737 research outputs found
Improved Quantum Communication Complexity Bounds for Disjointness and Equality
We prove new bounds on the quantum communication complexity of the
disjointness and equality problems. For the case of exact and non-deterministic
protocols we show that these complexities are all equal to n+1, the previous
best lower bound being n/2. We show this by improving a general bound for
non-deterministic protocols of de Wolf. We also give an O(sqrt{n}c^{log^*
n})-qubit bounded-error protocol for disjointness, modifying and improving the
earlier O(sqrt{n}log n) protocol of Buhrman, Cleve, and Wigderson, and prove an
Omega(sqrt{n}) lower bound for a large class of protocols that includes the
BCW-protocol as well as our new protocol.Comment: 11 pages LaTe
Causality and Cirel'son bounds
An EPR-Bell type experiment carried out on an entangled quantum system can
produce correlations stronger than allowed by local realistic theories. However
there are correlations that are no-signaling and are more non local than the
quantum correlations. Here we show that any correlations more non local than
those achievable in an EPR-Bell type experiment necessarily allow -in the
context of the quantum formalism- both for signaling and for generation of
entanglement. We use our approach to rederive Cirel'son bound for the CHSH
expression, and we derive a new Cirel'son type bound for qutrits. We discuss in
detail the interpretation of our approach.Comment: 5 page
Tsirelson's bound and supersymmetric entangled states
A superqubit, belonging to a -dimensional super-Hilbert space,
constitutes the minimal supersymmetric extension of the conventional qubit. In
order to see whether superqubits are more nonlocal than ordinary qubits, we
construct a class of two-superqubit entangled states as a nonlocal resource in
the CHSH game. Since super Hilbert space amplitudes are Grassmann numbers, the
result depends on how we extract real probabilities and we examine three
choices of map: (1) DeWitt (2) Trigonometric (3) Modified Rogers. In cases (1)
and (2) the winning probability reaches the Tsirelson bound
of standard quantum mechanics. Case (3)
crosses Tsirelson's bound with . Although all states used
in the game involve probabilities lying between 0 and 1, case (3) permits other
changes of basis inducing negative transition probabilities.Comment: Updated to match published version. Minor modifications. References
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Depth-Independent Lower bounds on the Communication Complexity of Read-Once Boolean Formulas
We show lower bounds of and on the
randomized and quantum communication complexity, respectively, of all
-variable read-once Boolean formulas. Our results complement the recent
lower bound of by Leonardos and Saks and
by Jayram, Kopparty and Raghavendra for
randomized communication complexity of read-once Boolean formulas with depth
. We obtain our result by "embedding" either the Disjointness problem or its
complement in any given read-once Boolean formula.Comment: 5 page
Berry phase for a spin 1/2 in a classical fluctuating field
The effect of fluctuations in the classical control parameters on the Berry
phase of a spin 1/2 interacting with a adiabatically cyclically varying
magnetic field is analyzed. It is explicitly shown that in the adiabatic limit
dephasing is due to fluctuations of the dynamical phase.Comment: 4 pages, 1 figure, published versio
Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities
Quantum theory imposes a strict limit on the strength of non-local
correlations. It only allows for a violation of the CHSH inequality up to the
value 2 sqrt(2), known as Tsirelson's bound. In this note, we consider
generalized CHSH inequalities based on many measurement settings with two
possible measurement outcomes each. We demonstrate how to prove Tsirelson
bounds for any such generalized CHSH inequality using semidefinite programming.
As an example, we show that for any shared entangled state and observables
X_1,...,X_n and Y_1,...,Y_n with eigenvalues +/- 1 we have | + <X_2
Y_1> + + + ... + - | <= 2 n
cos(pi/(2n)). It is well known that there exist observables such that equality
can be achieved. However, we show that these are indeed optimal. Our approach
can easily be generalized to other inequalities for such observables.Comment: 9 pages, LateX, V2: Updated reference [3]. To appear in Physical
Review
Substituting a qubit for an arbitrarily large number of classical bits
We show that a qubit can be used to substitute for an arbitrarily large
number of classical bits. We consider a physical system S interacting locally
with a classical field phi(x) as it travels directly from point A to point B.
The field has the property that its integrated value is an integer multiple of
some constant. The problem is to determine whether the integer is odd or even.
This task can be performed perfectly if S is a qubit. On the otherhand, if S is
a classical system then we show that it must carry an arbitrarily large amount
of classical information. We identify the physical reason for such a huge
quantum advantage, and show that it also implies a large difference between the
size of quantum and classical memories necessary for some computations. We also
present a simple proof that no finite amount of one-way classical communication
can perfectly simulate the effect of quantum entanglement.Comment: 8 pages, LaTeX, no figures. v2: added result on entanglement
simulation with classical communication; v3: minor correction to main proof,
change of title, added referenc
Energy Requirements for Quantum Data Compression and 1-1 Coding
By looking at quantum data compression in the second quantisation, we present
a new model for the efficient generation and use of variable length codes. In
this picture lossless data compression can be seen as the {\em minimum energy}
required to faithfully represent or transmit classical information contained
within a quantum state.
In order to represent information we create quanta in some predefined modes
(i.e. frequencies) prepared in one of two possible internal states (the
information carrying degrees of freedom). Data compression is now seen as the
selective annihilation of these quanta, the energy of whom is effectively
dissipated into the environment. As any increase in the energy of the
environment is intricately linked to any information loss and is subject to
Landauer's erasure principle, we use this principle to distinguish lossless and
lossy schemes and to suggest bounds on the efficiency of our lossless
compression protocol.
In line with the work of Bostr\"{o}m and Felbinger \cite{bostroem}, we also
show that when using variable length codes the classical notions of prefix or
uniquely decipherable codes are unnecessarily restrictive given the structure
of quantum mechanics and that a 1-1 mapping is sufficient. In the absence of
this restraint we translate existing classical results on 1-1 coding to the
quantum domain to derive a new upper bound on the compression of quantum
information. Finally we present a simple quantum circuit to implement our
scheme.Comment: 10 pages, 5 figure
Lower bounds in the quantum cell probe model
We introduce a new model for studying quantum data structure problems --- the "quantum cell probe model". We prove a lower bound for the static predecessor problem in the 'address-only' version of this model where, essentially, we allow quantum parallelism only over the 'address lines' of the queries. This model subsumes the classical cell probe model, and many quantum query algorithms like Grover's algorithm fall into this framework. We prove our lower bound by obtaining a round elimination lemma for quantum communication complexity. A similar lemma was proved by Miltersen, Nisan, Safra and Wigderson for classical communication complexity, but their proof does not generalise to the quantum setting. We also study the static membership problem in the quantum cell probe model. Generalising a result of Yao, we show that if the storage scheme is 'implicit', that is it can only store members of the subset and 'pointers', then any quantum query scheme must make \Omega(\log n) probes. We also consider the one-round quantum communication complexity of set membership and show tight bounds
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