Raj Gaurav Tripathi

Research

Quantum contextuality asks when observed quantum statistics resist a classical explanation that respects the equivalences between indistinguishable experiments. My MS thesis studies a matrix-based approach to this question, using probability matrices, nonnegative factorization and rank-separation methods. I am currently building and testing the framework on explicit examples; possible later directions include quantitative contextuality and typicality. My other work concerns entanglement in the cosmic microwave background and earlier research on cosmological magnetogenesis.

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Generalized quantum contextuality

MS thesis, supervised by Dr. Beata Zjawin (ICTQT, University of Gdańsk) Aug 2026 – present In progress

What is contextuality?

Quantum contextuality is a way of asking whether the probabilities predicted by quantum theory can be explained by a classical model of an underlying physical reality. The key idea is operational equivalence: two experimental procedures are operationally equivalent when no allowed experiment can distinguish them statistically. Generalized noncontextuality requires such procedures to receive the same representation in the underlying model. Contextuality is present when the observed statistics cannot be reproduced by any model satisfying all these requirements.

This is not simply a claim that quantum outcomes are random: classical probabilistic models can also be random. The issue is whether one consistent model can explain all the statistics while respecting the equivalences between procedures. The generalized framework applies this principle to preparations, measurement events and transformations, extending beyond the traditional focus on sharp measurements.

A simple quantum example. A qubit can be prepared by mixing the states \(|+\rangle\) and \(|-\rangle\) with equal probability, or by mixing \(|0\rangle\) and \(|1\rangle\) equally. These are different laboratory procedures, but both prepare the maximally mixed state:

\[\frac12 |+\rangle\langle+|+\frac12|-\rangle\langle-|=\frac{I}{2}=\frac12|0\rangle\langle0|+\frac12|1\rangle\langle1|.\]

Consequently, every quantum measurement gives the same outcome probabilities for both preparations. Preparation noncontextuality requires their underlying probability distributions over ontic states to be identical as well.

My MS thesis studies a recent linear-algebraic approach to this question. Instead of treating all the probabilities separately, the approach collects them into a matrix and asks whether its structure permits a noncontextual explanation. This leads to the COPE matrix, nonnegative matrix factorization and rank-separation methods.

For the mathematical details, expand Technical framework: COPE, ENMF and rank separation in the side panel.

Current work: building and testing the framework

The immediate assignment is to work through the linear-algebraic framework, its complexity results and rank-separation applications. I am constructing small prepare-and-measure examples, working through a classical toy model, and distinguishing the number of ontic states from the ordinary ranks of the factor matrices. I also plan to compare ENMF conclusions with linear-programming tests on the same scenarios.

Core reading: Shahandeh, Yianni and Doosti, A Unified Linear Algebraic Framework for Physical Models and Generalized Contextuality; Yianni and Shahandeh, Complexity of Contextuality; Doosti, Yianni and Shahandeh, Contextuality of All Optimal Quantum Cloning.

Potential research directions

The second-semester question remains to be selected in discussion with my supervisor. Two possible directions are:

  • Quantitative contextuality. Investigate whether nonnegative rank, ENNR or rank-separation properties yield new bounds or relations for existing measures, such as noise robustness or classical excess. Any proposed rank-based measure would need a clear interpretation and justified mathematical properties.
  • Typicality through ENMF. Study how ENMF-based criteria behave for carefully specified ensembles of quantum preparations and measurements. An initial step would be to compare ENMF and linear-programming verdicts on the same examples, then ask whether rank structure reveals anything beyond existing typicality results.

These are prospective directions, not established results or a final commitment. The aim is to identify a precise, feasible question after reproducing the existing framework and reviewing relevant quantitative approaches.

Technical framework: COPE, ENMF and rank separation

1. From an experiment to the COPE matrix

Consider a prepare-and-measure experiment with preparations \(P_i\), \(i=1,\ldots,I\), and measurements \(M_j\), \(j=1,\ldots,J\). Measurement \(M_j\) has outcomes \(k=1,\ldots,n_j\), with probabilities \(p(k\mid P_i,M_j)\) satisfying \(\sum_{k=1}^{n_j}p(k\mid P_i,M_j)=1\).

