Article Title]Observation of quantum effects on radiation reaction in strong fields

[1]E. E. Los

1, 2, 3]E. Gerstmayr

4]C. Arran

2]M. J. V. Streeter

1, 5]C. Colgan

4, 1]C. C. Cobo

1]B. Kettle

6]T. G. Blackburn

7]N. Bourgeois

2]L. Calvin

8]J. Carderelli

2]N. Cavanagh

7]S.J.D. Dann

9,10,11]A. Di Piazza

8]R. Fitzgarrald

12]A. Ilderton

11]C. H. Keitel

6]M. Marklund

13]P. McKenna

4]C. D. Murphy

1]Z. Najmudin

7,2]P. Parsons

7]P. P. Rajeev

7]D. R. Symes

11]M. Tamburini

8]A. G. R. Thomas

1, 14]J. C. Wood

15]M. Zepf

2]G. Sarri

4]C. P. Ridgers

1]S. P. D Mangles

*[1]The John Adams Institute for  Accelerator Science, Imperial College  London, Blackett Laboratory, London, SW72AZ, UK

[2]School of Mathematics and Physics, Queen's University Belfast, Belfast, BT7 1NN, UK

[3]Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA

[4]York Plasma Institute, University of York, School of Physics, Engineering and Technology, York, YO10 5DD, UK

[5]Tokamak Energy Ltd, Milton Park, OX14 4SD, UK

[6]Department of Physics, University of Gothenburg, Gothenburg, SE-41296, Sweden

[7]Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot, OX110QX, UK

[8]Gérard Mourou Center for  Ultrafast Optical Science, University of Michigan, Ann Arbor, MI 48109-2099, USA

[9]Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA

[10]Laboratory for Laser Energetics, University of Rochester, Rochester, New York, 14623, USA

[11]Max Planck Institute for Nuclear Physics, Saupfercheckweg 1, Heidelberg, 69117, Germany

[12]Higgs Centre, University of Edinburgh, Edinburgh, EH9 3FD, UK

[13]Department of Physics, SUPA, University of Strathclyde, Glasgow, G4 0NG, UK

[14]Deutsches Elektronen-Synchrotron DESY,  Hamburg, D-22607, Germany

[15]Helmholtz Institute Jena, Friedrich Schiller Universität Jena, Jena, 07743, Germany

§ MAIN

Radiation reaction describes the effective force experienced by an accelerated charge due to radiation emission. Quantum effects dominate charge dynamics and radiation production Fedotov_2023DiPiazza_2012 for charges accelerated by fields with strengths approaching the Schwinger field, $\mathbf{E_{sch}=}$ [detect-weight]1.3e18 Ritus_1985. Such fields exist in extreme astrophysical environments such as pulsar magnetospheres Timokhin_2010, may be accessed by high-power laser systems Willingale_2023,Doria_2020,Gan_2021, dense particle beams interacting with plasma Matheron_2023, crystals Wistisen_2018, and at the interaction point of next generation particle colliders Yakimenko_2019.

Classical radiation reaction theories do not limit the frequency of radiation emitted by accelerating charges and omit stochastic effects inherent in photon emission LL_1971_ClassicalTheoryFields, thus demanding a quantum
treatment. Two quantum radiation reaction models, the quantum-continuous Baier_1998 and quantum-stochastic Gonoskov_2022 models, correct the former issue, while only the quantum-stochastic model incorporates stochasticity Baier_1998. Such models are of fundamental importance, providing insight into the effect of the electron self-force on its dynamics in electromagnetic fields.

The difficulty of accessing conditions where quantum effects dominate inhibited previous efforts to observe quantum radiation reaction in charged particle dynamics with high significance.

We report the first direct, high significance ($>\bm{5\sigma}$) observation of strong-field radiation reaction on charged particles. Furthermore, we obtain strong evidence favouring the quantum radiation reaction models, which perform equivalently, over the classical model. Robust model comparison was facilitated by a novel Bayesian framework which inferred unknown collision parameters. This framework has widespread utility for experiments in which parameters governing lepton-laser collisions cannot be directly measured, including those utilising conventional accelerators.

In astrophysics, radiation reaction is predicted to limit electron-positron cascades which populate the magnetospheres of pulsars, magnetars and active black holes with plasma Timokhin_2010Philippov_2018 and can strongly affect reconnection in such plasmas Lyubarskii_1996,Petri_2012,Uzdensky_2014. Radiation reaction has been proposed as a dominant factor in gamma-burst generation Sultana_2013 and is expected to influence the dynamics of pair-plasmas Zhang_2020, including relativistic current sheets Jaroschek_2009.
Strong-field quantum radiation reaction may substantially affect the interaction point at high luminosity $>\SI[]{100}{\giga\electronvolt}$ class particle colliders Yakimenko_2019. Strong electromagnetic fields produced by multi-petawatt laser systems  Willingale_2023,Doria_2020,Gan_2021 will enable the exploration of compact particle acceleration Borghesi_2006Tajima_2020 and radiation generation Albert_2016 (e.g. via inelastic electron-photon scattering, termed Compton scattering Compton_1923) in higher-power regimes.

In strong-field environments, quantum radiation reaction is expected to dominate laser-solid target interactions Nakamura_2012Zhidkov_2002, ion-acceleration Capdessus_2015 and inverse Compton scattering, which has recently garnered considerable interest Salgado_2022Abramowicz_2021. The impact of radiation reaction on particle dynamics is characterised by the “dimensionless intensity parameter”, $a_0=\frac{E_Le}{\omega_L m_ec}$, and the “electron quantum parameter”, $\eta=E_{RF}/E_{sch}$, where $e$ and $m_e$ are the electron charge and mass respectively, $c$ is the speed of light in vacuum, $E_L$, $E_{RF}$ are the external electric field (laser) strengths in the laboratory and electron rest frames, respectively, $E_{sch}=\SI[]{1.3e18}{\volt\per\metre}$ is the Schwinger field and $\omega_L$ is the electric field frequency in the lab frame. When $a_0 \gtrsim 1$, both relativistic and multi-(laser) photon effects become important, while quantum effects become significant when $\eta>0.1$ Thomas_2012. The regime of strong classical radiation reaction is characterised by $\alpha a_0\eta\simeq 1$ and $\eta \ll 1$, where $\alpha$ is the fine structure constant DiPiazza_2012Blackburn_2020_b. Quantum effects dominate radiation reaction when $\alpha a_0\simeq 1$ and $\eta\gtrsim1$ DiPiazza_2012Blackburn_2020_b.

Previous all-optical experiments reported some evidence of radiation reaction in the strong-field classical-quantum transition regime, but to a much lower level of significance than the de facto $5\sigma$ standard due to data scarcity and large uncertainties Cole_2018Poder_2018. An alternative method, in which lepton beams were channelled through a crystal, measured only the emitted photon spectrum and was thus unable to directly measure the effect of radiation reaction on the lepton spectrum Wistisen_2018Nielsen_2020. Early inverse Compton scattering experiments Bula_1996 Bamber_1999 and observations of quantum effects in storage rings Sands_1960 were unable to access the field strengths required to observe strong-field quantum radiation reaction effects.
These challenges have prohibited the formation of an experimentally motivated consensus regarding the relative validity of different radiation reaction models.

In regimes dominated by classical radiation reaction, an electron emits many photons which each remove a small fraction of its energy Blackburn_2014; radiation emission is effectively continuous and its impact on electron motion is well-described by the classical Landau-Lifshitz equation LL_1971_ClassicalTheoryFields. In the strong-field quantum regime, interactions with the laser field must be treated non-perturbatively and are absorbed into electron basis states by quantizing the Dirac field in the presence of the laser field (Furry picture). Photon emission is described perturbatively with respect to these states Fedotov_2023DiPiazza_2012, and becomes stochastic, with single emissions
removing significant fractions of the electron energy DiPiazza_2012. Here, the “quantum-stochastic” model of radiation reaction invokes the locally constant field approximation (LCFA), which assumes emission events are point-like, depending only on local electric and magnetic fields, which are assumed to be constant over the timescale of photon emission Ritus_1985. Between emission events, electron motion is classical Duclous_2010,Arber_2015,Kirk_2009.

The “quantum-continuous” model, known as the semi-classical model in the literature, aims to incorporate quantum physics in a classical framework, treating radiation emission as continuous Ridgers_2017, but capturing the same rate of change of average electron momentum as the quantum-stochastic model via the inclusion of a correction term, the Gaunt factor Baier_1998.

All radiation reaction models predict net electron energy losses which increase with electron energy and $a_0$. However, quantum models prohibit the emission of photons with energies exceeding the electron energy, thus predicting lower energy losses than the classical model, which does not limit the frequency of emitted radiation. The quantum-stochastic model predicts spectral broadening, arising from probabilistic photon emission Neitz_2013,Ridgers_2017,Niel_2018. This is not captured by the classical and quantum-continuous models Neitz_2013,Ridgers_2017,Niel_2018.

Here we present the first direct observation of radiation reaction with $5\sigma$ significance in a strongly multi-photon regime where quantum effects are influential. We find strong evidence to favour the quantum-continuous and quantum-stochastic models over the classical model, while the quantum models perform comparably.

Notably, these findings clarify outstanding questions raised by previous experiments regarding the relative validity of different radiation reaction models Poder_2018Cole_2018.

The experimental setup is illustrated in figure <ref>a). Electron beams with mean energy $\approx$[separate-uncertainty]610(2) and shot-to-shot standard deviation [separate-uncertainty]40(1) (uncertainties calculated using bootstrapping due to non-normal distribution of data), generated using laser-driven wakefield acceleration Tajima_2020a, collided with a tightly focused, counter-propagating laser pulse with $I=\SI[separate-uncertainty]{1.0(2)e21}{\watt\per\centi\metre\squared}$, $\lambda_L=2\pi c/\omega_L=0.8\;\text{$\mu$m}$, $a_0= 21.4\pm1.8$ and $\langle\eta\rangle\leq0.13\pm0.02$ at the laser focus, sufficient to probe the quantum regime.

