129,845 research outputs found
Quantum Spectrum Testing
In this work, we study the problem of testing properties of the spectrum of a
mixed quantum state. Here one is given copies of a mixed state
and the goal is to distinguish whether 's
spectrum satisfies some property or is at least -far in
-distance from satisfying . This problem was promoted in
the survey of Montanaro and de Wolf under the name of testing unitarily
invariant properties of mixed states. It is the natural quantum analogue of the
classical problem of testing symmetric properties of probability distributions.
Here, the hope is for algorithms with subquadratic copy complexity in the
dimension . This is because the "empirical Young diagram (EYD) algorithm"
can estimate the spectrum of a mixed state up to -accuracy using only
copies. In this work, we show that given a
mixed state : (i) copies
are necessary and sufficient to test whether is the maximally mixed
state, i.e., has spectrum ; (ii)
copies are necessary and sufficient to test with
one-sided error whether has rank , i.e., has at most nonzero
eigenvalues; (iii) copies are necessary and
sufficient to distinguish whether is maximally mixed on an
-dimensional or an -dimensional subspace; and (iv) The EYD
algorithm requires copies to estimate the spectrum of
up to -accuracy, nearly matching the known upper bound. In
addition, we simplify part of the proof of the upper bound. Our techniques
involve the asymptotic representation theory of the symmetric group; in
particular Kerov's algebra of polynomial functions on Young diagrams.Comment: 70 pages, 6 figure
Multiple Quantum Hypothesis Testing Expressions and Classical-Quantum Channel Converse Bounds
Alternative exact expressions are derived for the minimum error probability
of a hypothesis test discriminating among quantum states. The first
expression corresponds to the error probability of a binary hypothesis test
with certain parameters; the second involves the optimization of a given
information-spectrum measure. Particularized in the classical-quantum channel
coding setting, this characterization implies the tightness of two existing
converse bounds; one derived by Matthews and Wehner using hypothesis-testing,
and one obtained by Hayashi and Nagaoka via an information-spectrum approach.Comment: Presented at the 2016 IEEE International Symposium on Information
Theory, July 10-15, 2016, Barcelona, Spai
General formulas for capacity of classical-quantum channels
The capacity of a classical-quantum channel (or in other words the classical
capacity of a quantum channel) is considered in the most general setting, where
no structural assumptions such as the stationary memoryless property are made
on a channel. A capacity formula as well as a characterization of the strong
converse property is given just in parallel with the corresponding classical
results of Verd\'{u}-Han which are based on the so-called information-spectrum
method. The general results are applied to the stationary memoryless case with
or without cost constraint on inputs, whereby a deep relation between the
channel coding theory and the hypothesis testing for two quantum states is
elucidated. no structural assumptions such as the stationary memoryless
property are made on a channel. A capacity formula as well as a
characterization of the strong converse property is given just in parallel with
the corresponding classical results of Verdu-Han which are based on the
so-called information-spectrum method. The general results are applied to the
stationary memoryless case with or without cost constraint on inputs, whereby a
deep relation between the channel coding theory and the hypothesis testing for
two quantum states is elucidated
Effective temperature for black holes
The physical interpretation of black hole's quasinormal modes is fundamental
for realizing unitary quantum gravity theory as black holes are considered
theoretical laboratories for testing models of such an ultimate theory and
their quasinormal modes are natural candidates for an interpretation in terms
of quantum levels. The spectrum of black hole's quasinormal modes can be
re-analysed by introducing a black hole's effective temperature which takes
into account the fact that, as shown by Parikh and Wilczek, the radiation
spectrum cannot be strictly thermal. This issue changes in a fundamental way
the physical understanding of such a spectrum and enables a re-examination of
various results in the literature which realizes important modifies on quantum
physics of black holes. In particular, the formula of the horizon's area
quantization and the number of quanta of area result modified becoming
functions of the quantum "overtone" number n. Consequently, the famous formula
of Bekenstein-Hawking entropy, its sub-leading corrections and the number of
microstates are also modified. Black hole's entropy results a function of the
quantum overtone number too. We emphasize that this is the first time that
black hole's entropy is directly connected with a quantum number. Previous
results in the literature are re-obtained in the limit n \to \infty.Comment: 10 pages,accepted for publication in Journal of High Energy Physics.
Comments are welcom
Quantum Algorithms for Learning and Testing Juntas
In this article we develop quantum algorithms for learning and testing
juntas, i.e. Boolean functions which depend only on an unknown set of k out of
n input variables. Our aim is to develop efficient algorithms:
- whose sample complexity has no dependence on n, the dimension of the domain
the Boolean functions are defined over;
- with no access to any classical or quantum membership ("black-box")
queries. Instead, our algorithms use only classical examples generated
uniformly at random and fixed quantum superpositions of such classical
examples;
- which require only a few quantum examples but possibly many classical
random examples (which are considered quite "cheap" relative to quantum
examples).
Our quantum algorithms are based on a subroutine FS which enables sampling
according to the Fourier spectrum of f; the FS subroutine was used in earlier
work of Bshouty and Jackson on quantum learning. Our results are as follows:
- We give an algorithm for testing k-juntas to accuracy that uses
quantum examples. This improves on the number of examples used
by the best known classical algorithm.
- We establish the following lower bound: any FS-based k-junta testing
algorithm requires queries.
- We give an algorithm for learning -juntas to accuracy that
uses quantum examples and
random examples. We show that this learning algorithms is close to optimal by
giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum
Information Processin
Tunable Quantum Beam Splitters for Coherent Manipulation of a Solid-State Tripartite Qubit System
Coherent control of quantum states is at the heart of implementing
solid-state quantum processors and testing quantum mechanics at the macroscopic
level. Despite significant progress made in recent years in controlling single-
and bi-partite quantum systems, coherent control of quantum wave function in
multipartite systems involving artificial solid-state qubits has been hampered
due to the relatively short decoherence time and lacking of precise control
methods. Here we report the creation and coherent manipulation of quantum
states in a tripartite quantum system, which is formed by a superconducting
qubit coupled to two microscopic two-level systems (TLSs). The avoided
crossings in the system's energy-level spectrum due to the qubit-TLS
interaction act as tunable quantum beam splitters of wave functions. Our result
shows that the Landau-Zener-St\"{u}ckelberg interference has great potential in
the precise control of the quantum states in the tripartite system.Comment: 24 pages, 3 figure
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