38 research outputs found
On the role of designs in the data-driven approach to quantum statistical inference
Designs, and in particular symmetric, informationally complete (SIC)
structures, play an important role in the quantum tomographic reconstruction
process and, by extension, in certain interpretations of quantum theory
focusing on such a process. This fact is due to the symmetry of the
reconstruction formula that designs lead to. However, it is also known that the
same tomographic task, albeit with a less symmetric formula, can be
accomplished by any informationally complete (non necessarily symmetric)
structure. Here we show that, if the tomographic task is replaced by a
data-driven inferential approach, the reconstruction, while possible with
designs, cannot by accomplished anymore by an arbitrary informationally
complete structure. Hence, we propose the data-driven inference as the arena in
which the role of designs naturally emerges. Our inferential approach is based
on a minimality principle according to which, among all the possible inferences
consistent with the data, the weakest should be preferred, in the sense of
majorization theory and statistical comparison.Comment: 6 pages, 1 figur
Quantum guesswork
The guesswork quantifies the minimum cost incurred in guessing the state of a
quantum ensemble, when only one state can be queried at a time. Here, we derive
the guesswork for a broad class of ensembles and cost functions.Comment: 6 page
The signaling dimension of physical systems
The signaling dimension of a physical system is the minimum dimension of a
classical channel that can reproduce the set of input-output correlations
attainable by the given system. Here we put the signaling dimension into
perspective by reviewing some of the main known results on the topic, starting
from Frenkel and Weiner's 2015 breakthrough showing that the signaling
dimension of any quantum system is equal to its Hilbert space dimension.Comment: 6 pages, Perspective on P. Frenkel, Quantum 6, 751 (2022
Quantum conditional operations
An essential element of classical computation is the "if-then" construct,
that accepts a control bit and an arbitrary gate, and provides conditional
execution of the gate depending on the value of the controlling bit. On the
other hand, quantum theory prevents the existence of an analogous universal
construct accepting a control qubit and an arbitrary quantum gate as its input.
Nevertheless, there are controllable sets of quantum gates for which such a
construct exists. Here we provide a necessary and sufficient condition for a
set of unitary transformations to be controllable, and we give a complete
characterization of controllable sets in the two dimensional case. This result
reveals an interesting connection between the problem of controllability and
the problem of extracting information from an unknown quantum gate while using
it.Comment: 7 page
Tight conic approximation of testing regions for quantum statistical models and measurements
Quantum statistical models (i.e., families of normalized density matrices)
and quantum measurements (i.e., positive operator-valued measures) can be
regarded as linear maps: the former, mapping the space of effects to the space
of probability distributions; the latter, mapping the space of states to the
space of probability distributions. The images of such linear maps are called
the testing regions of the corresponding model or measurement. Testing regions
are notoriously impractical to treat analytically in the quantum case. Our
first result is to provide an implicit outer approximation of the testing
region of any given quantum statistical model or measurement in any finite
dimension: namely, a region in probability space that contains the desired
image, but is defined implicitly, using a formula that depends only on the
given model or measurement. The outer approximation that we construct is
minimal among all such outer approximations, and close, in the sense that it
becomes the maximal inner approximation up to a constant scaling factor.
Finally, we apply our approximation formulas to characterize, in a semi-device
independent way, the ability to transform one quantum statistical model or
measurement into another.Comment: 9 pages, 1 figur