Experimental set-up and qualitative comparison of hits and nulls. a) One laser pulse, focused into a gas jet, drove a wakefield accelerator. A second, tightly focused, counter-propagating laser pulse collided with the electron beam which emitted gamma photons. The electron spectrometer consisted of a dipole magnet which dispersed the electron beam through a wire array onto two LANEX scintillating screens (green). A Caesium Iodide (CsI) profile screen and stack characterised the transverse profile and spectrum of the emitted gamma radiation, respectively. b) The simulated post-collision electron spectrum, normalised by integration, from a collision between a randomly selected null electron spectrum and a laser pulse with $a_0=11$ and gaussian transverse and longitudinal intensity profiles with full-width half-maxima (FWHM) []2.47 and []45, respectively. c) Measured hit and null electron spectra  (selected nulls occurred closest in time to corresponding hits), shown alongside corresponding gamma profile signals.

Figure <ref>b) illustrates two key signatures of radiation reaction, identified by simulating numerous collisions between an electron beam and a counter-propagating, focused laser pulse Los_2024. Compared to the pre-collision spectrum, the simulated post-collision spectrum has lower mean energy, $\langle E \rangle$, and a less pronounced peak above the $70^{th}$ percentile energy, $P_{70}$, where $P_{70}$ was calculated using electron spectra normalised by integration. The $70^{th}$ percentile energy is the energy under which $70\%$ of the electron beam charge lies. Changes in $P_{70}$ reflect the re-distribution of charge due to spectral broadening, if present. Spectral broadening may result from stochasticity, or if equally energetic electrons experience different laser intensities and hence energy losses due to the spatial and temporal dependence of laser intensity and electron beam energy.

In lieu of measured pre-collision electron spectra, misses and beam-off shots (nulls) were compared to successful collisions (hits). Section <ref> outlines identification of hits and nulls. Qualitatively, electron spectra in figure <ref>c) exhibit less pronounced high-energy peaks for hits than nulls, consistent with fewer electrons at higher energies. These peak heights are observed to decrease with increasing gamma yield. Some nulls have lower energies than their corresponding hits, illustrating the shot-to-shot variability in electron energy. We examined whether differences in $\langle E \rangle$ (figure <ref>a)) and $P_{70}$ (figure <ref>b)) between hits and nulls exceeded the shot-to-shot variation in electron beam energy and shape.

Some similarities between the hit and null distributions are expected due to the varying spatial and temporal alignment between the electron beam and the laser focus for hits. Misaligned collision shots (for which electron energy losses are smaller) are included, reducing the differences between hit and null populations.

[width=0.99, trim=0.25cm 0.2cm 0.2cm 6.15cm, clip]figures/peak_height_above_70th_percentile_mean_energy_hist_with_transverse_offset_shots_ymirror_20210620.png

(-2.2, 29)a)
(47.2, 29)b)

[width=1.0, trim=0.2cm 0.28cm 0.1cm 0.2cm, clip]figures/percentile_included_shots_20210620_with_transverse_offset_shots.png

Comparison of distributions of electron beam metrics and their scalings with normalised photon yield for hits and nulls. Distributions of a) mean energy and b) peak height above the $70^{th}$ percentile electron energy, $P_{70}$, for measured hits and nulls. $P_{70}$ indicates the prominence of the high energy peak in the spectrum. Plots of the complementary cumulative distribution functions of c) $\langle E \rangle$, and d) $P_{70}$. Average values of $\langle E \rangle$ and $P_{70}$, calculated for shots where the gamma yield normalised to $Q\langle\gamma\rangle^2$ exceeded a given threshold, are shown as functions of the corresponding percentile of the total population. The blue dashed line indicates average values for null shots. The number of hits remaining once the $50^{th}$, $90^{th}$ and $99^{th}$ percentile lowest yield shots are excluded (black, dashed) are indicated.

In figure <ref>a) a higher prevalence of electron spectra with lower $\langle E \rangle$ is evident for hits compared to nulls, while the reverse is true of electron spectra with higher $\langle E \rangle$. The hit distribution has mean energy $\SI[separate-uncertainty=true]{600(1)}{\mega\electronvolt}$, $5\sigma$ below the null distribution mean, $\SI[separate-uncertainty=true]{610(2)}{\mega\electronvolt}$,

indicating a significant reduction in $\langle E \rangle$ for hits. In figure <ref>b), $\langle P_{70}\rangle$ is $6\sigma$ lower for hits than nulls, which yield distribution means of $\langle P_{70}\rangle=\SI[separate-uncertainty]{2.28(2)}{\per\GeV}$ and $\langle P_{70}\rangle=\SI[separate-uncertainty]{2.47(2)}{\per\GeV}$, respectively. Two-sample Kolmogorov-Smirnov tests (selected for applicability to arbitrary distributions) applied to the distributions of $\langle E \rangle$ and $P_{70}$ returned p-values of $5\times10^{-7}$ and $4.1\times10^{-12}$, respectively, under the null hypotheses that hits and nulls originated from the same distribution. Thus, the null hypotheses for $\langle E \rangle$ and $P_{70}$ can be rejected at the $5\sigma$ and $7\sigma$ levels, respectively.

In the presence of radiation reaction, $\langle E \rangle$ and $P_{70}$ should decrease with increasing photon yield, once the pre-collision electron beam Lorentz factor, $\gamma$, and charge, $Q$, are corrected for. To see whether this relationship exists, shot selection was performed iteratively beginning with the full hit population. At each iteration, the threshold for hit selection (see section <ref>) was increased and the mean $\langle E \rangle$ and $\langle P_{70}\rangle$ were calculated. In figures <ref>c) and <ref>d), as the lowest yield shots are iteratively excluded, the mean $\langle E \rangle$ and $\langle P_{70}\rangle$ of the remaining hits decrease, consistent with the occurrence of radiation reaction.

The relationship between the percentile of shots included and mean $\langle E \rangle$ or $\langle P_{70}\rangle$ reflects the decreasing probability of colliding at a given laser intensity with increasing intensity. This results from the transverse and longitudinal jitter between the electron beam and the colliding laser and the diminishing laser waist as it focuses.

The differences between the $\langle E \rangle$ and $P_{70}$ hit and null distributions are consistent with electron energy loss, as is the inverse scaling between gamma yield and mean $\langle E \rangle$ or $\langle P_{70}\rangle$. Collectively, these findings constitute a highly significant observation of radiation reaction.

Various collision parameters, listed with their values in supplementary note <ref>, affect the laser intensity experienced by the electron beam and hence electron energy loss Los_2024. Accurate estimates of these parameters and their uncertainties are needed to perform model comparison.

To address this issue we developed a Bayesian inference framework Los_2024. This statistical technique utilises Bayes theorem (see equation <ref>) to iteratively update the posterior probability, $P(M|D)$, defined as the probability of a model, $M$, given data, $D$ Heard_2021.

\begin{equation}
\label{eqn:Bayes_theorem}
\end{equation}

The likelihood, $P(D|M)$, is the probability of observing the data given the model, the prior, $P(M)$, incorporates pre-inference fitting parameter information and $P(D)$, the probability of observing the data, is a constant.

In our implementation, radiation reaction models are parameterised by the electron beam duration, $z_d$, longitudinal displacement of the collision from the laser focus, $\tau_e$, and $a_0$, while the remaining parameters were fixed to avoid over-fitting and excessive computational run times Los_2024. Lack of on-shot fitting parameter measurements necessitated broad priors Los_2024. The inference procedure utilises one Markov chain Monte-Carlo (MCMC) Brooks_2011 per model to sample parameters from the prior, compute the corresponding posterior probabilities and converge upon the posterior optimum.

The ten highest gamma yield shots, normalised by $Q\langle\gamma\rangle^2$, were analysed to minimise the probability of transverse displacements between the electrons and the colliding laser, $r_d$, as the Bayesian framework assumed $r_d=0$. Constraints in computational resources limited the number of shots analysed (each inference required $\approx19200$CPU hours, 60GB per CPU). Measured and predicted post-collision electron and photon spectra are shown in figure <ref> for the highest normalised gamma yield shot. Each prediction arises from one set of MCMC parameters and has an associated posterior probability. A neural network predicted the distribution of pre-collision electron spectra given the measured laser wakefield driver energy, plasma density and longitudinal profile of plasma re-combination light Streeter_2023. The accuracy of the neural network motivated its use over summary statistics such as the mean null electron spectrum.
Additional measured and inferred post-collision electron spectra and photon spectrometer responses are provided in supplementary note <ref>.

[width=1.0, trim=0.25cm, 0.0cm, 0.19cm, 0.23cm,clip]figures/electrons_mcmc_no_chirp_err_inference_all_models_indv_spec_1.png

(74.5, 20.5)Quantum-stochastic

[width=1.0, trim=0.2cm, 0.2cm, 0.16cm, 0.22cm,clip]figures/gspec_inferred_vs_measured_CsI_response_normed_no_chirp_err_inference_all_models_indv_spec_1.png

Bayesian inference results for the highest gamma-yield shot normalised to $Q\langle\gamma\rangle^2$ (shot 2). Measured data (red) and predictions for the classical (green), quantum-continuous (blue), quantum-stochastic (magenta) models, which inferred $\langle \tilde{a}_{0}\rangle=6.2\pm1.0$ and $\sigma_{a_0}=1.2\pm0.3$, $\langle \tilde{a}_{0}\rangle=6.8\pm0.9$ and $\sigma_{a_0}=0.4\pm0.1$ and $\langle \tilde{a}_{0}\rangle=6.7\pm0.9$ and $\sigma_{a_0}=0.4\pm0.1$, respectively.
a) Measured and inferred post-collision electron spectra. For the former, $\langle E\rangle=\SI[separate-uncertainty]{564.1(103)}{\mega\electronvolt}$, $P_{70}=0.78\pm 0.01$. The distribution of pre-collision electron spectra predicted by the neural network (orange), for which $\langle E\rangle=\SI[separate-uncertainty]{574.8(107)}{\mega\electronvolt}$, $P_{70}=1.44\pm 0.01$, and its median (black).
b) Measured and inferred photon energy deposition as a function of propagation distance in the CsI photon spectrometer. The mean photon energy measured was [separate-uncertainty]63.3(58).

[width=1.0, trim=0.25cm, 0.28cm, 0.2cm, 0.2cm,clip]figures/likelihood_ratio_split_espec_gspec_total_cl_sc_cf_st_new_format.png

model comparison.

Bayes factors for individual shots (orange) and combined over ten shots (cyan) are shown. Regions of weak, substantial and strong evidence favouring a given model follow the half-log scale convention outlined by Kass and Raftery Kass_1995. The dashed grey line indicates equal performance of compared models.

