Optimal universal and state-dependent quantum cloning, Phys. Rev. A 57, 2368 (1998)

1998
[11]

R. F. Werner, Optimal cloning of pure states, Phys. Rev. A 58, 1827 (1998)

1998
[13]

Chiribella, On quantum estimation, quantum cloning and fi- nite quantum de finetti theorems, in Conference on Quantum Computation, Communication, and Cryptography (Springer,

G. Chiribella, On quantum estimation, quantum cloning and fi- nite quantum de finetti theorems, in Conference on Quantum Computation, Communication, and Cryptography (Springer,
[14]

A. J. Parzygnat, J. Fullwood, F. Buscemi, and G. Chiribella, Virtual quantum broadcasting, Phys. Rev. Lett. 132, 110203 (2024)

2024
[15]

J. F. Fitzsimons, J. A. Jones, and V . Vedral, Quantum correla- tions which imply causation, Sci. Rep. 5, 18281 (2015)

2015
[16]

A. J. Parzygnat and J. Fullwood, From time-reversal symmetry to quantum bayes’ rules, PRX Quantum 4, 020334 (2023)

2023
[17]

X. Liu, Q. Chen, and O. Dahlsten, Inferring the arrow of time in quantum spatiotemporal correlations, Phys. Rev. A109, 032219 (2024)

2024
[18]

X. Liu, Y . Qiu, O. Dahlsten, and V . Vedral, Quantum causal inference with extremely light touch, npj Quantum Inf 11, 54 (2025)

2025
[19]

X. Liu, Z. Jia, Y . Qiu, F. Li, and O. Dahlsten, Unification of spatiotemporal quantum formalisms: mapping between process and pseudo-density matrices via multiple-time states, New J. Phys. 26, 033008 (2024)

2024
[20]

M. Song, V . Narasimhachar, B. Regula, T. J. Elliott, and M. Gu, Causal classification of spatiotemporal quantum correlations, 6 Phys. Rev. Lett. 133, 110202 (2024)

2024
[21]

Fullwood and A

J. Fullwood and A. J. Parzygnat, On quantum states over time, Proc. R. Soc. A. 478, 20220104 (2022)

2022
[22]

Z. Jia, M. Song, and D. Kaszlikowski, Quantum space-time marginal problem: global causal structure from local causal in- formation, New J. Phys. 25, 123038 (2023)

2023
[23]

Fullwood, Quantum dynamics as a pseudo-density matrix, arXiv:2304.03954 (2023)

J. Fullwood, Quantum dynamics as a pseudo-density matrix, arXiv:2304.03954 (2023)

2023 arXiv
[24]

Bressanini, F

G. Bressanini, F. Hanif, H. Kwon, and M. Kim, Quantum ob- servables over time for information recovery, arXiv:2412.11659 (2024)

2024 arXiv
[25]

S. H. Lie and N. H. Y . Ng, Quantum state over time is unique, Phys. Rev. Res. 6, 033144 (2024)

2024
[26]

H. Liu, X. Liu, Q. Chen, Y . Qiu, V . Vedral, X. Nie, O. Dahlsten, and D. Lu, Experimental demonstration of quantum causal inference via noninvasive measurements, arXiv:2411.06051 (2024)

2024 arXiv
[27]

H. Liu, X. Liu, Q. Chen, Y . Qiu, V . Vedral, X. Nie, O. Dahlsten, and D. Lu, Quantum causal inference via scattering circuits in NMR, arXiv:2411.06052 (2024)

2024 arXiv
[28]

Fullwood and A

J. Fullwood and A. J. Parzygnat, Operator representation of spa- tiotemporal quantum correlations, arXiv:2405.17555 (2024)

2024 arXiv
[29]

Jia and D

Z. Jia and D. Kaszlikowski, The spatiotemporal doubled- density operator: A unified framework for analyzing spatial and temporal quantum processes, Advanced Quantum Technologies 7, 2400102 (2024)

2024
[30]

Fullwood, General covariance for quantum states over time, Letters in Mathematical Physics 114, 1 (2024)

J. Fullwood, General covariance for quantum states over time, Letters in Mathematical Physics 114, 1 (2024)

2024
[31]

Fiur ´aˇsek, Structural physical approximations of unphysical maps and generalized quantum measurements, Phys

J. Fiur ´aˇsek, Structural physical approximations of unphysical maps and generalized quantum measurements, Phys. Rev. A66, 052315 (2002)