The matrix of conditional outcome probabilities of events (COPE) collects these statistics into a single matrix \(C\). Each column corresponds to a preparation; each row corresponds to a measurement event. Writing \(m=\sum_{j=1}^{J}n_j\),

\[C_{(j,k),i}=p(k\mid P_i,M_j),\qquad C\in\mathbb{R}_{\geq0}^{m\times I},\qquad m=\sum_{j=1}^{J}n_j.\]

Here the pair \((j,k)\) is combined into a single row index. Schematically,

\[C=\begin{pmatrix} p(1\mid P_1,M_1)&\cdots&p(1\mid P_I,M_1)\\ \vdots&\ddots&\vdots\\ p(n_1\mid P_1,M_1)&\cdots&p(n_1\mid P_I,M_1)\\ \vdots&\ddots&\vdots\\ p(n_J\mid P_1,M_J)&\cdots&p(n_J\mid P_I,M_J) \end{pmatrix}.\]

2. Operational equivalences become matrix constraints

These equivalences are visible directly in \(C\): equivalent preparations produce identical columns, while equivalent measurement events produce identical rows. For example,

\[P_i\simeq P_{i'}\ \Longrightarrow\ C_{:,i}=C_{:,i'}.\]

If \(O_k^j\) denotes outcome \(k\) of measurement \(M_j\), equivalent events satisfy

\[O_k^j\simeq O_{k'}^{j'}\ \Longrightarrow\ C_{(j,k),:}=C_{(j',k'),:}.\]

Equivalences between mixtures impose linear relations rather than necessarily making individual columns equal. For instance,

\[\tfrac12P_1+\tfrac12P_2\simeq\tfrac12P_3+\tfrac12P_4\ \Longrightarrow\ \tfrac12(C_{:,1}+C_{:,2})=\tfrac12(C_{:,3}+C_{:,4}).\]

3. Ontological models as nonnegative factorizations

An ontological model supposes that a preparation \(P_i\) produces an underlying state \(\lambda\in\Lambda\) with probability \(\mu(\lambda\mid P_i)\). Given that state, the probability of outcome \(k\) for measurement \(M_j\) is the response function \(\xi(k\mid\lambda,M_j)\). The predicted statistics obey the law of total probability:

\[p(k\mid P_i,M_j)=\sum_{\lambda\in\Lambda}\xi(k\mid\lambda,M_j)\,\mu(\lambda\mid P_i).\]

Let \(K=|\Lambda|\) be the number of ontic states, and define the response and preparation matrices by

\[R_{(j,k),\lambda}:=\xi(k\mid\lambda,M_j),\qquad E_{\lambda i}:=\mu(\lambda\mid P_i).\]

The ontological-model equation is then the matrix factorization

\[\boxed{C=RE},\qquad R\in\mathbb{R}_{\geq0}^{m\times K},\quad E\in\mathbb{R}_{\geq0}^{K\times I}.\]

The columns of \(R\) are measurement-response profiles; the columns of \(E\) are probability distributions over ontic states for each preparation. A valid model also satisfies the probability normalizations \(\sum_{\lambda}E_{\lambda i}=1\) and \(\sum_{k=1}^{n_j}R_{(j,k),\lambda}=1\) for each preparation \(i\), measurement \(j\) and ontic state \(\lambda\).

4. Ordinary rank, nonnegative rank and ENMF

The ordinary matrix rank \(r=\operatorname{rank}(C)\) counts independent real-valued patterns in the observed probability profiles. It allows arbitrary real coefficients, including negative ones. The nonnegative rank imposes the extra requirement that both factors be nonnegative:

\[\operatorname{rank}_{+}(C):=\min\{K:C=RE,\ R\geq0,\ E\geq0\}.\]

It is the smallest number of nonnegative rank-one terms needed to construct \(C\), and corresponds to the minimum number of ontic states in a finite ontological representation, with the relevant probability normalizations. The inner dimension \(K\) must not be confused with the ordinary ranks of the factors: for every factorization \(C=RE\),

\[\operatorname{rank}(C)\leq\min\{\operatorname{rank}(R),\operatorname{rank}(E)\}\leq K.\]

An equirank nonnegative matrix factorization (ENMF) is a nonnegative factorization that also satisfies

\[\operatorname{rank}(C)=\operatorname{rank}(R)=\operatorname{rank}(E).\]

The common rank can be smaller than \(K\): the number of ontic labels need not equal the number of independent linear patterns. The equirank nonnegative rank (ENNR), \(\operatorname{ENNR}(C)\), is the minimum inner dimension \(K\) among ENMFs.

5. Rank separation as a route to contextuality

In the COPE/ENMF framework, the existence of a noncontextual ontological model is equivalent to the existence of an ENMF, subject to the framework's assumptions and probability constraints. Therefore, if one can prove that every nonnegative factorization of \(C\) has \(\operatorname{rank}(R)>\operatorname{rank}(C)\) or \(\operatorname{rank}(E)>\operatorname{rank}(C)\), no ENMF can exist. This rules out a noncontextual model within that framework and establishes contextuality. A rank mismatch in just one chosen factorization is not enough; the argument must rule out all admissible factorizations.