The gamma and electron spectra are equally important for model fitting; both determine the posterior optimum location in parameter space. The novel ability to analyse these diagnostics within one self-consistent framework is a key strength of this Bayesian approach.

In figure <ref>a), the $\approx\SI[]{0.9}{\giga\electronvolt}$ peak in the predicted pre-collision spectrum is not evident post-collision. While none of the models fully reproduce the post-collision electron spectrum, likely due to experimental collision parameters differing from their fixed values in the models, all three models accurately reproduce this peak, inferring $\langle E\rangle$ and $P_{70}$ within one standard error of the measured spectrum. This also suggests the models predict statistically indistinguishable post-collision electron spectra.

The gamma spectra, having fewer degrees of freedom than the electron spectra, produces less accurate inferences, as indicated by the inferred mean photon energies in figure <ref>b). However, the gamma spectra do not have substantially higher fractional uncertainties than the electron spectra. Consequently, the photon spectrum has a narrower posterior than the electron spectrum, and thus dominates the shape of the combined posterior. Model selection is performed using Bayes factors (see <ref> in section <ref>) which integrate the posterior over the parameter space. As the gamma spectrum determines the posterior shape, its contribution to the Bayes factor and model selection dominates that of the electron spectrum. The quantum models predict more accurate photon spectra with lower uncertainties than the classical model, as indicated by the mean photon energy uncertainties in figure <ref>b). This determines their relative performance.

In figure <ref>, individual shots yield weak (Bayes factor $<3.2$ Kass_1995), but consistent evidence favouring the quantum models over the classical model, increasing the credibility of the former. The Bayes factors for the quantum-continuous and quantum-stochastic models lie consistently around 1, signifying their comparable performance. The combined Bayes factors over 10 shots indicate strong evidence favouring the quantum-stochastic and quantum-continuous models over the classical model, but insufficient evidence to distinguish between the quantum models.

The maximum and minimum first moments of the collision distributions of $\eta$, $\langle\eta\rangle$, and effective $a_0$, $\langle \tilde{a}_0\rangle$, a function of all three inference parameters, inferred by the quantum-stochastic model across ten shots were $0.05\leq\langle\eta\rangle\leq0.1$ and $7\leq\langle \tilde{a}_0\rangle\leq13$, respectively. $\langle \tilde{a}_0\rangle$ and $\langle\eta\rangle$ are effective parameters, representing the collision conditions through their dependence on $z_d$ and $\tau_e$. Since $7\leq\langle \tilde{a}_0\rangle$, the maximum possible transverse offset, relative to the laser waist at focus, $w_0$, was $r_d\leq0.64w_0$, well below the threshold of $r_d=1.6w_0$ at which model differentiation is no longer accurate Los_2024, indicating accurate model selection is feasible.

In conclusion, we report the first direct, highly significant observation of radiation reaction and find strong evidence to favour two quantum models over a classical model. The onset of quantum effects and reduced accuracy of the Landau-Liftshiftz model observed for $0.05\leq\langle\eta\rangle\leq0.1$ and $7\leq\langle \tilde{a}_0\rangle\leq13$ motivate the use of quantum-corrected models in this regime. While spectral broadening was observed, insufficient knowledge of collision parameters and large uncertainties on predicted pre-collision electron spectra inhibited our ability to determine whether this arose due to stochasticity. Model differentiability would improve with stable, mono-energetic electron beams and strongly constraining priors motivated by collision parameter measurements. Lower variation in transverse alignment or in collisions with an expanded, higher-power laser would boost statistics at higher $a_0$, facilitating investigations of quantum phenomena over a greater range of $\eta$ and providing new insight into the validity of the quantum-continuous and quantum stochastic models in these regimes.

§ METHODS

§.§ Experimental Methods

The experiment was conducted using the dual-beam Gemini laser at the Central Laser Facility, Rutherford Appleton Laboratory, UK. An f/40 off-axis parabola was used to focus one arm of the linearly polarised, two-beam system to a transverse full-width half-maximum (FWHM) of the focal spot intensity of [separate-uncertainty]35(3)$\times$[separate-uncertainty]40.6(12). An off-shot Grenouille measurement of the FWHM duration of the laser intensity yielded [separate-uncertainty]59.5(25). The laser delivered [separate-uncertainty]6.8(06) to target, corresponding to an $a_0=1.0\pm0.15$ (standard deviation given).

The laser-wakefield drive beam was focused into a []15 supersonic gas jet with a trapezoidal density profile and []5 ramps. An average electron density of $\approx\SI[separate-uncertainty]{1.1(2)e18}{\per\centi\metre\cubed}$ was attained at the peak of the trapezoidal profile with He gas, doped with $1\%$ $N_2$ to induce ionisation injection McGuffey_2010.

The electron beam, and photons produced by inverse Compton scattering (ICS), propagated through the hole in the f/2 parabola, which had an acceptance angle of []42. A $\int B(x)dx$=[]0.4 dipole magnet was used to disperse the electron beam through a wire array onto two sequential LANEX screens which were imaged by two cooled 16-bit cameras. The electron spectrum was subsequently retrieved with the aid of a tracking algorithm which computed the trajectories of electrons through the magnetic field. The wire array and two screens allowed the degeneracy between the electron beam energy and pointing into the magnet to be de-convolved Clayton_2010Wang_2013Soloviev_2011. The uncertainty in the retrieved electron energy was $\zeta_{e}[\SI[]{}{\mega\electronvolt}]=32.45\times10^{-6}E^2[\SI[]{}{\mega\electronvolt}]$, where $E$ denotes electron energy.

The wakefield accelerator produced [separate-uncertainty]140(1) electron beams with mean and standard deviation energy [separate-uncertainty]610(2) and [separate-uncertainty]206(1), respectively (standard error given). Electrons with energies $<\SI{300}{\mega\electronvolt}$ could not be measured.
A radial source size of $<\SI[separate-uncertainty]{0.7(1)}{\micro\metre}$ was assumed, in line with previous measurements Kneip_2011Schnell_2012. The FWHM energy-dependent electron beam divergence, $\theta_D$, measured along the axis transverse to the dispersion plane by the LANEX screens which measured the electron spectrum, was $(b_1-b_2\sqrt{\gamma m_e[\SI[]{}{\giga\electronvolt}]})$, where $b_1=1.30^{+0.26}_{-0.19}\SI[]{}{\milli\radian}$, $b_2={0.26^{+0.24}_{-0.28}}\SI{}{\milli\radian\per\giga\electronvolt\tothe{1/2}}$. The axial symmetry of the electron beam divergence was confirmed using linear Thomson scattering Gerstmayr_2020.

The colliding laser pulse was focused at the rear of the gas jet by an f/2 parabola with a []25.4 on-axis hole. The laser intensity profile had transverse FWHM [separate-uncertainty]2.5(2)$\times$[separate-uncertainty]2.1(1) and FWHM duration [separate-uncertainty]45.0(25). Due to energy losses in the laser system, including the on-axis hole in the f/2 parabola, the energy on-target was [separate-uncertainty]6.13(2), yielding a peak $a_0=21.4\pm1.8$.

§.§ Spatial and Temporal Overlap of Laser pulses

A micron knife-edge []90 prism, imaged using a $\times 10$ microscope objective, was used to overlap the two laser pulses spatially and temporally, where the latter was achieved using spatial interferometry, as demonstrated previously Cole_2018Corvan_2016. By optimising the contrast of the interference pattern, the two laser pulses were synchronised to within $\pm$[]10. To time the colliding pulse, two additional effects needed to be corrected for, namely the reduced non-linear group velocity of the wakefield-driver laser pulse in the plasma and the longitudinal displacement of the electron beam from the wakefield driver by $N$ plasma wavelengths, where $N=\frac{1}{2}$ for an electron beam travelling at the dephasing limit with velocity close to $c$. Thus, the longitudinal collision position was shifted closer to the gas jet by $\delta_z$,

\begin{equation}
\delta_z=\frac{3d}{4}\frac{n_e}{n_{c}}+N\frac{\lambda_L}{2}\sqrt{\frac{n_{c}}{n_e}}
\end{equation}

where $n_e$ and $n_{c}=\frac{\epsilon_0 m_e\omega_L^2}{e^2}$ are the plasma and critical densities and $d$ is the distance from the upstream edge of the plasma to the injection point.

Once timed and aligned, references for optimal timing and alignment were taken using a spectral interferometer in the laser area and a diagnostic of the f/40 beam pointing, respectively. As fluctuations in temperature altered the temporal and spatial alignment between the two beams over the course of shooting, long-term drifts in spatial and temporal alignment could be corrected for by adjusting the tip and tilt of a mirror in the f/40 beamline, and by altering the path length of one of the laser arms, respectively.

This was implemented using an automated feedback loop. Thus, the remaining misalignment between the electron beam and the colliding laser resulted from shot-to-shot variation in the beam paths and the laser pointing due to vibrations. The temporal jitter between the two laser arms was assumed to be normally distributed with standard deviation $\pm$[]30. An additional source of uncertainty in the timing between the electron beam and colliding laser stems from the unknown value of $d$, which we assume to be uniformly distributed with lower and upper bounds []0 and []10, respectively. To correct for $\delta_z$, the path between the two beams was reduced by []20. Thus, the offset in timing between the two beams, $\Delta t\approx\frac{\delta_z}{c}$, $\SI[]{2.7}{\femto\second}\leq \Delta t\leq\SI[]{45.8}{\femto\second}$.
The standard deviations of the radial positions of the colliding laser and the electron beam due to pointing variations were measured to be [separate-uncertainty]0.53(26) and [separate-uncertainty]17.5(5) respectively, which correspond to a standard deviation in transverse alignment of [separate-uncertainty]17.5(6).

§.§ Gamma Radiation Diagnostics

The angular distribution of gamma radiation was measured using a [parse-numbers=false]50×50×10 profile screen consisting of [parse-numbers=false]1 ×1 ×10 CsI(Tl) crystals separated by []0.2 titanium oxide spacers. The front of the profile screen was coated with []0.5 titanium oxide. The profile screen was placed outside the vacuum chamber, [separate-uncertainty]2244(4) from the interaction, and was imaged using a cooled 16-bit CCD camera.

