Generalized Contextuality via �EquiRank NMF
Farid Shahandeh, Theo Yianni, Mina Doosti
[2406.19382] Characterizing Contextuality via Rank Separation with Applications to Cloning
[2506.09133] Complexity of Contextuality.
ESSLLI 2025, Theory and applications of sheaf theory, Bochum, August 8, 2025
story-tellers
is “history” real?
there are “facts”
I like to think that “physics” is “telling a story about continuously occurring phenomena.” And I don’t think there are “rules of nature.” We create rules to tell consistent stories.
prepare-measure scenario
Spekkens (2005), Chiribella et al. (2010), Barrett, PRA (2007)
prepare-measure scenario
Preparation | |
Turn on the laser (with characteristics X, Y, Z, …) AND align it such and such AND pass the beam through a polarizer. | |
Turn on the laser (with characteristics X, Y, Z, …) AND align it such and such AND reflect the beam off a polarizing beamsplitter. | |
Toss a coin AND do (1) above for heads and (2) above for tails. | |
… | |
Spekkens (2005), Chiribella et al. (2010), Barrett, PRA (2007)
prepare-measure scenario
Preparation | Measurement |
Turn on the laser (with characteristics X, Y, Z, …) AND align it such and such AND pass the beam through a polarizer. | Turn on the avalanche photodetector AND connect it to the oscilloscope (w. characteristics X, Y, Z) AND align it as such. |
Turn on the laser (with characteristics X, Y, Z, …) AND align it such and such AND reflect the beam off a polarizing beamsplitter. | Turn on the photon-number-resolving detector AND connect it to the oscilloscope (w. characteristics X’, Y’, Z’) AND align it as such AND coarse-grain the output as such. |
Toss a coin AND do (1) above for heads and (2) above for tails. | Turn the array of APDs AND connect them to oscilloscopes (w. characteristics X, Y, Z) AND align them as such AND postprocess the output as such. |
… | … |
Spekkens (2005), Chiribella et al. (2010), Barrett, PRA (2007)
Prepare-measure scenario
a sentence!
Operational structure
Chiribella et al. (2010)
a sentence!
Probabilistic structure
Chiribella et al. (2010)
matrix of conditional outcome probability evaluations
(COPE)
FS, Yianni, Doost (2406.19382), Harrigan et al. (2008)
Matrix of conditional outcome probability evaluations
(COPE)
FS, Yianni, Doost (2406.19382), Harrigan et al. (2008)
matrix of conditional outcome probability evaluations
(COPE)
preparation
outcome
FS, Yianni, Doost (2406.19382), Harrigan et al. (2008)
matrix of conditional outcome probability evaluations
(COPE)
preparation
outcome
FS, Yianni, Doost (2406.19382), Harrigan et al. (2008)
Models
Models:
OPT (Chiribella et al.)
Quotiented OPT (Chiribella et al.)
preGPT (FS, unpublished)
GPT (quotiented preGPT) (Hardy, Barrett, J&H, FS)
Quasiprobabilistic (Ferrie et al.)
Ontological (Spekkens)
Noncontextual ontological (Spekkens)
Models:
OPT (Chiribella et al.)
Quotiented OPT (Chiribella et al.)
preGPT (FS, unpublished)
GPT (quotiented preGPT) (Hardy, Barrett, J&H, FS)
Quasiprobabilistic (Ferrie et al.)
Ontological (Spekkens)
Noncontextual ontological (Spekkens)
projections,
set-valued embeddings,
general linear
transformations
Models:
OPT (Chiribella et al.)
Quotiented OPT (Chiribella et al.)
preGPT (FS, unpublished)
GPT (quotiented preGPT) (Hardy, Barrett, J&H, FS)
Quasiprobabilistic (Ferrie et al.)
Ontological (Spekkens)
Noncontextual ontological (Spekkens)
projections,
set-valued embeddings,
general linear
transformations
Models:
OPT (Chiribella et al.)
Quotiented OPT (Chiribella et al.)
preGPT (FS, unpublished)
GPT (quotiented preGPT) (Hardy, Barrett, J&H, FS)
Quasiprobabilistic (Ferrie et al.)
Ontological (Spekkens)
Noncontextual ontological (Spekkens)
Models: Decompositions of COPE matrix
Working example: The boxworld
Working example: The boxworld
preparation
Working example: The boxworld
preparation
Working example: The boxworld
preparation
outcome
SVD
GPT
SVD
GPT
SVD
GPT
SVD
GPT
GPT
GPT
GPT
every prep. and outcome has a unique representation!
NMF
Ontological model
FS, Yianni, Doost (2406.19382)
NMF
Ontological model
FS, Yianni, Doost (2406.19382)
NMF
Ontological model
FS, Yianni, Doost (2406.19382)
NMF
Ontological model
FS, Yianni, Doost (2406.19382)
NMF
Ontological model
FS, Yianni, Doost (2406.19382)
NMF
Ontological model
FS, Yianni, Doost (2406.19382)
Ontological model
Ontological model
Ontological model
Ontological model
Ontological model
Generalized contextuality means operationally indiscernible propositions are represented differently in the model.
FS, Yianni, Doost (2406.19382)
generalized contextuality
FS, Yianni, Doost (2406.19382)
Working example: The boxworld
for boxworld, we can satisfy either nonnegativity
or
equirank condition
boxworld does not admit a noncontextual ontological model
Yianni, FS (2506.09133)
Yianni, FS (2506.09133)
Finding equirank NMF is at least exponential in the rank of C. So, it’s very hard!
Yianni, FS (2506.09133)
Sheaf theory?
Agrios (2202.01379)
Bell scenario
A
B
x
y
Bell scenario
Williams, Doosti, FS (coming soon), Constantin (1510.02561)
Bell scenario
Williams, Doosti, FS (coming soon), Constantin (1510.02561)
incidence matrix
global section
Bell scenario
Williams, Doosti, FS (coming soon), Constantin (1510.02561)
Bell scenario
Williams, Doosti, FS (coming soon)
incidence matrix
global sections
more data/preparations
Bell scenario
Williams, Doosti, FS (coming soon)
inner (ontic) dimension is fixed
response functions are fixed
Bell scenario
Williams, Doosti, FS (coming soon)
incidence matrix
Bell scenario
Williams, Doosti, FS (coming soon), Constantin (1510.02561)
global section doesn’t exist
model is sheaf-contextual
Bell scenario
Williams, Doosti, FS (coming soon)
Bell scenario
Williams, Doosti, FS (coming soon)
rank-one response function
rank one epistemic state
Bell scenario
Williams, Doosti, FS (coming soon)
rank-one response function
rank one epistemic state
generalized noncontextual
sheaf-contextuality
generalized contextuality
Williams, Doosti, FS (coming soon)
take-home messages:
Lecturer in Computer Science, RHUL, UK
Closing Date: Friday 29 August 2025