40 research outputs found
A note on the joint measurability of POVMs and its implications for contextuality
The purpose of this note is to clarify the logical relationship between joint
measurability and contextuality for quantum observables in view of recent
developments [1-4].Comment: are welcome! This note may be revised further in due cours
Minimal state-dependent proof of measurement contextuality for a qubit
We show that three unsharp binary qubit measurements are enough to violate a
generalized noncontextuality inequality, the LSW inequality, in a
state-dependent manner. For the case of trine spin axes we calculate the
optimal quantum violation of this inequality. Besides, we show that unsharp
qubit measurements do not allow a state-independent violation of this
inequality. We thus provide a minimal state-dependent proof of measurement
contextuality requiring one qubit and three unsharp measurements. Our result
rules out generalized noncontextual models of these measurements which were
previously conjectured to exist. More importantly, this class of generalized
noncontextual models includes the traditional Kochen-Specker (KS) noncontextual
models as a proper subset, so our result rules out a larger class of models
than those ruled out by a violation of the corresponding KS-inequality in this
scenario.Comment: 9 pages, 3 figure
From the Kochen-Specker theorem to noncontextuality inequalities without assuming determinism
The Kochen-Specker theorem demonstrates that it is not possible to reproduce
the predictions of quantum theory in terms of a hidden variable model where the
hidden variables assign a value to every projector deterministically and
noncontextually. A noncontextual value-assignment to a projector is one that
does not depend on which other projectors - the context - are measured together
with it. Using a generalization of the notion of noncontextuality that applies
to both measurements and preparations, we propose a scheme for deriving
inequalities that test whether a given set of experimental statistics is
consistent with a noncontextual model. Unlike previous inequalities inspired by
the Kochen-Specker theorem, we do not assume that the value-assignments are
deterministic and therefore in the face of a violation of our inequality, the
possibility of salvaging noncontextuality by abandoning determinism is no
longer an option. Our approach is operational in the sense that it does not
presume quantum theory: a violation of our inequality implies the impossibility
of a noncontextual model for any operational theory that can account for the
experimental observations, including any successor to quantum theory.Comment: 5+8 pages, 4+3 figures. Comments are welcome
From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities
The Kochen-Specker theorem rules out models of quantum theory wherein
projective measurements are assigned outcomes deterministically and
independently of context. This notion of noncontextuality is not applicable to
experimental measurements because these are never free of noise and thus never
truly projective. For nonprojective measurements, therefore, one must drop the
requirement that an outcome is assigned deterministically in the model and
merely require that it is assigned a distribution over outcomes in a manner
that is context-independent. By demanding context-independence in the
representation of preparations as well, one obtains a generalized principle of
noncontextuality that also supports a quantum no-go theorem. Several recent
works have shown how to derive inequalities on experimental data which, if
violated, demonstrate the impossibility of finding a generalized-noncontextual
model of this data. That is, these inequalities do not presume quantum theory
and, in particular, they make sense without requiring an operational analogue
of the quantum notion of projectiveness. We here describe a technique for
deriving such inequalities starting from arbitrary proofs of the Kochen-Specker
theorem. It extends significantly previous techniques that worked only for
logical proofs, which are based on sets of projective measurements that fail to
admit of any deterministic noncontextual assignment, to the case of statistical
proofs, which are based on sets of projective measurements that do admit of
some deterministic noncontextual assignments, but not enough to explain the
quantum statistics.Comment: 14 pages, 4 figures, published versio
Fine's theorem, noncontextuality, and correlations in Specker's scenario
A characterization of noncontextual models which fall within the ambit of
Fine's theorem is provided. In particular, the equivalence between the
existence of three notions is made explicit: a joint probability distribution
over the outcomes of all the measurements considered, a
measurement-noncontextual and outcome-deterministic (or KS-noncontextual, where
'KS' stands for 'Kochen-Specker') model for these measurements, and a
measurement-noncontextual and factorizable model for them. A KS-inequality,
therefore, is implied by each of these three notions. Following this
characterization of noncontextual models that fall within the ambit of Fine's
theorem, non-factorizable noncontextual models which lie outside the domain of
Fine's theorem are considered. While outcome determinism for projective (sharp)
measurements in quantum theory can be shown to follow from the assumption of
preparation noncontextuality, such a justification is not available for
nonprojective (unsharp) measurements which ought to admit
outcome-indeterministic response functions. The Liang-Spekkens-Wiseman (LSW)
inequality is cited as an example of a noncontextuality inequality that should
hold in any noncontextual model of quantum theory without assuming
factorizability. Three other noncontextuality inequalities, which turn out to
be equivalent to the LSW inequality under relabellings of measurement outcomes,
are derived for Specker's scenario. The polytope of correlations admissible in
this scenario, given the no-disturbance condition, is characterized.Comment: 11 pages, comments welcome. Version 2 is essentially the published
versio
Relative volume of separable bipartite states
Every choice of an orthonormal frame in the d-dimensional Hilbert space of a
system corresponds to one set of all mutually commuting density matrices or,
equivalently, a classical statistical state space of the system; the quantum
state space itself can thus be profitably viewed as an SU(d) orbit of classical
state spaces, one for each orthonormal frame. We exploit this connection to
study the relative volume of separable states of a bipartite quantum system.