The energy deposition of gamma photons was measured using a [parse-numbers=false]50 ×50 ×150 dual-axis CsI(Tl) scintillator, comprised of alternating layers of horizontally and vertically oriented [parse-numbers=false]5 ×5 ×50 CsI(Tl) crystals which were held in place by a 3D printed []1 nylon frame and separated by []1 rubber spacers to prevent light leakage between crystals. Two cooled 16-bit CCD cameras imaged the scintillation light from above and laterally. The calorimeter was placed outside the vacuum chamber, [separate-uncertainty]3570(3) from the interaction.

Geant4 Allison_2016Agostinelli_2003 simulations were used to obtain the energy deposition in the calorimeter as a function of incident photon energy as demonstrated in Behm  et. al. Behm_2018. These simulations included the chamber geometry, large objects inside the chamber such as the dipole magnets and all materials placed in the beam path, including a []1 alumina laser block, a []25 Kapton window with a []375 Kevlar backing sheet and a []25 aluminium foil. Variations in the scintillation efficiency of the crystals and in the efficiency of the imaging system were characterised and subsequently compensated for by comparing the measured and simulated response of the calorimeter to bremsstrahlung generated by an electron beam propagating through a [separate-uncertainty]1.5(1) PTFE target with radiation length much less than the radiation length of a $\SI[]{1}{\giga\electronvolt}$ electron beam.

The ICS spectrum, $S_{ICS}$, has the characteristic shape

\begin{equation}
\label{eqn:gamma_spec}
S_{ICS}=A\left(\frac{E_\gamma}{E_{\gamma c}}\right)^{-\frac{2}{3}}e^{-\left(\frac{E_\gamma}{E_{\gamma c}}\right)}
\end{equation}

where $A$ and $E_{\gamma c}$ represent photon number and the critical energy of the spectrum, respectively. Bayesian inference was used to obtain the values of $A$ and $E_{\gamma c}$ for which the energy deposition calculated using equation <ref> fitted the measured energy deposition.

§.§ Shot Selection

Following the approach employed by Cole et. al. Cole_2018, the photon yield measured by the gamma profile diagnostic was used to identify hits and nulls. The total yield measured by the profile screen, $Y_{\gamma}$, is expected to scale with the total charge, $Q$ and expected value of $\gamma^2$, $\langle\gamma^2\rangle$ of the electron beam:

\begin{equation}
Y_{\gamma}=(C_{\rm{ICS}}a_0^2 +C_{\rm{BKG}}) Q\langle\gamma^2\rangle
\end{equation}

where the first and second terms describe the contributions of inverse Compton scattering (ICS) and background (e.g. due to bremsstrahlung) to the total yield, respectively, and $C_{\rm{ICS}}$, $C_{\rm{BKG}}$, are scaling constants. Multiple sets of shots in which the counter-propagating laser was not fired were taken to obtain the characteristic background scaling with $Q\langle\gamma^2\rangle$. Misses and hits, classified as shots which produced yields within $1\sigma$ and above $3\sigma$ of the background scaling respectively, are shown in Extended Material <ref>.

§.§ Analytical Methods

The Bayes factor, $P_X/P_Y$, for models $X$ and $Y$, used to perform model comparison, is defined as the integral over the marginalised posterior/likelihood

\begin{equation}
\label{eqn:bic}
\frac{P_X}{P_Y}=\frac{\int p(\phi_X|M_X)p(D|\phi_X, M_X)d\phi_X}{\int p(\phi_Y|M_Y)p(D|\phi_Y, M_Y)d\phi_Y}
%\rm{BIC}=\textit{k}\ln(\textit{n})-2\ln\left(\prod_\textit{i}^{\textit{n}} \hat{\textit{L}}_{i} \right) %\hat{L}_{i} \right)
\end{equation}

where $\phi_X$, $\phi_Y$ are the parameter vectors which characterise models $M_X$ and $M_Y$, respectively. The integrals in equation <ref> do not have analytic solutions and are challenging to compute numerically due to the complex shape of the posterior distribution. Therefore, the Bayes factor was approximated using leave-one-out cross-validation with Pareto-smoothed importance sampling (LOO-PSIS) Vehtari_2017 available from the python package arviz arviz_2019.
As the posterior probabilities for each shot are independent, their product yields the total Bayes factor. A full, detailed account of the Bayesian inference framework and the forward models used therein is provided in Los_2024.

§ SUPPLEMENTARY NOTE

§.§ Free and Fixed Parameter Selection, Bayesian Inference Test Cases

A number of parameters (including electron beam source size and chirp, laser duration, transverse offset, etc) are not fitted by the Bayesian procedure and have therefore been assigned fixed values in the forward model. Several factors underpin the decision to fix these parameters. These are as follows:

* The expected effect of variation. This incorporates both the probability of parameter variation by a given amount and the impact of this variation on the post-collision observables, namely the electron and gamma spectra. Parameters fixed due to the small expected effect of their variation include the laser duration and focal spot size and the electron beam chirp.

* Shot selection. The ten shots which produced the highest CsI profile screen yields, normalised to $Q\langle\gamma\rangle^2$, were analysed using the Bayesian framework. As the laser intensity decreases most steeply with transverse (as opposed to longitudinal) misalignment, by analysing the highest yield shots the probability that the transverse offset is large is reduced. As the Bayesian analysis inferred values of $7\leq\langle \tilde{a}_0\rangle\leq13$, the highest transverse offset possible for these shots was consistently within one spot size.

* Degeneracy. If changes in two (or more) collision parameters engender similar alterations in the post-collision observables, it is possible to fix one of these parameters and vary the second to reproduce the effect of the fixed parameter. For example, if the electron beam has finite divergence, varying the longitudinal position of the collision alters the size of the electron beam at the collision. This produces post-collision observables similar to those obtained by varying the electron beam source size. Degeneracy allows changes in the electron beam source size, divergence and transverse offset from the laser focus to be compensated for by free parameters (laser energy, longitudinal offset of the collision from focus, electron beam duration).
The laser energy was chosen as a free parameter to enable the Bayesian inference procedure to tackle shot-to-shot variations therein.

The laser, electron and collision parameters which were measured, estimated or inferred based on previous measurements, are summarised alongside their assigned values in the forward models in tables  <ref>, <ref> and <ref>, respectively.

Laser parameters    Experiment     Value in forward model

Energy on target ([])       $6.13\pm 0.02$     Free parameter

FWHM transverse waist ([])     $\SI[separate-uncertainty]{2.5(2)}{}\times\SI[separate-uncertainty]{2.1(1)}{}$     2.47

FWHM duration ([])     $45\pm3$     45

Measured laser parameters.

Electron beam property     Experiment   Value in forward model

Duration (standard deviation) ([])     [separate-uncertainty]14(14)     Free parameter

Transverse source size    [separate-uncertainty]0.68(13)     0.68

(standard deviation) ([])

Distance from end of gas jet to    0.0   0.0

electron beam initial position ([])

Total electron charge ([])    [separate-uncertainty]140(1)    Normalised

FWHM divergence ([])     $(b_1-b_2\sqrt{\gamma m_e[\SI[]{}{\giga\electronvolt}]})$    $(b_1-b_2\sqrt{\gamma m_e[\SI[]{}{\giga\electronvolt}]})$

Measured or estimated electron beam parameters. The electron beam source size has been estimated from previous measurements Schnell_2012, while the electron beam duration was obtained from particle-in-cell simulations using the code FBPIC Lehe_2016. The constants $b_1=1.30^{+0.26}_{-0.19}~\SI[]{}{\milli\radian}$, $b_2={0.26^{+0.24}_{-0.28}}~\SI{}{\milli\radian\per\giga\electronvolt\tothe{1/2}}$.

Collision parameters    Experiment     Value in forward model

Transverse displacement of     $\pm17.5$     0.0

collision from focus ([])

Temporal displacement of     $\pm N(0, 30)$     Free parameter

collision from focus ([])     $+ U(3, 46)$

The expected transverse and temporal alignment of the electron beam and the colliding laser and the expected shot-to-shot jitter in the above parameters. $U$ and $N$ denote uniform and normal distributions, respectively.

§ SUPPLEMENTARY NOTE

§.§ Bayesian Inference Additional Results

Results of the Bayesian inference procedure applied to nine shots, ordered by the corresponding normalised gamma profile signal with the brightest shots first.
In figure <ref>, the low energy peak in the measured post-collision (red) electron spectrum is not present in the pre-collision spectrum (orange, median shown in black) predicted by the neural network, likely due to a deficiency of spectra with this feature in the training data.

[width=0.99, trim=0.1cm, 0.28cm, 0.25cm, 0.26cm,clip]figures/electrons_mcmc_no_chirp_err_inference_all_models_indv_spec_4.png
(74.5, 20.5)Quantum-stochastic

[width=0.99, trim=0.1cm, 0.25cm, 0.21cm, 0.2cm,clip]figures/gspec_inferred_vs_measured_CsI_response_normed_no_chirp_err_inference_all_models_indv_spec_4.png

Bayesian inference results for shot 5.  Similar to figure <ref>. The distribution of $\tilde{a}_{0}$ inferred by the classical, quantum-continuous and quantum-stochastic models had mean and standard deviation, $\langle \tilde{a}_{0}\rangle=7.5\pm1.2$ and $\sigma_{a_0}=1.9\pm0.5$, $\langle \tilde{a}_{0}\rangle=8.9\pm1.2$ and $\sigma_{a_0}=2.8\pm0.6$ and $\langle \tilde{a}_{0}\rangle=9.4\pm1.4$ and $\sigma_{a_0}=0.9\pm0.2$, respectively. a) The measured post-collision electron spectra, with mean $\langle E\rangle=\SI[separate-uncertainty]{548(10)}{\mega\electronvolt}$ and $\langle P_{70} \rangle=0.88\pm 0.01$ and pre-collision spectra predicted by the neural network, for which mean $\langle E\rangle=\SI[separate-uncertainty]{593(11)}{\mega\electronvolt}$ and $\langle P_{70} \rangle=1.66\pm 0.02$, are shown alongside the predicted post-collision electron spectra. b) The measured post-collision photon spectrum, with mean energy $\SI[separate-uncertainty]{62.4(66)}{\mega\electronvolt}$ and predicted photon spectra.

