Quantum Blackwell–Sherman–Stein Theorem and Related Results

Kaniowski, Krzysztof; Lubnauer, Katarzyna; Łuczak, Andrzej

doi:10.1142/s1230161213500170

Cited by 5 publications

(3 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We remark that we can construct the theory developed in this section based instead on the loss functional defined by [4,32,43]). Now we prove some elementary properties of the gain functional.…”

Section: Gain Functional and The Blackwell-sherman-stein (Bss) Theoremmentioning

confidence: 99%

Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements

Kuramochi

2020

Preprint

View full text Add to dashboard Cite

We introduce the post-processing preorder and equivalence relations for general measurements on a possibly infinite-dimensional general probabilistic theory described by an order unit Banach space E with a Banach predual. We define the measurement space M(E) as the set of post-processing equivalence classes of continuous measurements on E. We define the weak topology on M(E) as the weakest topology in which the state discrimination probabilities for any finite-label ensembles are continuous and show that M(E) equipped with the convex operation corresponding to the probabilistic mixture of measurements can be regarded as a compact convex set regularly embedded in a locally convex Hausdorff space. We also prove that the measurement space M(E) is infinite-dimensional except when the system is 1-dimensional and give a characterization of the post-processing monotone affine functional. We apply these general results to the problems of simulability and incompatibility of measurements. We show that the robustness measures of unsimulability and incompatibility coincide with the optimal ratio of the state discrimination probability of measurement(s) relative to that of simulable or compatible measurements, respectively. The latter result for incompatible measurements generalizes the recent result for finite-dimensional quantum measurements. Throughout the paper, the fact that any weakly * continuous measurement can be arbitrarily approximated in the weak topology by a post-processing increasing net of finite-outcome measurements is systematically used to reduce the discussions to finite-outcome cases.

show abstract

Section: Gain Functional and The Blackwell-sherman-stein (Bss) Theoremmentioning

confidence: 99%

Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements

Kuramochi

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Another challenging problem is the extension of these results to infinite dimensional Hilbert spaces, or to general von Neumann algebras. Some partial results in this direction were obtained in [16]. Although the methods used in [14,15] rely on finite dimensions, it seems plausible that some of the useful properties of the norms can be extended also to this case.…”

Section: The Randomization Criterion For Quantum Experimentsmentioning

confidence: 99%

Comparison of quantum channels and statistical experiments

Jenčová

2016

2016 IEEE International Symposium on Information Theory (ISIT)

View full text Add to dashboard Cite

For a pair of quantum channels with the same input space, we show that the possibility of approximation of one channel by postprocessings of the other channel can be characterized by comparing the success probabilities for the two ensembles obtained as outputs for any ensemble on the input space coupled with an ancilla. This provides an operational interpretation to a natural extension of Le Cam's deficiency to quantum channels. In particular, we obtain a version of the randomization criterion for quantum statistical experiments. The proofs are based on some properties of the diamond norm and its dual, which are of independent interest.

show abstract

“…We can also ask whether the directed-completeness holds for other preorder relations for statistical experiments, for example that induced by the statistical morphisms. 23,24 As mentioned in Ref. 24 (Section 9), the equivalence by morphism can be defined for general probabilistic theories (GPTs) and it may be natural to consider this problem in the GPT setting.…”

Section: B Ideal Quantum Linear Amplifiermentioning

confidence: 99%

Directed-completeness of quantum statistical experiments in the randomization order

Kuramochi

2018

Preprint

View full text Add to dashboard Cite

A parametrized family of normal states on a von Neumann algebra is called a statistical experiment, which generalizes the corresponding concepts in classical statistics and finite-dimensional quantum systems. We introduce randomization preorder and equivalence relations for statistical experiments with a fixed parameter set and for normal channels with a fixed input space by post-processing completely positive channels. In this paper, we prove that the set of equivalence classes of statistical experiments or those of normal channels is an upper and lower directed-complete partially ordered set with respect to the randomization order, i.e. any increasing or decreasing net of statistical experiments or channels has its supremum or infimum in the randomization order. We also show that if the outcome space of each statistical experiment or channel of a randomization-monotone net is commutative, the outcome space of the supremum or infimum can also be taken to be commutative.We consider two examples of homogeneous Markov processes of channels on infinitedimensional separable Hilbert spaces, namely block-diagonalization with irrational translation and ideal quantum linear amplifier channels, and explicitly derive their infima. Throughout the paper, the concept of channel conjugation is used to obtain results for decreasing channels from those for increasing channels.

show abstract

Quantum Blackwell–Sherman–Stein Theorem and Related Results

Cited by 5 publications

References 5 publications

Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements

Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements

Comparison of quantum channels and statistical experiments

Directed-completeness of quantum statistical experiments in the randomization order

Contact Info

Product

Resources

About