2002
[32]

J. K. Korbicz, M. L. Almeida, J. Bae, M. Lewenstein, and A. Ac ´ın, Structural approximations to positive maps and entanglement-breaking channels, Phys. Rev. A 78, 062105 (2008)

2008
[33]

Lim, Y .-S

H.-T. Lim, Y .-S. Kim, Y .-S. Ra, J. Bae, and Y .-H. Kim, Experi- mental realization of an approximate partial transpose for pho- tonic two-qubit systems, Phys. Rev. Lett. 107, 160401 (2011)

2011
[34]

Z. Liu, X. Zhang, Y .-Y . Fei, and Z. Cai, Virtual channel purifi- cation, arXiv:2402.07866 (2024)

2024 arXiv
[35]

F. Wei, Z. Liu, G. Liu, Z. Han, D.-L. Deng, and Z. Liu, Sim- ulating non-completely positive actions via exponentiation of hermitian-preserving maps, npj Quantum Information 10, 134 (2024)

2024
[36]

Kandala, K

A. Kandala, K. Temme, A. D. C ´orcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Error mitigation extends the com- putational reach of a noisy quantum processor, Nature567, 491 (2019)

2019
[37]

Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y . Li, J. R. McClean, and T. E. O’Brien, Quantum error mitiga- tion, Rev. Mod. Phys. 95, 045005 (2023)

2023
[38]

Yamamoto, S

K. Yamamoto, S. Endo, H. Hakoshima, Y . Matsuzaki, and Y . Tokunaga, Error-mitigated quantum metrology via virtual purification, Phys. Rev. Lett. 129, 250503 (2022)

2022
[39]

X. Yuan, B. Regula, R. Takagi, and M. Gu, Virtual quantum resource distillation, Phys. Rev. Lett. 132, 050203 (2024)

2024
[40]

Takagi, X

R. Takagi, X. Yuan, B. Regula, and M. Gu, Virtual quantum re- source distillation: General framework and applications, Phys. Rev. A 109, 022403 (2024)

2024
[41]

Zhang, Y

T. Zhang, Y . Zhang, L. Liu, X.-X. Fang, Q.-X. Zhang, X. Yuan, and H. Lu, Experimental virtual distillation of entanglement and coherence, Phys. Rev. Lett. 132, 180201 (2024)

2024
[42]

Yamamoto, Y

K. Yamamoto, Y . Matsuzaki, Y . Suzuki, Y . Tokunaga, and S. Endo, Virtual entanglement purification via noisy entangle- ment, arXiv:2411.10024 (2024)

2024 arXiv
[43]

C. Zhu, Y . Mo, Y .-A. Chen, and X. Wang, Reversing unknown quantum processes via virtual combs for channels with limited information, Phys. Rev. Lett. 133, 030801 (2024)

2024
[44]

Gui-Lu, General quantum interference principle and duality- computer, Commun

L. Gui-Lu, General quantum interference principle and duality- computer, Commun. Theor. Phys. 45, 825 (2006)

2006
[45]

A. M. Childs and N. Wiebe, Hamiltonian simulation using lin- ear combinations of unitary operations, Quantum Inf. Comput. 11, 901 (2012)

2012
[46]

H. Liu, Z. Liu, S. Chen, X. Nie, X. Liu, and D. Lu, Certifying quantum temporal correlation via randomized measurements: Theory and experiment, Phys. Rev. Lett. 134, 040201 (2025)

2025
[47]

L. M. K. Vandersypen and I. L. Chuang, Nmr techniques for quantum control and computation, Rev. Mod. Phys. 76, 1037 (2005)

2005
[48]

D. G. Cory, A. F. Fahmy, and T. F. Havel, Ensemble quantum computing by nmr spectroscopy, PNAS 94, 1634 (1997)

1997
[49]

See Supplemental Material at [url] for more details on theoreti- cal analysis of our method, and experimental procedures
[50]

Bruss, A

D. Bruss, A. Ekert, and C. Macchiavello, Optimal universal quantum cloning and state estimation, Phys. Rev. Lett.81, 2598 (1998)

1998
[51]

De Martini, V

F. De Martini, V . Buˇzek, F. Sciarrino, and C. Sias, Experimental realization of the quantum universal not gate, Nature 419, 815 (2002)

2002
[52]