[width=0.99, trim=0.1cm, 0.20cm, 0.20cm, 0.22cm,clip]figures/electrons_mcmc_no_chirp_err_inference_all_models_indv_spec_3.png
(74.5, 20.5)Quantum-stochastic

[width=0.99, trim=0.1cm, 0.25cm, 0.20cm, 0.2cm,clip]figures/gspec_inferred_vs_measured_CsI_response_normed_no_chirp_err_inference_all_models_indv_spec_3.png

Bayesian inference results for shot 4. Similar to figure <ref>. The distribution of $\tilde{a}_{0}$ inferred by the classical, quantum-continuous and quantum-stochastic models had mean and standard deviation, $\langle \tilde{a}_{0}\rangle=6.2\pm1.1$ and $\sigma_{a_0}=1.2\pm0.3$, $\langle \tilde{a}_{0}\rangle=6.7\pm0.8$ and $\sigma_{a_0}=0.4\pm0.1$ and $\langle \tilde{a}_{0}\rangle=6.7\pm0.9$ and $\sigma_{a_0}=0.5\pm0.1$, respectively. a) The measured post-collision electron spectrum, with $\langle E \rangle=\SI[separate-uncertainty]{546(10)}{\mega\electronvolt}$ and $P_{70}=0.95\pm 0.01$. The neural network predictions for the pre-collision electron spectrum have $\langle E \rangle=\SI[separate-uncertainty]{580(11)}{\mega\electronvolt}$ and $P_{70}=1.54\pm 0.01$. b) The measured photon spectrum, with mean energy [separate-uncertainty]72(23).

[width=0.99, trim=0.1cm, 0.20cm, 0.20cm, 0.22cm,clip]figures/electrons_mcmc_no_chirp_err_inference_all_models_indv_spec_7.png
(74.5, 20.5)Quantum-stochastic

[width=0.99, trim=0.1cm, 0.25cm, 0.20cm, 0.2cm,clip]figures/gspec_inferred_vs_measured_CsI_response_normed_no_chirp_err_inference_all_models_indv_spec_7.png

Bayesian inference results for shot 8. Similar to figure <ref>. The distribution of $\tilde{a}_{0}$ inferred by the classical, quantum-continuous and quantum-stochastic models had mean and standard deviation, $\langle \tilde{a}_{0}\rangle=6.3\pm1.0$ and $\sigma_{a_0}=0.3\pm0.1$, $\langle \tilde{a}_{0}\rangle=7.7\pm1.1$ and $\sigma_{a_0}=0.6\pm0.2$ and $\langle \tilde{a}_{0}\rangle=7.7\pm1.0$ and $\sigma_{a_0}=0.5\pm0.1$, respectively.
a) The measured post-collision electron spectrum, with $\langle E \rangle=\SI[separate-uncertainty]{557(10)}{\mega\electronvolt}$ and $P_{70}=0.95\pm 0.01$. The neural network predictions for the pre-collision electron spectrum have $\langle E \rangle=\SI[separate-uncertainty]{580(11)}{\mega\electronvolt}$ and $P_{70}=1.51\pm 0.01$. b) The measured photon spectrum, with mean energy [separate-uncertainty]67(6).

[width=0.99, trim=0.1cm, 0.20cm, 0.20cm, 0.22cm,clip]figures/electrons_mcmc_no_chirp_err_inference_all_models_indv_spec_8.png
(74.5, 20.5)Quantum-stochastic

[width=0.99, trim=0.1cm, 0.25cm, 0.20cm, 0.2cm,clip]figures/gspec_inferred_vs_measured_CsI_response_normed_no_chirp_err_inference_all_models_indv_spec_8.png

Bayesian inference results for shot 9. Similar to figure <ref>. The distribution of $\tilde{a}_{0}$ inferred by the classical, quantum-continuous and quantum-stochastic models had mean and standard deviation, $\langle \tilde{a}_{0}\rangle=7.1\pm1.1$ and $\sigma_{a_0}=0.3\pm1.0$, $\langle \tilde{a}_{0}\rangle=9.3\pm1.0$ and $\sigma_{a_0}=2.1\pm0.5$ and $\langle \tilde{a}_{0}\rangle=9.1\pm1.2$ and $\sigma_{a_0}=0.7\pm0.1$, respectively. a) The measured post-collision electron spectrum, with $\langle E \rangle=\SI[separate-uncertainty]{574(11)}{\mega\electronvolt}$ and $P_{70}=0.67\pm 0.05$. The neural network predictions for the pre-collision electron spectrum have $\langle E \rangle=\SI[separate-uncertainty]{578(11)}{\mega\electronvolt}$ and $P_{70}=1.55\pm 0.01$. b) The measured photon spectrum, with mean energy [separate-uncertainty]73(7).

[width=0.99, trim=0.1cm, 0.20cm, 0.20cm, 0.22cm,clip]figures/electrons_mcmc_no_chirp_err_inference_all_models_indv_spec_5.png
(74.5, 20.5)Quantum-stochastic

[width=0.99, trim=0.1cm, 0.25cm, 0.20cm, 0.2cm,clip]figures/gspec_inferred_vs_measured_CsI_response_normed_no_chirp_err_inference_all_models_indv_spec_5.png

Bayesian inference results for shot 6. Similar to figure <ref>. The distribution of $\tilde{a}_{0}$ inferred by the classical, quantum-continuous and quantum-stochastic models had mean and standard deviation, $\langle \tilde{a}_{0}\rangle=6.5\pm1.1$ and $\sigma_{a_0}=1.3\pm0.2$, $\langle \tilde{a}_{0}\rangle=8.1\pm1.1$ and $\sigma_{a_0}=0.9\pm0.2$ and $\langle \tilde{a}_{0}\rangle=6.9\pm0.8$ and $\sigma_{a_0}=0.5\pm0.1$, respectively. a) The measured post-collision electron spectrum, with $\langle E \rangle=\SI[separate-uncertainty]{562(10)}{\mega\electronvolt}$ and $P_{70}=1.30\pm 0.01$. The neural network predictions for the pre-collision electron spectrum have $\langle E \rangle=\SI[separate-uncertainty]{577(11)}{\mega\electronvolt}$ and $P_{70}=1.56\pm 0.01$. b) The measured photon spectrum, with mean energy [separate-uncertainty]62(7).

[width=0.99, trim=0.1cm, 0.20cm, 0.20cm, 0.22cm,clip]figures/electrons_mcmc_no_chirp_err_inference_all_models_indv_spec_9.png
(74.5, 20.5)Quantum-stochastic

[width=0.99, trim=0.1cm, 0.25cm, 0.20cm, 0.2cm,clip]figures/gspec_inferred_vs_measured_CsI_response_normed_no_chirp_err_inference_all_models_indv_spec_9.png

Bayesian inference results for shot 10. Similar to figure <ref>. The distribution of $\tilde{a}_{0}$ inferred by the classical, quantum-continuous and quantum-stochastic models had mean and standard deviation, $\langle \tilde{a}_{0}\rangle=10.6\pm1.4$ and $\sigma_{a_0}=8.6\pm2.3$, $\langle \tilde{a}_{0}\rangle=14.5\pm2.2$ and $\sigma_{a_0}=7.7\pm1.7$ and $\langle \tilde{a}_{0}\rangle=13.3\pm1.9$ and $\sigma_{a_0}=8.6\pm2.3$, respectively. a) The measured post-collision electron spectrum, with $\langle E \rangle=\SI[separate-uncertainty]{617(12)}{\mega\electronvolt}$ and $P_{70}=1.23\pm 0.03$. The neural network predictions for the pre-collision electron spectrum have $\langle E \rangle=\SI[separate-uncertainty]{607(12)}{\mega\electronvolt}$ and $P_{70}=1.63\pm 0.04$. b) The measured photon spectrum, with mean energy [separate-uncertainty]68(7).

[width=0.99, trim=0.1cm, 0.20cm, 0.20cm, 0.22cm,clip]figures/electrons_mcmc_no_chirp_err_inference_all_models_indv_spec_2.png
(74.5, 20.5)Quantum-stochastic

[width=0.99, trim=0.1cm, 0.25cm, 0.20cm, 0.2cm,clip]figures/gspec_inferred_vs_measured_CsI_response_normed_no_chirp_err_inference_all_models_indv_spec_2.png

Bayesian inference results for shot 3. Similar to figure <ref>. The distribution of $\tilde{a}_{0}$ inferred by the classical, quantum-continuous and quantum-stochastic models had mean and standard deviation, $\langle \tilde{a}_{0}\rangle=6.8\pm1.3$ and $\sigma_{a_0}=1.4\pm0.4$, $\langle \tilde{a}_{0}\rangle=7.6\pm1.0$ and $\sigma_{a_0}=0.5\pm0.1$ and $\langle \tilde{a}_{0}\rangle=7.5\pm0.7$ and $\sigma_{a_0}=0.5\pm0.1$, respectively. a) The measured post-collision electron spectrum, with $\langle E \rangle=\SI[separate-uncertainty]{565(10)}{\mega\electronvolt}$ and $P_{70}=1.147\pm 0.001$. The neural network predictions for the pre-collision electron spectrum have $\langle E \rangle=\SI[separate-uncertainty]{582(11)}{\mega\electronvolt}$ and $P_{70}=1.59\pm 0.01$. b) The measured photon spectrum, with mean energy [separate-uncertainty]63(14).

[width=0.99, trim=0.1cm, 0.20cm, 0.20cm, 0.22cm,clip]figures/electrons_mcmc_no_chirp_err_inference_all_models_indv_spec_0.png
(74.5, 20.5)Quantum-stochastic

[width=0.99, trim=0.1cm, 0.25cm, 0.20cm, 0.2cm,clip]figures/gspec_inferred_vs_measured_CsI_response_normed_no_chirp_err_inference_all_models_indv_spec_0.png

Bayesian inference results for shot 1. Similar to figure <ref>. The distribution of $\tilde{a}_{0}$ inferred by the classical, quantum-continuous and quantum-stochastic models had mean and standard deviation, $\langle \tilde{a}_{0}\rangle=6.7\pm1.1$ and $\sigma_{\tilde{a}_{0}}=1.1\pm0.3$, $\langle \tilde{a}_{0}\rangle=8.1\pm1.2$ and $\sigma_{\tilde{a}_{0}}=1.1\pm0.3$, $\langle \tilde{a}_{0}\rangle=7.8\pm1.1$ and $\sigma_{\tilde{a}_{0}}=0.7\pm0.2$, respectively. a) The measured post-collision electron spectrum, with $\langle E \rangle=\SI[separate-uncertainty]{496(8)}{\mega\electronvolt}$ and $P_{70}=1.021\pm 0.001$. The neural network predictions for the pre-collision electron spectrum have $\langle E \rangle=\SI[separate-uncertainty]{573(11)}{\mega\electronvolt}$ and $P_{70}=1.60\pm 0.01$. b) The measured photon spectrum, with mean energy [separate-uncertainty]64(6).

