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Cited by 61Quantum Information and CryptographyQuantum Computing Algorithms and ArchitectureQuantum Mechanics and Applications

Scalable on-chip quantum state tomography

Open access
Titchener, James G., Gräfe, Markus, Heilmann, René, Solntsev, Alexander S., Szameit, Alexander, Sukhorukov, Andrey A.·2018·npj Quantum Information
DOI 10.1038/s41534-018-0063-5
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Abstract

Quantum information systems are on a path to vastly exceed the complexity of any classical device. The number of entangled qubits in quantum devices is rapidly increasing, and the information required to fully describe these systems scales exponentially with qubit number. This scaling is the key benefit of quantum systems, however it also presents a severe challenge. To characterize such systems typically requires an exponentially long sequence of different measurements, becoming highly resource demanding for large numbers of qubits. Here we propose and demonstrate a novel and scalable method for characterizing quantum systems based on expanding a multi-photon state to larger dimensionality. We establish that the complexity of this new measurement technique only scales linearly with the number of qubits, while providing a tomographically complete set of data without a need for reconfigurability. We experimentally demonstrate an integrated photonic chip capable of measuring two- and three-photon quantum states with statistical reconstruction fidelity of 99.71%.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper proposes and experimentally demonstrates a scalable on-chip quantum state tomography method in which a fixed linear optical transformation expands an N-photon state to many output modes, making measurement complexity scale linearly with photon number. An integrated photonic chip reconstructs two-photon states with statistical reconstruction fidelity 99.71%, and simulations support three-photon GHZ tomography.

Key findings

  • A static linear optical circuit with M outputs can perform tomographically complete measurement of N-photon states; the required number of output ports scales linearly with N and approaches M = N for large photon numbers, unlike the exponential reconfiguration of standard tomography.
  • The laser-written chip measured two-photon anti-bunched and N00N states: anti-bunched fidelity was 94.0% (limited by the photon source’s 93% Hong–Ou–Mandel visibility), N00N-state fidelities exceeded 94% for phases φ ≈ 0 and φ ≈ π/2, and the statistical reconstruction fidelity was 99.71%.
  • The SPSA reconstruction algorithm achieved 99.67% average fidelity after 1500 iterations over 500 random density matrices and recovered the mixed state ρmix = (ρanti-bunched + ρN00N)/2.
  • Three-photon GHZ tomography was simulated through the measured transfer function with Gaussian noise at 5% of the peak correlation value; the recovered density matrix closely matched the GHZ state. The device condition number was ≃5, compared with ≃9 for typical tomography.

Methods

  • Theory: an N-photon density matrix is mapped by a fixed unitary from two input ports to M output waveguides; coincidences are measured with non-photon-number-resolving detectors. Reconstruction is possible when the number of measurable N-photon correlations, M!/(M − N)!, exceeds the 2^{2N} real parameters of the density matrix.
  • Experiment: photon pairs at 815 nm were generated by type-I spontaneous parametric down-conversion with 93% visibility and coupled into a laser-written fused-silica waveguide chip; output two-photon coincidence rates were approximately 30 Hz with about 20 s integration time.
  • Reconstruction: simultaneous perturbation stochastic approximation (SPSA) minimized the least-squares distance between measured and guessed correlations, with the density matrix parameterized as ρ(t) = T†(t)T(t)/Tr[T†(t)T(t)] and run for 1500 iterations.

Limitations

  • Stated in the paper: the number of photon detections required for reconstruction still scales exponentially, and processing measurement data would be extremely resource demanding for large photon numbers; SPSA is used to avoid direct inversion.
  • Stated in the paper: the anti-bunched-state fidelity of 94.0% is limited by the photon source quality, whose Hong–Ou–Mandel visibility is limited at 93%.
  • Evident from the text: the three-photon GHZ demonstration is simulated using the experimentally determined transfer function and added Gaussian noise, not an experimental three-photon measurement; the experimental demonstrations are two-photon only.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1038/s41534-018-0063-5

Cites

41 of 41 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

12 of 14 cited sources checked · 5 supported · 0 flagged · accuracy not yet measured

  • Interpretationreasoning

    Due to its simplicity and lack of tunable elements, this approach to quantum measurement is uniquely suited to integration with on-chip superconducting single-photon detectors for a fully on-chip tomography scheme.