What this means for my thesis. I am first reproducing the framework on small, explicit examples: constructing COPE matrices, computing ranks, investigating ENMFs, and comparing conclusions with linear-programming tests. The research question for the next stage will be chosen after this foundation is established.
Key references
  • R. W. Spekkens, Contextuality for preparations, transformations, and unsharp measurements, Phys. Rev. A 71, 052108 (2005), arXiv:quant-ph/0406166.
  • F. Shahandeh, T. Yianni and M. Doosti, A Unified Linear Algebraic Framework for Physical Models and Generalized Contextuality, arXiv:2512.10000.
  • T. Yianni and F. Shahandeh, Complexity of Contextuality, arXiv:2506.09133.
  • M. Doosti, T. Yianni and F. Shahandeh, Contextuality of All Optimal Quantum Cloning, arXiv:2406.19382.
  • D. Schmid et al., Characterization of noncontextuality in the framework of generalized probabilistic theories, PRX Quantum 2, 010331 (2021).
  • J. H. Selby et al., A linear program for testing nonclassicality and an open-source implementation, arXiv:2204.11905.
  • V. P. Rossi et al., Typicality of Contextuality, arXiv:2510.20722.
  • N. Gillis, Nonnegative Matrix Factorization, SIAM (2020), for background on nonnegative matrix factorizations.

Entanglement through gravity in the CMB

Independent side project; discussions with Prof. Marios Christodoulou (IQOQI Vienna) Aug 2026 – presentLiterature stage

During inflation, the expanding background drives each Fourier mode of the curvature perturbation ζ into a two-mode squeezed state, in which modes k and −k are entangled in the Einstein–Podolsky–Rosen sense. Could that entanglement be witnessed in CMB data, and would a positive result say anything about gravity rather than about the inflaton field?

The question was suggested to me by Marios Christodoulou, whose work concerns testing the quantum nature of gravity with table-top experiments. I have no results of my own yet and am working toward a conference poster. Background: my inflationary cosmology notes and slides.

The obstruction

CMB maps are effectively position-type, equal-time measurements, while a witness needs a non-commuting, momentum-like variable as well. Outside the horizon ζ is conserved, and its conjugate momentum is suppressed by the same squeezing that makes ζ so robust an observable (Maldacena; Martin and Vennin). Christodoulou suggested that differences of measurements at different redshifts might supply the missing variable, which connects to Leggett–Garg-type tests, where one system is measured at several times.

Questions I plan to work on

  • Whether unequal-time observables, such as redshift tomography or Leggett–Garg-type constructions, can supply a usable non-commuting variable for ζ.
  • Whether Ireland and Vennin's Wigner-negativity result for the EFT Goldstone mode transfers to the curvature perturbation at finite wavenumber.
  • Whether the delocalised-source construction of Chen and Giacomini, which goes beyond the Newtonian potential in the laboratory, has a cosmological analogue. I have not found prior work on this.
Reading notes and references
  • Decoherence and non-Gaussianity. Squeezing alone does not decohere a closed system (Hsiang and Hu), and Gaussian squeezed states have non-negative Wigner functions, so genuine nonclassicality must come from non-Gaussian structure (Ireland and Vennin) or from environmental effects (Burgess et al.).
  • Framing. The (k, −k) split is not a privileged bipartition: a rotation to standing-wave modes makes the same state a product state. The squeezing is driven by a classical background, as in parametric down-conversion with a classical pump. Real-space entanglement is generic to any quantum field, and equal-time statistics of ζ alone always have a classical stand-in, because the observables commute.
  • The inference to gravity. Anastopoulos and Hu argue that in the weak-field regime such experiments probe only gravity's constraint sector, in analogy with the Coulomb term in QED. Whether entanglement implies a quantum mediator is also under debate following Aziz and Howl (Nature, 2025).

References

  • J. Maldacena, A model with cosmological Bell inequalities, Fortsch. Phys. 64, 10 (2016), arXiv:1508.01082.
  • J. Martin, V. Vennin, Obstructions to Bell CMB Experiments, Phys. Rev. D 96, 063501 (2017), arXiv:1706.05001.
  • J.-T. Hsiang, B.-L. Hu, No Intrinsic Decoherence of Inflationary Cosmological Perturbations, Universe 8, 27 (2022), arXiv:2112.04092.
  • C. P. Burgess, R. Holman, G. Kaplanek, J. Martin, V. Vennin, Minimal decoherence from inflation, JCAP 07 (2023) 022, arXiv:2211.11046.
  • A. Ireland, V. Vennin, When inflationary perturbations refuse to classicalise: the role of non-Gaussianity in Wigner negativity, JCAP 07 (2026) 042, arXiv:2601.22219.
  • C. Anastopoulos, B.-L. Hu, Comment on "A Spin Entanglement Witness for Quantum Gravity" and on "Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity", arXiv:1804.11315.
  • L.-Q. Chen, F. Giacomini, Quantum effects in gravity beyond the Newton potential from a delocalised quantum source, Phys. Rev. X 15, 031063 (2025), arXiv:2402.10288.
  • S. Aziz, R. Howl, Classical theories of gravity produce entanglement, Nature 646, 813 (2025).

