We determine the optimal measurement that maximizes the average information gain about the state of a qubit system. The qubit is prepared in one of two known states with known prior probabilities. To ...treat the problem analytically we employ the formalism developed for the maximum confidence quantum state discrimination strategy and obtain the POVM which optimizes the information gain for the entire parameter space of the system. We show that the optimal measurement coincides exactly with the minimum-error quantum measurement only for two pure states, or when the two states have the same Bloch radius or they are on the same diagonal of the Bloch disk.
In this paper, we estimate the difference |
E
h
(
Z
n
)
−
E
h
(
Y
)
|
between the expectations of real finite Lipschitz function
h
of the sum
Z
n
= (
X
1
+ ⋯ +
X
n
)
/B
n
, where
B
n
2
=
E
(
X
1
+ ⋯ ...+
X
n
)
2
>
0, and a standard normal random variable
Y
, where real centered random variables
X
1
,X
2
,
… satisfy the
φ
-mixing condition, defined between the “past” and “ future”, or are
m
-dependent. In particular cases, under the condition
∑
r
=
1
∞
r
φ
(
r
)
<
∞
or
∑
r
=
1
∞
r
φ
1
/
2
(
r
)
<
∞
, the obtained upper bounds for
φ
-mixing random variables are of order
O
(
n
−
1
/
2
). In addition, we refine the previously known upper bounds of order
O
((
m
+ 1)
1+
δ
L
2+
δ,n
), where
L
2+
δ,n
is the Lyapunov fraction of order 2 +
δ
, for
m
-dependent random variables, supplementing them with explicit constants. We also separately present the case of independent r.v.s.
This paper analyzes the information disclosure problems originated in economics through the lens of information theory. Such problems are radically different from the conventional communication ...paradigms in information theory since they involve different objectives for the encoder and the decoder, which are aware of this mismatch and act accordingly. This leads, in our setting, to a hierarchical communication game, where the transmitter announces an encoding strategy with full commitment, and its distortion measure depends on a private information sequence whose realization is available at the transmitter. The receiver decides on its decoding strategy that minimizes its own distortion based on the announced encoding map and the statistics. Three problem settings are considered, focusing on the quadratic distortion measures, and jointly Gaussian source and private information: compression, communication, and the simple equilibrium conditions without any compression or communication. The equilibrium strategies and associated costs are characterized. The analysis is then extended to the receiver side information setting and the major changes in structure of optimal strategies are identified. Finally, an extension of the results to the broader context of decentralized stochastic control is presented.
The root-mean-squared error (RMSE) and mean absolute error (MAE) are widely used metrics for evaluating models. Yet, there remains enduring confusion over their use, such that a standard practice is ...to present both, leaving it to the reader to decide which is more relevant. In a recent reprise to the 200-year debate over their use, Willmott and Matsuura (2005) and Chai and Draxler (2014) give arguments for favoring one metric or the other. However, this comparison can present a false dichotomy. Neither metric is inherently better: RMSE is optimal for normal (Gaussian) errors, and MAE is optimal for Laplacian errors. When errors deviate from these distributions, other metrics are superior.
More than 50 years ago, John Bell proved that no theory of nature that obeys locality and realism can reproduce all the predictions of quantum theory: in any local-realist theory, the correlations ...between outcomes of measurements on distant particles satisfy an inequality that can be violated if the particles are entangled. Numerous Bell inequality tests have been reported; however, all experiments reported so far required additional assumptions to obtain a contradiction with local realism, resulting in 'loopholes'. Here we report a Bell experiment that is free of any such additional assumption and thus directly tests the principles underlying Bell's inequality. We use an event-ready scheme that enables the generation of robust entanglement between distant electron spins (estimated state fidelity of 0.92 ± 0.03). Efficient spin read-out avoids the fair-sampling assumption (detection loophole), while the use of fast random-basis selection and spin read-out combined with a spatial separation of 1.3 kilometres ensure the required locality conditions. We performed 245 trials that tested the CHSH-Bell inequality S ≤ 2 and found S = 2.42 ± 0.20 (where S quantifies the correlation between measurement outcomes). A null-hypothesis test yields a probability of at most P = 0.039 that a local-realist model for space-like separated sites could produce data with a violation at least as large as we observe, even when allowing for memory in the devices. Our data hence imply statistically significant rejection of the local-realist null hypothesis. This conclusion may be further consolidated in future experiments; for instance, reaching a value of P = 0.001 would require approximately 700 trials for an observed S = 2.4. With improvements, our experiment could be used for testing less-conventional theories, and for implementing device-independent quantum-secure communication and randomness certification.
Let
μ
N
be the empirical measure associated to a
N
-sample of a given probability distribution
μ
on
R
d
. We are interested in the rate of convergence of
μ
N
to
μ
, when measured in the Wasserstein ...distance of order
p
>
0
. We provide some satisfying non-asymptotic
L
p
-bounds and concentration inequalities, for any values of
p
>
0
and
d
≥
1
. We extend also the non asymptotic
L
p
-bounds to stationary
ρ
-mixing sequences, Markov chains, and to some interacting particle systems.
In this paper, we investigate the precise large deviations for sums of straight phi-mixing and UND random variables with long-tailed distributions. The asymptotic relations for non random sum and ...random sum of random variables with long-tailed distributions are obtained.
Let
{
X
,
X
n
,
n
≥
1
}
be a sequence of pairwise NQD identically distributed random variables and
{
b
n
,
n
≥
1
}
be a sequence of positive constants. In this article, we study the strong laws of ...large numbers for the sequence
{
X
,
X
n
,
n
≥
1
}
, under the general moment condition
∑
n
=
1
∞
P
(
|
X
|
>
b
n
/
log
n
)
<
∞
, which improve some known results.
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