[width=0.99, trim=0.1cm, 0.20cm, 0.20cm, 0.22cm,clip]figures/electrons_mcmc_no_chirp_err_inference_all_models_indv_spec_6.png
(74.5, 20.5)Quantum-stochastic

[width=0.99, trim=0.1cm, 0.25cm, 0.20cm, 0.2cm,clip]figures/gspec_inferred_vs_measured_CsI_response_normed_no_chirp_err_inference_all_models_indv_spec_6.png

Bayesian inference results for shot 7. Similar to figure <ref>. The distribution of $\tilde{a}_{0}$ inferred by the classical, quantum-continuous and quantum-stochastic models had mean and standard deviation, $\langle \tilde{a}_{0}\rangle=6.6\pm0.9$ and $\sigma_{a_0}=1.4\pm0.3$, $\langle \tilde{a}_{0}\rangle=7.2\pm0.5$ and $\sigma_{a_0}=0.4\pm0.1$ and $\langle \tilde{a}_{0}\rangle=7.2\pm$0.7 and $\sigma_{a_0}=0.5\pm0.1$, respectively. a) The measured post-collision electron spectrum, with $\langle E \rangle=\SI[separate-uncertainty]{525(9)}{\mega\electronvolt}$ and $P_{70}=0.996\pm 0.004$. The neural network predictions for the pre-collision electron spectrum have $\langle E \rangle=\SI[separate-uncertainty]{563(10)}{\mega\electronvolt}$ and $P_{70}=1.57\pm 0.01$. b) The measured photon spectrum, with mean energy [separate-uncertainty]77(30).

§ EXTENDED MATERIAL

§.§ Shot Selection

[width=0.7, trim=0.2cm, 0.2cm, 14.5cm, 0.2cm,clip]figures/gamma_tot_counts_vs_QG2_nulls_beam_off_fitted.png

The shot selection procedure is illustrated. The majorities of nulls (blue) (i.e. combined misses and beam-off shots) lie within $1\sigma$ (cyan, dashed) of the scaling of background gamma yield with $Q\langle\gamma^2\rangle$ (cyan, continuous). The small fraction of nulls which lie above this threshold are beam-off shots.
The populations of hits (red) lies $3\sigma$ (orange, dashed) above the background scaling. A log-linear scale (threshold at $1.5\times10^5$) was used on the y-axis to allow hits and misses to be clearly viewed on the same figure as the hits are at least two orders of magnitude brighter than the nulls. The shots analysed using the Bayesian framework are encircled (black).


The Optimization Problem of Quantum Discord
And Quantumness in Quantum Information theory

Chitradeep G. Hazra , IIT Delhi

###### Abstract

In a pair of correlated quantum systems, a measurement in one corresponds to a
change in the state of the other. In the process, information is lost.
Measurement along which set of projectors would accompany minimum loss in
information content is the optimization problem of quantum discord. This
optimization problem needs to be addressed in any computation of quantum
discord and is also an important aspect of our understanding of any classical
to quantum transition. The zero discord condition is a useful tool to explore
this but it is also asymmetric in the way that it allows for conditional
measurements in the complimentary Hilbert space. The problem of seeing how the
measurement of a certain observable changes the distribution of the another
requires another projector to be introduced which removes the freedom of
conditional measurements. In such a context one could show that quantum
discord minimizes at the diagonal basis of the reduced density matrices. We
present a mathematical proof for the above statement and produce an analytical
expression for quantum discord for a general m x n density matrix.

Introduction: A seminal paper in 2001 by Ollivier and Zurech[1] established
that quantum correlations could exist beyond entanglement through what we
today know as quantum discord. While shedding light on such new correlations,
the paper also establishes a new frontier in exploring classical to quantum
transitions in terms of information content. A quantum state upon measurement
undergoes decoherence and loses information content that isn’t seen in
classical systems and this loss is being addressed as quantum discord[1]. The
states achieved upon minimizing this loss are called the most classical states
or Pointer States[2]. Ever since its discovery quantum discord is being used
for quantum speed up over classical algorithms[3,4,5], entanglement
distillation[6,7], quantum teleportation[5], superdense coding[5] and in many
other areas of quantum computation. In all such applications however,
researchers had to compute quantum discord through a fairly difficult[8]
optimization process over all possible sets of measurement by which discord is
defined. This computation is known to be NP complete[8] and the running time
of any algorithm grows exponentially with the size of the input. Analytical
results[9,10,11] for simple low dimensional states are known but the standard
method of optimization with state parameters that is prevalent in the
community runs into increasing number of transcedental equations that are not
easy to solve. Calculational difficulties aside, the optimization problem of
quantum discord is also deeply intertwined with our understanding of any form
of departure from classicality. The problem asks us to find the sets of
measurement that shall accompany minimum information loss. A classical system
should therefore correspond to zero discord.

In this letter the author wishes to address the important open problem of
minimizing discord in the product space of two correlated systems with more
emphasis given towards the resulting correlations in their observables. With
such an intention, we introduce a new measure for discord, perform the
required optimization for an in general $m\times n$ matrix with the help of
standard information inequalities and produce an analytical expression of
quantum discord that is symmetric. This is followed by a proof that the new
measure is stronger than the standard version in detecting quantum
superpositions and demonstrate that their character is exactly similar for
symmetric states. The new zero discord condition helps us to establish a
strong equivalence between discord and non-commutivity while the solution to
the optimization problem provides a clear identification scheme for finding
pointer states. This is the first generalized analytical expression for
discord to the best of our knowledge and we believe it could be of immense use
to researchers in the community.

Background: The formal development of quantum discord in the literature opens
with the insight that mutual information $I(S:A)$ between two systems S and A
that is symmetric in a classical setting fails to be so in a quantum context
with ${J(S:A)}_{\\{{\Pi}^{A}_{j}\\}}$ where the set of projectors along which
the measurement is being made needs to be specified.

$I(S:A)=H(A)+H(S)-H(S,A)$
${J(S:A)}_{\\{{\Pi}^{A}_{j}\\}}=H(S)-H(S|A)_{\\{{\Pi}^{A}_{j}\\}}$

The difference between the two gives us discord
$\delta(S:A)_{\\{{\Pi}^{A}_{j}\\}}$ for a specific set of projectors that in
turn speaks of the information loss as the difference in the information
content of the pre/post-measurement states. Discord between S and A is then
defined through an optimization over the entire projector space of A.

$\displaystyle\delta(S:A)_{\\{{\Pi}^{A}_{j}\\}}$
$\displaystyle=I(S:A)-{J(S:A)}_{\\{{\Pi}^{A}_{j}\\}}$
$\displaystyle=H(A)+H(S|A)_{\\{{\Pi}^{A}_{j}\\}}-H(S,A)$
$\displaystyle\delta(S:A)$
$\displaystyle=min(\delta(S:A)_{\\{{\Pi}^{A}_{j}\\}})$

Motivations for a new measure: The zero discord condition[1] is
$\rho_{\scriptscriptstyle{S,A}}=\rho_{\scriptscriptstyle{S,A}}^{\scriptscriptstyle{D}}=\sum_{j}\Pi^{\scriptscriptstyle{A}}_{j}\rho_{\scriptscriptstyle{S,A}}\Pi^{\scriptscriptstyle{A}}_{j}$;
the equivalence of the pre-measurement and post-measurement states. A
$\rho_{\scriptscriptstyle{S,A}}$ of the form
$\sum_{j}P_{j}\rho^{j}_{\scriptscriptstyle{S}}\times\Pi^{\scriptscriptstyle{A}}_{j}$
would satisfy such a condition and with a spectral decomposition one could
arrive at the following relation.

$\rho_{\scriptscriptstyle{S,A}}=\sum_{i,j}\Pi^{\scriptscriptstyle{S}}_{j,i}\Pi^{\scriptscriptstyle{A}}_{j}\rho_{\scriptscriptstyle{S,A}}\Pi^{\scriptscriptstyle{A}}_{j}\Pi^{\scriptscriptstyle{S}}_{j,i}$

Zero discord is equivalent to finding the set of measurements
$\\{\Pi^{\scriptscriptstyle{A}}_{j},\Pi^{\scriptscriptstyle{S}}_{j,i}\\}$ for
which the density matrix is not disturbed and no information is lost for any
other observer unaware of the outcomes. However, the set of measurements
$\\{\Pi^{\scriptscriptstyle{S}}_{j,i}\\}$ in S for a given $j$ is dependent on
the outcome of the measurement in A. In other words, what outcome one gets for
A determines which observable one needs to measure in S. A more classical
measure for correlations should answer the question of how much the
distribution of a certain particular observable in S changes based on the
outcome of the measurement in A which leads us to define a mutual information
of the form ${J(S:A)}_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$, where

${J(S:A)}_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}=H(S)-H(S|A)_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$

$H(S|A)_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$ is defined as
$H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{S|\\{\Pi^{A}_{j}\\}}}})_{\\{\Pi^{S}_{i}\\}}$.
While there is gain in information about S upon conditioning with A by
$H(\rho_{\scriptscriptstyle{S}})-H(\rho_{\scriptscriptstyle{S}|\\{\Pi^{A}_{j}\\}})$,
there is loss in information about S upon projection along $\\{\Pi^{S}_{i}\\}$
by
$H(\rho_{\scriptscriptstyle{S}|\\{\Pi^{A}_{j}\\}})-H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{S|\\{\Pi^{A}_{j}\\}}}})_{\\{\Pi^{S}_{i}\\}}$.
${J(S:A)}_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$ is then the sum of both,
expressing the amount of information gained about a certain observable in S
from a measurement in A. The extra information that is lost upon projection is
exclusively a quantum phenomena which would go to zero in the classical case.
This establishes the otherwise classical equivalence of the two mutual
informations ${J(S:A)}_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$ and $I(S:A)$ which
was the original starting point of Ollivier and Zurech in 2001. In such a
setting, one could define a discord $\alpha(S:A)$ as follows,

$\displaystyle\alpha(S:A)_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$
$\displaystyle=I(S:A)-{J(S:A)}_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$
$\displaystyle=H(A)+H(S|A)_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}-H(S,A)$
$\displaystyle\alpha(S:A)$
$\displaystyle=min(\alpha(S:A)_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}})$

$\alpha(S:A)_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$ is the information loss when
a series of successive measurements is made of two particular observables in
the respective Hilbert Spaces of A, and then in S. $\alpha(S:A)$ is then the
optimization of that over the two projector spaces.

