106 research outputs found
Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication
This paper studies the difficulty of discriminating between an arbitrary
quantum channel and a "replacer" channel that discards its input and replaces
it with a fixed state. We show that, in this particular setting, the most
general adaptive discrimination strategies provide no asymptotic advantage over
non-adaptive tensor-power strategies. This conclusion follows by proving a
quantum Stein's lemma for this channel discrimination setting, showing that a
constant bound on the Type I error leads to the Type II error decreasing to
zero exponentially quickly at a rate determined by the maximum relative entropy
registered between the channels. The strong converse part of the lemma states
that any attempt to make the Type II error decay to zero at a rate faster than
the channel relative entropy implies that the Type I error necessarily
converges to one. We then refine this latter result by identifying the optimal
strong converse exponent for this task. As a consequence of these results, we
can establish a strong converse theorem for the quantum-feedback-assisted
capacity of a channel, sharpening a result due to Bowen. Furthermore, our
channel discrimination result demonstrates the asymptotic optimality of a
non-adaptive tensor-power strategy in the setting of quantum illumination, as
was used in prior work on the topic. The sandwiched Renyi relative entropy is a
key tool in our analysis. Finally, by combining our results with recent results
of Hayashi and Tomamichel, we find a novel operational interpretation of the
mutual information of a quantum channel N as the optimal type II error exponent
when discriminating between a large number of independent instances of N and an
arbitrary "worst-case" replacer channel chosen from the set of all replacer
channels.Comment: v3: 35 pages, 4 figures, accepted for publication in Communications
in Mathematical Physic
Quantum state discrimination bounds for finite sample size
In the problem of quantum state discrimination, one has to determine by
measurements the state of a quantum system, based on the a priori side
information that the true state is one of two given and completely known
states, rho or sigma. In general, it is not possible to decide the identity of
the true state with certainty, and the optimal measurement strategy depends on
whether the two possible errors (mistaking rho for sigma, or the other way
around) are treated as of equal importance or not. Results on the quantum
Chernoff and Hoeffding bounds and the quantum Stein's lemma show that, if
several copies of the system are available then the optimal error probabilities
decay exponentially in the number of copies, and the decay rate is given by a
certain statistical distance between rho and sigma (the Chernoff distance, the
Hoeffding distances, and the relative entropy, respectively). While these
results provide a complete solution to the asymptotic problem, they are not
completely satisfying from a practical point of view. Indeed, in realistic
scenarios one has access only to finitely many copies of a system, and
therefore it is desirable to have bounds on the error probabilities for finite
sample size. In this paper we provide finite-size bounds on the so-called Stein
errors, the Chernoff errors, the Hoeffding errors and the mixed error
probabilities related to the Chernoff and the Hoeffding errors.Comment: 31 pages. v4: A few typos corrected. To appear in J.Math.Phy
On the quantum Renyi relative entropies and related capacity formulas
We show that the quantum -relative entropies with parameter
can be represented as generalized cutoff rates in the sense
of [I. Csiszar, IEEE Trans. Inf. Theory 41, 26-34, (1995)], which provides a
direct operational interpretation to the quantum -relative entropies.
We also show that various generalizations of the Holevo capacity, defined in
terms of the -relative entropies, coincide for the parameter range
, and show an upper bound on the one-shot epsilon-capacity of
a classical-quantum channel in terms of these capacities.Comment: v4: Cutoff rates are treated for correlated hypotheses, some proofs
are given in greater detai
Entropy growth of shift-invariant states on a quantum spin chain
We study the entropy of pure shift-invariant states on a quantum spin chain.
Unlike the classical case, the local restrictions to intervals of length
are typically mixed and have therefore a non-zero entropy which is,
moreover, monotonically increasing in . We are interested in the asymptotics
of the total entropy. We investigate in detail a class of states derived from
quasi-free states on a CAR algebra. These are characterised by a measurable
subset of the unit interval. As the entropy density is known to vanishes,
is sublinear in . For states corresponding to unions of finitely many
intervals, is shown to grow slower than . Numerical
calculations suggest a behaviour. For the case with infinitely many
intervals, we present a class of states for which the entropy increases
as where can take any value in .Comment: 18 pages, 2 figure
Quantum hypothesis testing with group symmetry
The asymptotic discrimination problem of two quantum states is studied in the
setting where measurements are required to be invariant under some symmetry
group of the system. We consider various asymptotic error exponents in
connection with the problems of the Chernoff bound, the Hoeffding bound and
Stein's lemma, and derive bounds on these quantities in terms of their
corresponding statistical distance measures. A special emphasis is put on the
comparison of the performances of group-invariant and unrestricted
measurements.Comment: 33 page
Stationary quantum source coding
In this paper the quantum source coding theorem is obtained for a completely
ergodic source. This results extends Shannon's classical theorem as well as
Schumacher's quantum noiseless coding theorem for memoryless sources. The
control of the memory effects requires earlier results of Hiai and Petz on high
probability subspaces.Comment: 8 page
Structure of sufficient quantum coarse-grainings
Let H and K be Hilbert spaces and T be a coarse-graining from B(H) to B(K).
Assume that density matrices D_1 and D_2 acting on H are given. In the paper
the consequences of the existence of a coarse-graining S from B(K) to B(H)
satisfying ST(D_1)=D_1 and ST(D_2)=D_2 are given. (This condition means the
sufficiency of T for D_1 and D_2.) Sufficiency implies a particular
decomposition of the density matrices. This decomposition allows to deduce the
exact condition for equality in the strong subadditivity of the von Neumann
entropy.Comment: 13 pages, LATE
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