3,796 research outputs found
Duality of Quantum Coherence and Path Distinguishability
We derive a generalized wave-particle duality relation for arbitrary
multipath quantum interference phenomena. Beyond the conventional notion of the
wave nature of a quantum system, i.e., the interference fringe visibility, we
introduce a quantifier as the normalized quantum coherence, recently defined in
the framework of quantum information theory. To witness the particle nature, we
quantify the path distinguishability or the which-path information based on
unambiguous quantum state discrimination. Then, the Bohr complementarity
principle for multipath quantum interference can be stated as a duality
relation between the quantum coherence and the path distinguishability. For
two-path interference, the quantum coherence is identical to the interference
fringe visibility, and the relation reduces to the well-known complementarity
relation. The duality relation continues to hold in the case where mixedness is
introduced due to possible decoherence effects.Comment: 6 pages, 0 figures, close to the published versio
Two-sided estimates of minimum-error distinguishability of mixed quantum states via generalized Holevo-Curlander bounds
We prove a concise factor-of-2 estimate for the failure rate of optimally
distinguishing an arbitrary ensemble of mixed quantum states, generalizing work
of Holevo [Theor. Probab. Appl. 23, 411 (1978)] and Curlander [Ph.D. Thesis,
MIT, 1979]. A modification to the minimal principle of Cocha and Poor
[Proceedings of the 6th International Conference on Quantum Communication,
Measurement, and Computing (Rinton, Princeton, NJ, 2003)] is used to derive a
suboptimal measurement which has an error rate within a factor of 2 of the
optimal by construction. This measurement is quadratically weighted and has
appeared as the first iterate of a sequence of measurements proposed by Jezek
et al. [Phys. Rev. A 65, 060301 (2002)]. Unlike the so-called pretty good
measurement, it coincides with Holevo's asymptotically optimal measurement in
the case of nonequiprobable pure states. A quadratically weighted version of
the measurement bound by Barnum and Knill [J. Math. Phys. 43, 2097 (2002)] is
proven. Bounds on the distinguishability of syndromes in the sense of
Schumacher and Westmoreland [Phys. Rev. A 56, 131 (1997)] appear as a
corollary. An appendix relates our bounds to the trace-Jensen inequality.Comment: It was not realized at the time of publication that the lower bound
of Theorem 10 has a simple generalization using matrix monotonicity (See [J.
Math. Phys. 50, 062102]). Furthermore, this generalization is a trivial
variation of a previously-obtained bound of Ogawa and Nagaoka [IEEE Trans.
Inf. Theory 45, 2486-2489 (1999)], which had been overlooked by the autho
Distinguishability times and asymmetry monotone-based quantum speed limits in the Bloch ball
For both unitary and open qubit dynamics, we compare asymmetry monotone-based
bounds on the minimal time required for an initial qubit state to evolve to a
final qubit state from which it is probabilistically distinguishable with fixed
minimal error probability (i.e., the minimal error distinguishability time).
For the case of unitary dynamics generated by a time-independent Hamiltonian,
we derive a necessary and sufficient condition on two asymmetry monotones that
guarantees that an arbitrary state of a two-level quantum system or a separable
state of two-level quantum systems will unitarily evolve to another state
from which it can be distinguished with a fixed minimal error probability
. This condition is used to order the set of qubit states
based on their distinguishability time, and to derive an optimal release time
for driven two-level systems such as those that occur, e.g., in the
Landau-Zener problem. For the case of non-unitary dynamics, we compare three
lower bounds to the distinguishability time, including a new type of lower
bound which is formulated in terms of the asymmetry of the uniformly
time-twirled initial system-plus-environment state with respect to the
generator of the Stinespring isometry corresponding to the dynamics,
specifically, in terms of ,
where .Comment: 13 pages, 4 figure
Thermal breakdown of coherent backscattering: a case study of quantum duality
We investigate coherent backscattering of light by two harmonically trapped
atoms in the light of quantitative quantum duality. Including recoil and
Doppler shift close to an optical resonance, we calculate the interference
visibility as well as the amount of which-path information, both for zero and
finite temperature.Comment: published version with minor changes and an added figur
A Theoretical Analysis of NDCG Type Ranking Measures
A central problem in ranking is to design a ranking measure for evaluation of
ranking functions. In this paper we study, from a theoretical perspective, the
widely used Normalized Discounted Cumulative Gain (NDCG)-type ranking measures.
Although there are extensive empirical studies of NDCG, little is known about
its theoretical properties. We first show that, whatever the ranking function
is, the standard NDCG which adopts a logarithmic discount, converges to 1 as
the number of items to rank goes to infinity. On the first sight, this result
is very surprising. It seems to imply that NDCG cannot differentiate good and
bad ranking functions, contradicting to the empirical success of NDCG in many
applications. In order to have a deeper understanding of ranking measures in
general, we propose a notion referred to as consistent distinguishability. This
notion captures the intuition that a ranking measure should have such a
property: For every pair of substantially different ranking functions, the
ranking measure can decide which one is better in a consistent manner on almost
all datasets. We show that NDCG with logarithmic discount has consistent
distinguishability although it converges to the same limit for all ranking
functions. We next characterize the set of all feasible discount functions for
NDCG according to the concept of consistent distinguishability. Specifically we
show that whether NDCG has consistent distinguishability depends on how fast
the discount decays, and 1/r is a critical point. We then turn to the cut-off
version of NDCG, i.e., NDCG@k. We analyze the distinguishability of NDCG@k for
various choices of k and the discount functions. Experimental results on real
Web search datasets agree well with the theory.Comment: COLT 201
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