    “Due to its simplicity and lack of tunable elements, this approach to quantum measurement is uniquely suited to integration with on-chip super-conducting single-photon detectors 37 for a fully on-chip tomography scheme.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [37] Najafi, F. et al. On-chip detection of non-classical light by scalable integration of single-photon detectors. Nat. Commun. 6, 5873 (2015). source
  • Findingcitation

    The paper experimentally demonstrates tomography of spatially entangled mixed or pure states of two indistinguishable photons using a specially designed on-chip laser-written waveguide circuit.

    “We experimentally demonstrate our approach by performing tomography of spatially entangled mixed or pure states of two indistinguishable photons, using a specially designed on-chip laser-written waveguide circuit.27”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [27] Meany, T. et al. Laser written circuits for quantum photonics. Laser Photon. Rev. 9, 363–384 (2015). source

      The paper’s section 6 describes process tomography of a polarization-encoded CNOT circuit, while section 4.2 discusses verifying W-state entanglement as an alternative to state tomography. Neither passage describes tomography of spatially entangled mixed or pure states of two indistinguishable photons. A search for that specific experiment in our copy found no supporting passage. (LLM, conf=0.76)

      Experimental — accuracy not yet measured
  • Premisecitation

    Even if all measurements can be completed, reconstructing the density matrix from measurement data becomes computationally challenging for high qubit-number states.

    “Finally, even if all the measurements can be completed, the task of reconstructing the density matrix from measurement data becomes computationally challenging for high qubit-number states.6”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [6] Banaszek, K., Cramer, M. & Gross, D. Focus on quantum tomography. New J. Phys. 15, 125020 (2013). source
  • Findingcitation

    With the appropriate optical transformation, multiplexed detection can allow full quantum state tomography.

    “Here we demonstrate experimentally that, with the appropriate optical transformation, multiplexed detection can also allow full quantum state tomography, and we present a comprehensive theoretical analysis of multi-photon case building on our original proposal for two-photon tomography with static measurement setups 25”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [25] Titchener, J. G., Solntsev, A. S. & Sukhorukov, A. A. Two-photon tomography using on-chip quantum walks. Opt. Lett. 41, 4079–4082 (2016). source
  • Interpretationcitation

    This provides a significantly simpler and more stable platform for three-photon tomography compared to previous realizations.

    “This provides a significantly simpler and more stable platform for three-photon tomography compared to previous realizations.35”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [35] Hamel, D. R. et al. Direct generation of three-photon polarization entanglement. Nat. Photonics 8, 801–807 (2014). source

      The introduction (p. 1) describes direct triplet production without outcome post-selection and says improved detection rates make three-photon tomography feasible where previous rates did not. The State Production and Characterization section (p. 2) reports completed tomography, but the paper does not establish that its platform is simpler or more stable than previous realizations; it identifies imperfect phase stabilization as a source of error. (LLM, conf=0.84)

      Experimental — accuracy not yet measured
    • Rests on an assumption: The experimentally determined transfer function Û accurately characterizes the chip's transformation for three-photon states.
  • May combine two claims. 20
  • Classified as probably not a substantial claim. 7

  • May combine two claims.

    Interpretationcitation

    The approach removes the need to build complex reconfigurable measurement systems, allowing full quantum tomography with just static linear optical circuits, which can easily be implemented on photonic chips.

    “In the context of photonic quantum states our approach also removes the need to build complex reconfigurable measurement systems, 8 instead allowing full quantum tomography with just static linear optical circuits, which can easily be implemented on photonic chips.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [8] James, D. F. V., Kwiat, P. G., Munro, W. J. & White, A. G. Measurement of qubits. Phys. Rev. A 64, 052312 (2001). source

  • May combine two claims.

    Interpretationcitation

    The equal spacing of vectors around the Bloch sphere gives the device maximum robustness to errors in the tomography procedure, by essentially realizing non-local measurements.