C. Xi, X. Liu, H. Liu, K. Huang, X. Long, D. Ebler, X. Nie, O. Dahlsten, and D. Lu, Experimental validation of enhanced information capacity by quantum switch in accordance with thermodynamic laws, Phys. Rev. Lett. 133, 040401 (2024)

2024
[53]

Huang, C

K. Huang, C. Xi, X. Long, H. Liu, Y .-a. Fan, X. Wang, Y . Zheng, Y . Feng, X. Nie, and D. Lu, Experimental realiza- tion of self-contained quantum refrigeration, Phys. Rev. Lett. 132, 210403 (2024)

2024
[54]

C. A. Ryan, M. Laforest, J. C. Boileau, and R. Laflamme, Ex- perimental implementation of a discrete-time quantum random walk on an nmr quantum-information processor, Phys. Rev. A 72, 062317 (2005)

2005
[55]

Negrevergne, R

C. Negrevergne, R. Somma, G. Ortiz, E. Knill, and R. Laflamme, Liquid-state nmr simulations of quantum many- body problems, Phys. Rev. A 71, 032344 (2005)

2005
[56]

Knill, R

E. Knill, R. Laflamme, R. Martinez, and C. Negrevergne, Benchmarking quantum computers: The five-qubit error cor- recting code, Phys. Rev. Lett. 86, 5811 (2001)

2001
[57]

Khaneja, T

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr ¨uggen, and S. J. Glaser, Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172, 296 (2005)

2005
[58]

Micadei, J

K. Micadei, J. P. Peterson, A. M. Souza, R. S. Sarthour, I. S. Oliveira, G. T. Landi, T. B. Batalh˜ao, R. M. Serra, and E. Lutz, Reversing the direction of heat flow using quantum correlations, Nat. Commun. 10, 2456 (2019)

2019
[59]

Oliveira, R

I. Oliveira, R. Sarthour Jr, T. Bonagamba, E. Azevedo, and J. C. Freitas, NMR quantum information processing (Elsevier, 2011)

2011
[60]

X. Nie, X. Zhu, K. Huang, K. Tang, X. Long, Z. Lin, Y . Tian, C. Qiu, C. Xi, X. Yang, J. Li, Y . Dong, T. Xin, and D. Lu, Experimental realization of a quantum refrigerator driven by indefinite causal orders, Phys. Rev. Lett. 129, 100603 (2022)

2022
[61]

Long, W.-T

X. Long, W.-T. He, N.-N. Zhang, K. Tang, Z. Lin, H. Liu, X. Nie, G. Feng, J. Li, T. Xin, Q. Ai, and D. Lu, Entanglement- enhanced quantum metrology in colored noise by quantum zeno effect, Phys. Rev. Lett. 129, 070502 (2022)

2022
[62]

Y .-a. Fan, X. Li, S. Wei, Y . Li, X. Long, H. Liu, X. Nie, J. Ng, and D. Lu, Solving non-hermitian physics for optical manipu- lation on a quantum computer, Light Sci. Appl. 14, 132 (2025). 7

2025
[63]

Z. Lin, H. Liu, K. Tang, Y . Liu, L. Che, X. Long, X. Wang, Y .-a. Fan, K. Huang, X. Yang,et al., Hardware-efficient quan- tum principal component analysis for medical image recogni- tion, Front. Phys. 19, 51202 (2024)

2024
[64]

X. Nie, X. Zhu, Y .-a. Fan, X. Long, H. Liu, K. Huang, C. Xi, L. Che, Y . Zheng, Y . Feng, X. Yang, and D. Lu, Self-consistent determination of single-impurity anderson model using hybrid quantum-classical approach on a spin quantum simulator, Phys. Rev. Lett. 133, 140602 (2024)

2024
[65]

M. M. Wilde, From classical to quantum shannon theory, arXiv:1106.1445 (2011)

2011
[66]

Choi, Completely positive linear maps on complex ma- trices, Linear Alg

M.-D. Choi, Completely positive linear maps on complex ma- trices, Linear Alg. Appl. 10, 285 (1975)

1975
[67]

Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep

A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 275 (1972)

1972
[68]

Buˇzek and M

V . Buˇzek and M. Hillery, Quantum copying: Beyond the no- cloning theorem, Phys. Rev. A 54, 1844 (1996)

1996
[69]