Minimization in Projector Space: Minimization of $\alpha$ depends on the
minimization of $H(S|A)_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$. At this point we
would like to introduce a lemma, trying to understand what this term means
explicitly. The Hilbert spaces of S and A are spanned by the states
$\\{|n_{i}>\\}$ and $\\{|k_{j}>\\}$ respectively. The notation $P(x,y)$ stands
for the probability of some state $|x>$ to collapse at $|y>$.

Lemma 1
${H(S|A)}_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}=H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{S,A}}})_{\\{\Pi^{S}_{i}\times\Pi^{A}_{j}\\}}-H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{A}}})_{\\{\Pi^{A}_{j}\\}}=D(P(SA,n_{i}k_{j})||P(A,k_{j}))$

Proof:
$H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{S|\\{\Pi^{A}_{j}\\}}}})_{\\{\Pi^{S}_{i}\\}}=\sum_{j}P(A,k_{j})H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{S|\Pi^{A}_{j}}}})_{\\{\Pi^{S}_{i}\\}}\\\
=-\sum_{j,i}P(A,k_{j})P(S,n_{i}|A,k_{j})log(P(S,n_{i}|A,k_{j}))\\\
=-\sum_{j,i}P(SA,n_{i}k_{j})log(P(SA,n_{i}k_{j})/P(A,k_{j}))\\\
=H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{S,A}}})_{\\{\Pi^{S}_{i}\times\Pi^{A}_{j}\\}}-H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{A}}})_{\\{\Pi^{A}_{j}\\}}$

The next lemma helps us establish that any form of measurement or projection
in the Hilbert space is always accompanied with a loss in information content.
It is zero only when the projection is along the diagonal basis.

Lemma 2 $H(\rho^{\scriptscriptstyle{D}}_{\\{{\Pi_{j}\\}}})\geq H(\rho)$

Proof: $\\{\lambda_{i}\\}$ are the eigenvalues of $\rho$ with eigenstates
$\\{\bar{k}_{i}\\}$ while projection is taken along $\\{k_{j}\\}$. We apply
Jensen’s inequality[14] to complete the proof.
$H(\rho^{\scriptscriptstyle{D}}_{\\{{\Pi_{j}\\}}})=H(\sum_{j,i}\Pi_{j}\lambda_{i}\bar{\Pi}_{i}\Pi_{j})=H(\sum_{j,i}P(k_{j},\bar{k}_{i})\lambda_{i}\Pi_{j})\\\
=-\sum_{j}(\sum_{i}P(k_{j},\bar{k}_{i})\lambda_{i})log(\sum_{i}P(k_{j},\bar{k}_{i})\lambda_{i})\\\
\geq-\sum_{j,i}P(k_{j},\bar{k}_{i})\lambda_{i}log\lambda_{i}=-\sum_{i}\lambda_{i}log\lambda_{i}=H(\rho)$

Theorem 1 Minimization \- $\alpha(S:A)_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$
for a separable density matrix $\rho_{\scriptscriptstyle{S,A}}$ minimizes
along the projectors of the diagonal basis of its reduced density matrices.

Proof: The density matrix we choose to work with is a mixture of otherwise
classical density matrices of the form
$\sum_{j}P_{j}\Pi^{\scriptscriptstyle{S}}_{j}\times\Pi^{\scriptscriptstyle{A}}_{j}$.
The resultant matrix is of the form -

$\rho_{\scriptscriptstyle{S,A}}=\sum_{i,j}P_{i}P^{i}_{j}\Pi^{\scriptscriptstyle{S}}_{i,j}\times\Pi^{\scriptscriptstyle{A}}_{i,j}$

In this case, the minimization of $\alpha$ by lemma 1 is just that of
$D(P(SA,n_{i^{\prime}}k_{j^{\prime}}||P(A,k_{j^{\prime}}))$, as in
$D(P_{i^{\prime}j^{\prime}}||P_{j^{\prime}}))$. The reduced density matrices
have eigenbasis $\\{\bar{{\Pi}}^{A}_{j^{\prime\prime}}\\}$ and
$\\{\bar{{\Pi}}^{S}_{i^{\prime\prime}}\\}$.

$\displaystyle P_{i^{\prime}j^{\prime}}$
$\displaystyle=\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},k_{j^{\prime}})$
(1) $\displaystyle\rho_{\scriptscriptstyle{A}}$
$\displaystyle=\sum_{i,j}P_{i}P^{i}_{j}\Pi^{\scriptscriptstyle{A}}_{i,j}=\sum_{j^{\prime\prime}}\bar{P}_{j^{\prime\prime}}\bar{\Pi}^{\scriptscriptstyle{A}}_{j^{\prime\prime}}$
$\displaystyle\bar{P}_{j^{\prime\prime}}$
$\displaystyle=\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})$
(2) $\displaystyle P_{j^{\prime}}$
$\displaystyle=\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},k_{j^{\prime}})$ (3)
$\displaystyle P_{j^{\prime}}$
$\displaystyle=\sum_{j^{\prime\prime}}\bar{P}_{j^{\prime\prime}}P(\bar{k}_{j^{\prime\prime}},k_{j^{\prime}})$
$\displaystyle=\sum_{i,j,j^{\prime\prime}}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_{j^{\prime\prime}},k_{j^{\prime}})$
(4)

We arrived at (4) using (2). Comparing (4) and (3) gives us the following
equations that we shall use in differentiating the states of the diagonal
basis from the rest.

$\displaystyle P(k_{ij},k_{j^{\prime}})$
$\displaystyle=\sum_{j^{\prime\prime}}P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_{j^{\prime\prime}},k_{j^{\prime}})$
(5) $\displaystyle P(n_{ij},n_{i^{\prime}})$
$\displaystyle=\sum_{i^{\prime\prime}}P(n_{ij},\bar{n}_{i^{\prime\prime}})P(\bar{n}_{i^{\prime\prime}},n_{i^{\prime}})$
(6)

It is important to notice here that the chain rule multiplication of
probabilities in (5) and (6) followed by a summation over the exclusive
alternatives wouldn’t work for any other basis other than
$\\{|\bar{n}_{i^{\prime\prime}}>\\}$ or $\\{|\bar{k}_{j^{\prime\prime}}>\\}$
because of the interference terms[15]. Equations (5) and (6) ask us to
recognize that the states of S and A are essentially a classical mixture only
in their diagonal basis and no other. Now that we have all the required
probabilities in order, let us minimize.

We use equations (1) and (3) to find
${H(S|A)}_{\\{{\Pi}^{A}_{j^{\prime}},{\Pi}^{S}_{i^{\prime}}\\}}$, introduce
(5) at the next step and employ log-sum inequality[16] with respect to the
index $\\{j^{\prime\prime}\\}$. The $j^{\prime}$-terms within the logarithm
cancel out and a summation over $\\{j^{\prime}\\}$ gives us the minimization
at ${\\{\bar{{\Pi}}^{A}_{j^{\prime\prime}}}\\}$.

$\displaystyle{H(S|A)}_{\\{{\Pi}^{A}_{j^{\prime}},{\Pi}^{S}_{i^{\prime}}\\}}$
$\displaystyle=-\sum_{i^{\prime},j^{\prime}}P_{i^{\prime}j^{\prime}}log(\frac{P_{i^{\prime}j^{\prime}}}{P_{j^{\prime}}})$
$\displaystyle=-\sum_{i^{\prime},j^{\prime}}({\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},k_{j^{\prime}})})log(\frac{\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},k_{j^{\prime}})}{\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},{k}_{j^{\prime}})})$
$\displaystyle=-\sum_{i^{\prime},j^{\prime}}({\sum_{i,j,j^{\prime\prime}}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_{j^{\prime\prime}},k_{j^{\prime}})})log(\frac{\sum_{i,j,j^{\prime\prime}}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_{j^{\prime\prime}},k_{j^{\prime}})}{\sum_{i,j,j^{\prime\prime}}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_{j^{\prime\prime}},k_{j^{\prime}})})$
$\displaystyle\geq-\sum_{i^{\prime},j^{\prime},j^{\prime\prime}}({\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_{j^{\prime\prime}},k_{j^{\prime}})})log(\frac{\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_{j^{\prime\prime}},k_{j^{\prime}})}{\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})P(\bar{k}_{j^{\prime\prime}},k_{j^{\prime}})})$
$\displaystyle\geq-\sum_{i^{\prime},j^{\prime\prime}}({\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})})log(\frac{\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})}{\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})})$
$\displaystyle\geq{H(S|A)}_{\\{\bar{{\Pi}}^{A}_{j^{\prime\prime}},{\Pi}^{S}_{i^{\prime}}\\}}$

We use equations (1) and (3) to find
${H(S|A)}_{\\{\bar{{\Pi}}^{A}_{j^{\prime\prime}},{\Pi}^{S}_{i^{\prime}}\\}}$,
introduce (6) at the next step and employ Jensen’s inequality for concave
functions[14] with respect to the distribution
$\\{P(\bar{n}_{i^{\prime\prime}},n_{i^{\prime}})\\}$ with the varying index as
$\\{{i^{\prime\prime}}\\}$. A summation over $\\{i^{\prime}\\}$ then gives us
the minimization at ${\\{\bar{{\Pi}}^{S}_{i^{\prime\prime}}}\\}$ and completes
the proof.