    “The equal spacing of vectors around the Bloch sphere gives the device maximum robustness to errors in the tomography procedure, 29 by essentially realizing non-local measurements.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [29] Foreman, M. R., Favaro, A. & Aiello, A. Optimal frames for polarization state reconstruction. Phys. Rev. Lett. 115, 263901 (2015). source

      The paper links optimal measurement frames on the Poincaré sphere to reduced error amplification in polarization reconstruction (pp. 1–3). It describes the Bloch sphere only as a geometric parallel that may inform two-level quantum tomography (p. 4); it does not say that the device achieves robustness by realizing non-local measurements. (LLM, conf=0.86)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Interpretationcitation

    The method offers the same computational advantage as self-guided tomography, but without the need for complex reconfigurable measurements that previously restricted this approach to only pure states.

    “Importantly, our method offers the same computational advantage as self-guided tomography, but without a need for complex reconfigurable measurements which so far restricted this approach to only pure states.18”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [18] Chapman, R. J., Ferrie, C. & Peruzzo, A. Experimental demonstration of self-guided quantum tomography. Phys. Rev. Lett. 117, 040402 (2016). source

      The paper says SGQT avoids computationally expensive postprocessing and optimizes projection measurements, including entangling measurements for the original algorithm. It also says its implementation estimates the pure state with greatest overlap and leaves demonstration on arbitrary mixed states to future work. It does not say that the need for complex reconfigurable measurements is what restricted SGQT to pure states. (LLM, conf=0.84)

      Experimental — accuracy not yet measured

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Tomography procedures involving some global quantum measurements have increased error robustness relative to using only local qubit measurements, and thus can be completed in less time.

    “tomography procedures involving some global quantum measurements have increased error robustness relative to using only local qubit measurements, and thus can be completed in less time.20”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [20] Miranowicz, A. et al. Optimal two-qubit tomography based on local and global measurements: Maximal robustness against errors as described by condition numbers. Phys. Rev. A 90, 062123 (2014). source

      Section VI (p. 8) says the approach combining local and global measurements is more robust against errors than a local-only approach. Section IV (p. 4) describes its condition-number-based robustness, but the paper does not establish that the procedure can therefore be completed in less time; searches for shorter measurement time and faster tomography did not find that link. (LLM, conf=0.88)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Premisecitation

    To characterize an N-qubit state, 2^(2N) different observables are measured, so the measurement apparatus must be reconfigured exponentially many times, which is impractical for large states.

    “To characterize an Nqubit state, 2 2N different observables are measured, 8 thus the measurement apparatus must be reconfigured exponentially many times, which is impractical for large states.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [8] James, D. F. V., Kwiat, P. G., Munro, W. J. & White, A. G. Measurement of qubits. Phys. Rev. A 64, 052312 (2001). source

  • May combine two claims.

    Findingcitation

    The complexity of the transformation only scales linearly with the number of photons in the state, in contrast to the exponential scaling in the number of measurements required in usual tomography.

    “Importantly, the complexity of the transformation only scales linearly with the number of photons in the state, in contrast to the exponential scaling in the number of measurements required in usual tomography.36”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [36] White, A. G., James, D. F. V., Eberhard, P. H. & Kwiat, P. G. Nonmaximally entangled states: Production, characterization, and utilization. Phys. Rev. Lett. 83, 3103–3107 (1999). source

      The tomography discussion (p. 2) says a full analysis requires d²−1 parameters and gives 3 and 15 analyzer settings for single- and two-photon states, supporting the contrast with the growing measurement demands of tomography. It does not describe the cited transformation or say that its complexity scales linearly with photon number. (LLM, conf=0.87)

      Experimental — accuracy not yet measured

  • May combine two claims.

    Findingcitation

    The experimentally realized device has a condition number of approximately 5, which is better than the condition number of approximately 9 for typical tomography.

    “The experimentally realized device has a condition number of ≃5, which is better than the condition number of ≃9 for typical tomography.20”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [20] Miranowicz, A. et al. Optimal two-qubit tomography based on local and global measurements: Maximal robustness against errors as described by condition numbers. Phys. Rev. A 90, 062123 (2014). source

      The Introduction (p. 1) says that a smaller condition number makes tomography more robust, supporting the direction of the comparison. But the cited paper does not establish a condition number of approximately 5 for the experimentally realized device or approximately 9 for typical tomography; its Experimental Setups section (p. 8) says an experimental realization of its proposed protocol will be presented elsewhere. (LLM, conf=0.78)

      Experimental — accuracy not yet measured

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data

    Typical two-photon coincidence rates after propagating through the chip were approximately 30 Hz, and an integration time of around 20 s was used for coincidence measurements.