Magnetic fields from ultralight dark matter

Undergraduate research, supervised by Prof. Koushik Dutta (IISER Kolkata) Aug 2025 – Jul 2026Completed

The origin of cosmic magnetic fields is unknown. This project asked whether an oscillating ultralight dark matter field can generate them after recombination. Brandenberger, Fröhlich and Jiao find of order 10−15 G at 1 Mpc; Sharma, Majumdar and Sachdeva then showed that the residual ionization left after recombination makes the gauge field heavily overdamped and suppresses the resonance. Two workarounds have since been proposed.

My derivation notes: UL-Scalar DM (PDF) and ALP-DM (PDF), also listed on the Notes page.

What I did. I worked through this literature line by line and wrote step-by-step derivation notes for both the scalar and the ALP couplings (linked above). I also studied the finite-conductivity critique closely and drafted proposals for follow-up directions, some of which are listed below.

Directions I would pursue next

  • Repeating the conductivity analysis for the channels it has not yet been applied to: the scalar/dilaton coupling, the narrow-band resonance, and the non-Abelian cascade.
  • Following the ionization fraction through recombination, to see whether a brief window exists where σ/H is low enough for efficient resonance, instead of using the frozen-out relic value.
  • Treating dark matter density fluctuations as noise on the driving term, which should enlarge the Floquet exponent and may soften the conductivity bound.
Mechanism, obstruction and references

Observational bounds. Blazar observations point to a lower bound of roughly 10−17 G on scales of about 0.1 Mpc and above, inferred from the absence of expected GeV–TeV cascade emission; CMB observations give an upper bound of roughly 10−9 G. Inflationary and phase-transition mechanisms both struggle to deliver a large enough coherence length.

Mechanism. Dark matter is an axion-like field ϕ coupled to photons through ϕ F F̃. Once the Hubble rate falls below the field's mass, ϕ oscillates, making one circular polarization of the gauge field tachyonically unstable during part of each period. The resonance produces a helical magnetic field peaked near kc ≃ g m ϕ0 a, after recombination, when Thomson scattering no longer screens the photon. The field at 1 Mpc depends steeply on the coupling and on a backreaction parameter F, the fraction of the dark matter energy that can drain into photons before the resonance shuts off; the original paper leaves F free. Follow-up papers extend the idea to scalar (dilaton-like) and narrow-resonance channels and to a non-Abelian dark matter condensate.

Obstruction. Recombination is never complete. A residual free-electron fraction of about 10−4 gives the post-recombination universe a conductivity σ that exceeds the Hubble rate by roughly 26 orders of magnitude. The gauge-field equation is then heavily overdamped, and the growth rate is suppressed by a factor of order H/σ instead of growing exponentially; a growth rate above the Hubble rate would require a coupling several orders of magnitude above what observations allow. Sharma et al. confirm this numerically with the Pencil Code. Two proposed ways around it are to move the resonance to a dark photon and transfer the field by kinetic mixing (Alexander et al.), and to produce the field earlier, during radiation domination, using a delayed oscillation (Gerlach, Schmitt and Schwaller).

References

  • R. Brandenberger, J. Fröhlich, H. Jiao, Cosmological Magnetic Fields from Ultralight Dark Matter, Phys. Rev. Lett. 136, 031001 (2026), arXiv:2502.19310.
  • V. Kamali, R. Brandenberger, Late Time Magnetogenesis from Ultralight Scalar Dark Matter, arXiv:2601.00443.
  • S. Brahma, R. Brandenberger, Parametric Resonance and Backreaction Effects in Magnetogenesis from Ultralight Dark Matter, arXiv:2602.04285.
  • Daniel, Kamali, Brandenberger, Indirect Parametric Resonance of the Electromagnetic Field Driven by an Oscillating SU(2) Dark Matter Condensate, arXiv:2605.20333.
  • Sharma, Majumdar, Sachdeva, Suppressed Magnetogenesis from Ultralight Dark Matter due to Finite Conductivity, arXiv:2604.17230.
  • Alexander, Baksh, Edmond, Sun, Symbiotic Magnetogenesis during Radiation Domination, arXiv:2606.25171.
  • Gerlach, Schmitt, Schwaller, Audible Axion Magnetogenesis: Linking Intergalactic Magnetic Fields and Gravitational Waves, arXiv:2605.21092.

Collaborations

I welcome discussions about physics and philosophy. My CV has the full record. You can reach me at rajgauravmirp@gmail.com.