Buscemi, M

F. Buscemi, M. Dall’Arno, M. Ozawa, and V . Vedral, Di- rect observation of any two-point quantum correlation function, arXiv:1312.4240 (2013)

2013 arXiv
[70]

Duan and G.-C

L.-M. Duan and G.-C. Guo, Probabilistic cloning and identifi- cation of linearly independent quantum states, Phys. Rev. Lett. 80, 4999 (1998)

1998
[71]

H. Chen, D. Lu, B. Chong, G. Qin, X. Zhou, X. Peng, and J. Du, Experimental demonstration of probabilistic quantum cloning, Phys. Rev. Lett. 106, 180404 (2011)

2011
[72]

Tang, Quantum principal component analysis only achieves an exponential speedup because of its state preparation assump- tions, Phys

E. Tang, Quantum principal component analysis only achieves an exponential speedup because of its state preparation assump- tions, Phys. Rev. Lett. 127, 060503 (2021)

2021
[73]

Cotler, S

J. Cotler, S. Choi, A. Lukin, H. Gharibyan, T. Grover, M. E. Tai, M. Rispoli, R. Schittko, P. M. Preiss, A. M. Kaufman, M. Greiner, H. Pichler, and P. Hayden, Quantum virtual cool- ing, Phys. Rev. X 9, 031013 (2019). 8 Supplementary Material for Experimental Virtual Quantum Br...

2019
[74]

Meanwhile, Qubits Q3 and Q4 are entangled into the state ∣Φ+⟩ using a combination of rotation operations Ry( π

followed by a gradient-field pulse Gz. Meanwhile, Qubits Q3 and Q4 are entangled into the state ∣Φ+⟩ using a combination of rotation operations Ry( π
[75]

Figures 8(b-d) illustrate the initialization of all initial states used in the experiments

and CNOT gates. Figures 8(b-d) illustrate the initialization of all initial states used in the experiments. For the initialization shown in Fig. 8(d), the mixed state ρ = p∣+⟩⟨+∣+ (1− p)I/2 is prepared using two single-qubit gates and a gradient-field pulse. The first single-q...
[76]

A quantum state can be decomposed in the complete Pauli bsis ∏n i=1⊗σi 0,x,y,z with the Pauli matrices σ0 = I, σx, σy, and σz

can be used to measure the observable σxIσ xσx by transferring it to σzIσ zσx. A quantum state can be decomposed in the complete Pauli bsis ∏n i=1⊗σi 0,x,y,z with the Pauli matrices σ0 = I, σx, σy, and σz. The expectation values of each Pauli basis can be measured in the exper...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [12]

A. W. Harrow, The church of the symmetric subspace, arXiv:1308.6595 (2013)

arXiv 2013

[2] [1]

Wehner, D

S. Wehner, D. Elkouss, and R. Hanson, Quantum internet: A vision for the road ahead, Science 362, eaam9288 (2018)

2018

[3] [2]

The experimental results align closely with the theoretical predictions, demon- strating the success and reliability of our circuit and the post- processing procedure. V alidation of optimality of the universal cloner B+.—We design and implement an experiment to validate the univer- sal cloner B+ as the closest CPTP map from a specially de- signed set of ...

work page

[4] [3]

The Choi state CB is constructed by d(p0CB+− p1CB−)= CB

Finally, the Choi state of B can be constructed as follows: set θ = π/2 and perform the pro- jective measurement {P0, P1} on the probe, which yields the collapsed states of the remaining three qubits, CB+ and CB−, with probabilities p0 and p1, respectively (see SM [47]). The Choi state CB is constructed by d(p0CB+− p1CB−)= CB. We experimentally construct ...

work page

[5] [4]

Caleffi, M

M. Caleffi, M. Amoretti, D. Ferrari, J. Illiano, A. Manzalini, and A. S. Cacciapuoti, Distributed quantum computing: a survey, Computer Networks 254, 110672 (2024)

work page 2024

[6] [5]

A. S. Cacciapuoti, M. Caleffi, F. Tafuri, F. S. Cataliotti, S. Gher- ardini, and G. Bianchi, Quantum internet: Networking chal- lenges in distributed quantum computing, IEEE Network 34, 137 (2019)

work page 2019

[7] [6]

Zhang and Q

Z. Zhang and Q. Zhuang, Distributed quantum sensing, Quan- tum Sci. Technol. 6, 043001 (2021)

work page 2021

[8] [7]

C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys. 89, 035002 (2017)

2017