    “The typical two-photon coincidence rates after propagating through the chip were approximately 30 Hz, and an integration time of around 20 s was used for coincidence measurements.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given

    The fabricated waveguides exhibit low propagation losses (<0.3 dB/cm) and almost vanishing birefringence (Δn_H,V ≈ 10^-6).

    “Such waveguides exhibit low propagation losses (<0.3 dB/cm) and almost vanishing birefringence (Δn H,V ≈ 10 -6 ).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The reconstructed N00N states have phases determined to be φ ≈ 0 and φ ≈ π/2.

    “the corresponding reconstructed N00N states, with phases determined to be ϕ ' 0 in Fig. 3b and ϕ ' π=2 in Fig. 3c.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingown data

    The real part of the density matrix was recovered using the simulated correlations from (b) for a three-photon GHZ state.

    “Real part of the density matrix that was recovered using the simulated correlations from (b)”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    The approach is based on interfering all the photons through a special unitary transformation, thus can be optimized to incorporate nonlocal measurement, allowing the error robustness to be increased.

    “Our approach is based on interfering all the photons through a special unitary transformation, thus can be optimized to incorporate nonlocal measurement, allowing the error robustness to be increased.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Study design and conductnone given

    To ensure that the calculated statistical fidelity accurately represents the noise in the system, the measured correlations were recorded with a moving average of 13 s over a total time of 60 s, and the average statistical fidelity was calculated for each 13 s window, then the mean value was taken.

    “To ensure that the calculated statistical fidelity accurately represents the noise in the system, the measured correlations were recorded with a moving average of 13 s over a total time of 60 s and the average statistical fidelity was calculated for each 13 s window, then the mean value was taken.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationnone given

    For reconstruction using experimentally measured data, there will naturally be errors in the amplitude of the measured correlation elements, in particular due to shot noise.

    “For reconstruction using experimentally measured data there will naturally be errors in the amplitude of the measured correlation elements, in particular due to shot noise.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation

    The measurement permits information about the spectral overlap of the pair of photons because the observed correlations exhibit a generalized form of Hong-Ou-Mandel interference.

    “this measurement permits us to get information about the spectral overlap of the pair of photons, since the observed correlations exhibit a generalized form of Hong-Ou-Mandel interference, [30][31][32]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [30][31][32]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationown data

    The 94.0% fidelity of the anti-bunched state is limited by the quality of the photon source, whose Hong-Ou-Mandel visibility is limited at 93%.

    “This value is actually limited by the quality of the photon source (which Hong-Ou-Mandel visibility is limited at 93%)”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Study design and conductown data

    The reconstruction accuracy was determined using the experimentally measured device transfer function.

    “We determine the reconstruction accuracy using the "experimentally measured" device transfer function.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    In the approach, the full complement of measurements are performed simultaneously, and thus the most important correlation detections for reconstructing a particular state naturally have the highest count rates.

    “This is because in our approach, the full complement of measurements are performed simultaneously, and thus the most important correlation detections for reconstructing a particular state naturally have the highest count rates.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    Because the N00N phase shift is double the value that would accumulate classically, it is highly sensitive to the environment.

    “because the shift is double the value that would accumulate classically 33 it is highly sensitive to the environment.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [33] Ou, Z. Y., Zou, X. Y., Wang, L. J. & Mandel, L. Experiment on nonclassical fourth-order interference. Phys. Rev. A 42, 2957–2965 (1990). source

  • May combine two claims.

    Interpretationreasoning

    The minimum number of output ports required scales linearly with the number of photons and approaches M = N for large photon numbers.

    “We establish that the minimum number of output ports required scales linearly with the number of photons and approaches M = N for large photon numbers, see red”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given

    The approach is compatible with computationally scalable reconstruction, avoiding resource intensive direct inversion.

    “Furthermore, we show that our approach is compatible with computationally scalable reconstruction, avoiding resource intensive direct inversion.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given

    If detectors can distinguish photons but cannot resolve photon numbers, then there are M!/(M-N)! different N-photon correlations which can be measured, while the density matrix is described by 2^(2N) real parameters.