While the two-qubit case is studied in considerable analytic detail, for higher
dimensional systems we fall back on Monte Carlo. Several new insights seem to
emerge from our study.Comment: Essentially the published versio
Joint measurability structures realizable with qubit measurements: incompatibility via marginal surgery
Measurements in quantum theory exhibit incompatibility, i.e., they can fail
to be jointly measurable. An intuitive way to represent the (in)compatibility
relations among a set of measurements is via a hypergraph representing their
joint measurability structure: its vertices represent measurements and its
hyperedges represent (all and only) subsets of compatible measurements.
Projective measurements in quantum theory realize (all and only) joint
measurability structures that are graphs. On the other hand, general
measurements represented by positive operator-valued measures (POVMs) can
realize arbitrary joint measurability structures. Here we explore the scope of
joint measurability structures realizable with qubit POVMs. We develop a
technique that we term marginal surgery to obtain nontrivial joint
measurability structures starting from a set of compatible measurements. We
show explicit examples of marginal surgery on a special set of qubit POVMs to
construct joint measurability structures such as -cycle and -Specker
scenarios for any integer . We also show the realizability of various
joint measurability structures with vertices. In particular, we
show that all possible joint measurability structures with vertices are
realizable. We conjecture that all joint measurability structures are
realizable with qubit POVMs. This contrasts with the unbounded dimension
required in R. Kunjwal et al., Phys. Rev. A 89, 052126 (2014). Our results also
render this previous construction maximally efficient in terms of the required
Hilbert space dimension. We also obtain a sufficient condition for the joint
measurability of any set of binary qubit POVMs which powers many of our results
and should be of independent interest.Comment: 37 pages, 32 figures, 1 table, comments welcome
Contextuality in entanglement-assisted one-shot classical communication
We consider the problem of entanglement-assisted one-shot classical
communication. In the zero-error regime, entanglement can increase the one-shot
zero-error capacity of a family of classical channels following the strategy of
Cubitt et al., Phys. Rev. Lett. 104, 230503 (2010). This strategy uses the
Kochen-Specker theorem which is applicable only to projective measurements. As
such, in the regime of noisy states and/or measurements, this strategy cannot
increase the capacity. To accommodate generically noisy situations, we examine
the one-shot success probability of sending a fixed number of classical
messages. We show that preparation contextuality powers the quantum advantage
in this task, increasing the one-shot success probability beyond its classical
maximum. Our treatment extends beyond Cubitt et al. and includes, for example,
the experimentally implemented protocol of Prevedel et al., Phys. Rev. Lett.
106, 110505 (2011). We then show a mapping between this communication task and
a corresponding nonlocal game. This mapping generalizes the connection with
pseudotelepathy games previously noted in the zero-error case. Finally, after
motivating a constraint we term context-independent guessing, we show that
contextuality witnessed by noise-robust noncontextuality inequalities obtained
in R. Kunjwal, Quantum 4, 219 (2020), is sufficient for enhancing the one-shot
success probability. This provides an operational meaning to these inequalities
and the associated hypergraph invariant, the weighted max-predictability,
introduced in R. Kunjwal, Quantum 3, 184 (2019). Our results show that the task
of entanglement-assisted oneshot classical communication provides a fertile
ground to study the interplay of the Kochen-Specker theorem, Spekkens
contextuality, and Bell nonlocality.Comment: 22 pages, 3 figures, some discussion added, presentation improved.
Comments welcome
Almost Quantum Correlations are Inconsistent with Specker's Principle
Ernst Specker considered a particular feature of quantum theory to be
especially fundamental, namely that pairwise joint measurability of sharp
measurements implies their global joint measurability
(https://vimeo.com/52923835). To date, Specker's principle seemed incapable of
singling out quantum theory from the space of all general probabilistic
theories. In particular, its well-known consequence for experimental
statistics, the principle of consistent exclusivity, does not rule out the set
of correlations known as almost quantum, which is strictly larger than the set
of quantum correlations. Here we show that, contrary to the popular belief,
Specker's principle cannot be satisfied in any theory that yields almost
quantum correlations.Comment: 17 pages + appendix. 5 colour figures. Comments welcom