$\displaystyle{H(S|A)}_{\\{\bar{{\Pi}}^{A}_{j^{\prime\prime}},{\Pi}^{S}_{i^{\prime}}\\}}$
$\displaystyle=-\sum_{i^{\prime},j^{\prime\prime}}({\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})})log(\frac{\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})}{\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})})$
$\displaystyle=-\sum_{i^{\prime},j^{\prime\prime}}({\sum_{i,j,i^{\prime\prime}}P_{i}P^{i}_{j}P(n_{ij},\bar{n}_{i^{\prime\prime}})P(\bar{n}_{i^{\prime\prime}},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})})log(\frac{\sum_{i,j,i^{\prime\prime}}P_{i}P^{i}_{j}P(n_{ij},\bar{n}_{i^{\prime\prime}})P(\bar{n}_{i^{\prime\prime}},n_{i^{\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})}{\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})})$
$\displaystyle\geq-\sum_{i^{\prime},j^{\prime\prime},i^{\prime\prime}}P(\bar{n}_{i^{\prime\prime}},n_{i^{\prime}})({\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},\bar{n}_{i^{\prime\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})})log(\frac{\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},\bar{n}_{i^{\prime\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})}{\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})})$
$\displaystyle\geq-\sum_{i^{\prime\prime},j^{\prime\prime}}({\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},\bar{n}_{i^{\prime\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})})log(\frac{\sum_{i,j}P_{i}P^{i}_{j}P(n_{ij},\bar{n}_{i^{\prime\prime}})P(k_{ij},\bar{k}_{j^{\prime\prime}})}{\sum_{i,j}P_{i}P^{i}_{j}P(k_{ij},\bar{k}_{j^{\prime\prime}})})$
$\displaystyle\geq{H(S|A)}_{\\{\bar{{\Pi}}^{A}_{j^{\prime\prime}},\bar{{\Pi}}^{S}_{i^{\prime\prime}}\\}}$

The most general $m\times n$ separable density matrix
$\sum_{i}P_{i}\rho^{i}_{\scriptscriptstyle{S}}\times\rho^{i}_{\scriptscriptstyle{A}}$
upon decomposition would give
$\sum_{i,j,l}P_{i}P^{i}_{j}P^{i}_{l}\Pi^{\scriptscriptstyle{S}}_{i,j}\times\Pi^{\scriptscriptstyle{A}}_{i,l}$.
The extra index $l$ is being ignored for simplification. The reader is still
encouraged to check for herself that even in the most general case the proof
gets through.

Theorem 2 Symmetrization \- $\alpha(S:A)=\alpha(A:S)$

Proof:
$\alpha(S:A)=\alpha(S:A)_{\\{\bar{{\Pi}}^{A}_{j},\bar{{\Pi}}^{S}_{i}\\}}\\\
=H(A)+H(S|A)_{\\{\bar{\Pi}^{A}_{j},\bar{\Pi}^{S}_{i}\\}}-H(S,A)\\\
=H(A)+H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{S,A}}})_{\\{\bar{\Pi}^{S}_{i}\times\bar{\Pi}^{A}_{j}\\}}-H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{A}}})_{\\{\bar{\Pi}^{A}_{j}\\}}-H(S,A)\\\
=H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{S,A}}})_{\\{\bar{\Pi}^{S}_{i}\times\bar{\Pi}^{A}_{j}\\}}-H(S,A)$

The symmetry that broke down at $J(S:A)_{\\{{\Pi}^{A}_{j},{\Pi}^{S}_{i}\\}}$
is again restored in $\alpha(S:A)$ while giving us an analytical expression of
discord in equation (7).

$\displaystyle\alpha(S:A)$
$\displaystyle=H(\rho^{\scriptscriptstyle{D}}_{\scriptscriptstyle{{S,A}}})_{\\{\bar{\Pi}^{S}_{i}\times\bar{\Pi}^{A}_{j}\\}}-H(S,A)$
(7)

This is the expression we had strived for; notice the projectors in equation
(7) are a function of the given density matrix. In such a context referring
back to lemma 2 helps us recognize: (i) The zero discord condition-
$\alpha(S:A)=0$ when the projectors
${\\{\bar{{\Pi}}^{S}_{i}\times\bar{{\Pi}}^{A}_{j}\\}}$ becomes the diagonal
basis of the given density matrix, restricting it to the form
$\sum_{i,j}P_{ij}{{\Pi}}^{S}_{i}\times\Pi^{A}_{j}$ with product eigenstates
such that $[\rho_{\scriptscriptstyle{{S,A}}},N\times K]=0$ for some N and K.
(ii) The strongness condition- $\alpha(S:A)\geq\delta(S:A)$. This is a direct
consequence of lemma 2 since $\alpha$ only differs from $\delta$ by an extra
projection. The zero discord condition of $\alpha(S:A)$ in (i) assigns both
the discords $\delta(S:A)$ and $\delta(A:S)$ to be zero together. Hence, one
could consider $\alpha(S:A)$ a stronger measure of quantum discord. (iii)
Identification of Pointer States: The minimization of $\alpha$ at the diagonal
basis of the reduced density matrices make the corresponding states a strong
candidate for Pointer States. This is in alignment with lemma 2 where we saw
that measurement along the diagonal basis doesn’t lead to any information loss
and hence, are most classical. Equation (7) in that sense, talks about the
closest one could get to such a scenario in the product space when the
diagonal basis of the joint density matrix is essentially entangled for a
nonzero discord.

Instances and Discussions: A comparative study of both the discords is being
done as we see it play out for real systems.

$\alpha$ for the Werner states
$\rho_{w}=\frac{1}{4}(I+x\vec{\sigma_{1}}.\vec{\sigma_{2}})$

$\alpha(x)=\frac{1+x}{4}log\frac{1+x}{4}+\frac{1-3x}{4}log\frac{1-3x}{4}-\frac{1-x}{2}log\frac{1-x}{4}$

The curve for $\alpha(x)$ behaves just as $\delta(x)$ provided in [17].

$\alpha$ for the example in Zurech’01
$\rho_{z}=\frac{1}{2}(|00><00|+|11><11|)+\frac{z}{2}(|00><11|+|11><00|)$

$\alpha(z)=\frac{1+z}{2}log\frac{1+z}{2}+\frac{1-z}{2}log\frac{1-z}{2}+log2$

The curve for $\alpha$ that we have obtained corresponds to the spine of the
plot given in Zurech’01[1]. The variation over the projector space by $\theta$
and how it minimizes at that spine follows from theorem 1. So it seems
$\alpha$ and $\delta$ agree fairly well. For asymmetric states in S and A,
they of course shouldn’t, contributions from both $\delta(S:A)$ and
$\delta(A:S)$ play into $\alpha$ because of the symmetrization.

In Closing: A stronger measure for discord is formalized to address the
quantumness in correlations between a pair of observables rather than the
entire systems. The optimization problem is then solved for an in general
$m\times n$ matrix to produce an analytical expression of discord that is
already symmetric. This helps us to identify the most classical states
(pointer states) as the diagonal basis of the reduced density matrices that
when measured would disturb the systems the least with minimum loss in
information content. This being the synopsis of our main results, one can’t
help but wonder about the exact scope of theorem 1. We chose to work with a
separable density matrix because it is known now that entanglement isn’t
necessary for a nonzero discord[1]. Having said that the proof heavily relies
on the separability of $\rho_{S,A}$. If such a theorem cannot be established
for entangled states at all then one could devise a new separability criterion
from it which from where we stand, seems highly unlikely. At the end, the
confidence in theorem 1 is reinforced via equation (7) which establishes a
strong equivalence between discord and non-commutivity. Creation of discord in
terms of information becomes breakdown of commutation in terms of geometry.

Acknowledgement: The author wishes to thank Prof. V. Ravishankar for his
helpful comments and for introducing the student to discord.

References:
$[1]$ Ollivier, H. and Zurek, W. (2001) ”Quantum Discord: A Measure of the
Quantumness of Correlations” Phys. Rev. Lett. 88, 017901

$[2]$ Zurek, W. (2003). ”Decoherence, einselection and the quantum origins of
the classical”. Reviews of Modern Physics. 75 (3): 715–775

$[3]$ Datta, A. Shaji, A and Caves,C. (2008) ”Quantum discord and the power of
one qubit” Phys. Rev. Lett., 100(5):050502

$[4]$ Henderson, L. and Vedral, V. (2001) ”Classical quantum and total
correlations” J. Phys. A: Math. Gen., 34(35):6899

$[5]$ Madhok, V. and Datta, A. (2013) ”Quantum discord as a resource in
quantum communication” Int. J. Mod. Phys. B, 27(1–3):1345041

$[6]$ Streltsov, A. Kampermann, H. and Bruss, D. ( 2011) ”Linking quantum
discord to entanglement in a measurement” Phys. Rev. Lett., 106(16):160401

$[7]$ Devetak, I. and Winter, A. (2004) ”Distilling common randomness from
bipartite quantum states” IEEE Trans. Inf. Theory, 50(12):3183–3196

$[8]$ Huang, Y. (2014) ”Computing quantum discord is NP-complete” New Journal
of Physics 16, 033027

$[9]$ Ali, M. Rau, A. and Alber, G. (2010) ”Quantum discord for two-qubit X
states” Phys. Rev. A, 81(4):042105

$[10]$ Girolami, D. and Adesso, G. ”Quantum discord for general two-qubit
states: Analytical progress” Phys. Rev. A 83, 052108

$[11]$ Vinjanampathy, S. and Rau, A. (2012) ”Quantum discord for qubit–qudit
systems” J. of Phys. A: Mathematical and Theoretical

$[12]$ Lu, X. Ma, J. Xi, Z. and Wang, X. (2011) ”Optimal measurements to
access classical correlations of two-qubit states” Phys. Rev. A, 83(1):012327

$[13]$ Maziero, J. Celeri, L. and Serra, R. (2010) ”Suitability of symmetric
and asymmetric versions of the quantum discord” arXiv:1004.2082[quant-ph]

$[14]$ Cover, T. and Thomas, J. (1991) ”Elements of Information Theory”
Chapter-16, Theorem-16.1.2

$[15]$ Feynman, R. (1948) ”Space-Time Approach to Non-Relativistic Quantum
Mechanics” Rev. Mod. Phys. 20, 367

$[16]$ Cover, T. and Thomas, J. (1991) ”Elements of Information Theory”
Chapter-16, Theorem-16.1.1

$[17]$ Díaz-Solórzano, S. and Castro, E. (2018) ”Exact analytical solution of
Entanglement of Formation and Quantum Discord for Werner state and Generalized
Werner-Like states” arXiv:1802.05877 [quant-ph], Figure-2