    “If the detectors can distinguish photons (i.e., by arrival time for time-bin encoded states), but cannot resolve photon numbers, then there are M!/(M -N)! different N-photon correlations which can be measured, while the density matrix is described by 2 2N real parameters.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductreasoning

    The diagonal correlation elements were set to zero because photon number resolved detection was not used, and thus these elements were not measured.

    “For this normalization the diagonal correlation elements (corresponding to detection of two photons at the same detector) are set to zero since we do not use photon number resolved detection, and thus do not measure these elements.”
    Quote in the paperNot a substantial claim
23 of 23

  • The paper relies on this without support.

    Many of the expectation values measured in standard tomography are vanishingly small and thus contribute little useful information.

    “Furthermore many of the expectation values measured will be vanishingly small, and thus contribute little useful information.”

  • The paper relies on this without support.

    All existing approaches rely on a common measurement paradigm whereby different characteristics of a system are measured sequentially, thus they become exponentially complex to implement as the number of parameters in the density matrix scales exponentially with qubit number.

    “However all these approaches rely on a common measurement paradigm, whereby different characteristics of a system are measured sequentially, thus they become exponentially complex to implement as the number of parameters in the density matrix scales exponentially with qubit number.”

  • The paper relies on this without support.

    The number of photon detections required for reconstruction scales exponentially and processing measurement data would be extremely resource demanding for large photon numbers.

    “although the number of detectors and waveguides in the circuit scales linearly with the number of qubits to be measured, the number of photon detections required for reconstruction still necessarily scales exponentially (as in standard quantum state tomography) and processing of measurement data would be extremely resource demanding for large photon numbers.”

  • The paper relies on this without support.

    The experimentally determined transfer function Û accurately characterizes the chip's transformation for three-photon states.

    “We demonstrate this using the experimentally determined transfer function, Û, and simulate the propagation of a three-photon Greenberger-Horne-Zeilinger (GHZ) state (Fig. 5a) through the chip.”
4 of 4

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • N00N-state correlations are assigned to the wrong figure

    Minor

    The text locates the time-varying two-photon correlations in Fig. 2b,c, but the Fig. 3 caption identifies its panels b and c as the N00N-state results and says their second column contains the measured correlations.

1 of 1

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Authors
Titchener, J. G., Gräfe, M., Heilmann, R., Solntsev, A. S., Szameit, A., & Sukhorukov, A. A.
Affiliations
  • Australian National University (AU)
  • Fraunhofer Institute for Applied Optics and Precision Engineering (DE)
  • Friedrich Schiller University Jena (DE)
  • University of Technology Sydney (AU)
  • University of Rostock (DE)
Corresponding author
Titchener, James G.
ORCID
Venue
npj Quantum Information
Publisher
Springer Science and Business Media LLC
Published
2018-03-08
Volume
4
Issue
1
ISSN
2056-6387 (ISSN-L)
DOI
10.1038/s41534-018-0063-5
OpenAlex
W2560259994
Semantic Scholar
0a65005407e17a078aef0d98f65c9b574a30f5f7
Cited by
61
References
36
Citations by year
2026: 2 · 2025: 5 · 2024: 5 · 2023: 8 · 2022: 5 · 2021: 11 · 2020: 9 · 2019: 13 · 2018: 3
FWCI
5.07
Type
journal article
Language
en
Open access
Licence
Locations
  • npj Quantum Information (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
  • Figshare (submittedVersion, cc-by)
  • Publikationsdatenbank der Fraunhofer-Gesellschaft (Fraunhofer-Gesellschaft) (submittedVersion, other-oa)
  • ANU Open Research (Australian National University) (submittedVersion, cc-by)
  • Fraunhofer-Publica (Fraunhofer-Gesellschaft) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Alexander von Humboldt-Stiftung
  • Deutscher Akademischer Austauschdienst
  • Deutsche Forschungsgemeinschaft (SZ 276/12‐1, 574/13-1, BL574/13-1)
  • Australian Research Council (DP130100135, DE180100070, DP160100619, 2013 5659/002-001)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
Keywords
quantum state tomography, multi-photon states, integrated photonics, entangled qubits, two-photon states

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