7,071 research outputs found
Equality Alone Does not Simulate Randomness
The canonical problem that gives an exponential separation between deterministic and randomized communication complexity in the classical two-party communication model is "Equality". In this work we show that even allowing access to an "Equality" oracle, deterministic protocols remain exponentially weaker than randomized ones. More precisely, we exhibit a total function on n bits with randomized one-sided communication complexity O(log n), but such that every deterministic protocol with access to "Equality" oracle needs Omega(n) cost to compute it.
Additionally we exhibit a natural and strict infinite hierarchy within BPP, starting with the class P^{EQ} at its bottom
Zero-error channel capacity and simulation assisted by non-local correlations
Shannon's theory of zero-error communication is re-examined in the broader
setting of using one classical channel to simulate another exactly, and in the
presence of various resources that are all classes of non-signalling
correlations: Shared randomness, shared entanglement and arbitrary
non-signalling correlations. Specifically, when the channel being simulated is
noiseless, this reduces to the zero-error capacity of the channel, assisted by
the various classes of non-signalling correlations. When the resource channel
is noiseless, it results in the "reverse" problem of simulating a noisy channel
exactly by a noiseless one, assisted by correlations. In both cases, 'one-shot'
separations between the power of the different assisting correlations are
exhibited. The most striking result of this kind is that entanglement can
assist in zero-error communication, in stark contrast to the standard setting
of communicaton with asymptotically vanishing error in which entanglement does
not help at all. In the asymptotic case, shared randomness is shown to be just
as powerful as arbitrary non-signalling correlations for noisy channel
simulation, which is not true for the asymptotic zero-error capacities. For
assistance by arbitrary non-signalling correlations, linear programming
formulas for capacity and simulation are derived, the former being equal (for
channels with non-zero unassisted capacity) to the feedback-assisted zero-error
capacity originally derived by Shannon to upper bound the unassisted zero-error
capacity. Finally, a kind of reversibility between non-signalling-assisted
capacity and simulation is observed, mirroring the famous "reverse Shannon
theorem".Comment: 18 pages, 1 figure. Small changes to text in v2. Removed an
unnecessarily strong requirement in the premise of Theorem 1
New Bounds for the Garden-Hose Model
We show new results about the garden-hose model. Our main results include
improved lower bounds based on non-deterministic communication complexity
(leading to the previously unknown bounds for Inner Product mod 2
and Disjointness), as well as an upper bound for the
Distributed Majority function (previously conjectured to have quadratic
complexity). We show an efficient simulation of formulae made of AND, OR, XOR
gates in the garden-hose model, which implies that lower bounds on the
garden-hose complexity of the order will be
hard to obtain for explicit functions. Furthermore we study a time-bounded
variant of the model, in which even modest savings in time can lead to
exponential lower bounds on the size of garden-hose protocols.Comment: In FSTTCS 201
Entanglement and non-locality are different resources
Bell's theorem states that, to simulate the correlations created by
measurement on pure entangled quantum states, shared randomness is not enough:
some "non-local" resources are required. It has been demonstrated recently that
all projective measurements on the maximally entangled state of two qubits can
be simulated with a single use of a "non-local machine". We prove that a
strictly larger amount of this non-local resource is required for the
simulation of pure non-maximally entangled states of two qubits
with
.Comment: 8 pages, 3 figure
Shared Randomness and Quantum Communication in the Multi-Party Model
We study shared randomness in the context of multi-party number-in-hand
communication protocols in the simultaneous message passing model. We show that
with three or more players, shared randomness exhibits new interesting
properties that have no direct analogues in the two-party case.
First, we demonstrate a hierarchy of modes of shared randomness, with the
usual shared randomness where all parties access the same random string as the
strongest form in the hierarchy. We show exponential separations between its
levels, and some of our bounds may be of independent interest. For example, we
show that the equality function can be solved by a protocol of constant length
using the weakest form of shared randomness, which we call "XOR-shared
randomness."
Second, we show that quantum communication cannot replace shared randomness
in the k-party case, where k >= 3 is any constant. We demonstrate a promise
function GP_k that can be computed by a classical protocol of constant length
when (the strongest form of) shared randomness is available, but any quantum
protocol without shared randomness must send n^Omega(1) qubits to compute it.
Moreover, the quantum complexity of GP_k remains n^Omega(1) even if the "second
strongest" mode of shared randomness is available. While a somewhat similar
separation was already known in the two-party case, in the multi-party case our
statement is qualitatively stronger:
* In the two-party case, only a relational communication problem with similar
properties is known.
* In the two-party case, the gap between the two complexities of a problem
can be at most exponential, as it is known that 2^(O(c)) log n qubits can
always replace shared randomness in any c-bit protocol. Our bounds imply that
with quantum communication alone, in general, it is not possible to simulate
efficiently even a three-bit three-party classical protocol that uses shared
randomness.Comment: 14 pages; v2: improved presentation, corrected statement of Theorem
2.1, corrected typo
Non-locality and Communication Complexity
Quantum information processing is the emerging field that defines and
realizes computing devices that make use of quantum mechanical principles, like
the superposition principle, entanglement, and interference. In this review we
study the information counterpart of computing. The abstract form of the
distributed computing setting is called communication complexity. It studies
the amount of information, in terms of bits or in our case qubits, that two
spatially separated computing devices need to exchange in order to perform some
computational task. Surprisingly, quantum mechanics can be used to obtain
dramatic advantages for such tasks.
We review the area of quantum communication complexity, and show how it
connects the foundational physics questions regarding non-locality with those
of communication complexity studied in theoretical computer science. The first
examples exhibiting the advantage of the use of qubits in distributed
information-processing tasks were based on non-locality tests. However, by now
the field has produced strong and interesting quantum protocols and algorithms
of its own that demonstrate that entanglement, although it cannot be used to
replace communication, can be used to reduce the communication exponentially.
In turn, these new advances yield a new outlook on the foundations of physics,
and could even yield new proposals for experiments that test the foundations of
physics.Comment: Survey paper, 63 pages LaTeX. A reformatted version will appear in
Reviews of Modern Physic
Simulating typical entanglement with many-body Hamiltonian dynamics
We study the time evolution of the amount of entanglement generated by one
dimensional spin-1/2 Ising-type Hamiltonians composed of many-body
interactions. We investigate sets of states randomly selected during the time
evolution generated by several types of time-independent Hamiltonians by
analyzing the distributions of the amount of entanglement of the sets. We
compare such entanglement distributions with that of typical entanglement,
entanglement of a set of states randomly selected from a Hilbert space with
respect to the unitarily invariant measure. We show that the entanglement
distribution obtained by a time-independent Hamiltonian can simulate the
average and standard deviation of the typical entanglement, if the Hamiltonian
contains suitable many-body interactions. We also show that the time required
to achieve such a distribution is polynomial in the system size for certain
types of Hamiltonians.Comment: Revised, 11 pages, 7 figure
Simulating Quantum Correlations with Finite Communication
Assume Alice and Bob share some bipartite -dimensional quantum state. A
well-known result in quantum mechanics says that by performing two-outcome
measurements, Alice and Bob can produce correlations that cannot be obtained
locally, i.e., with shared randomness alone. We show that by using only two
bits of communication, Alice and Bob can classically simulate any such
correlations. All previous protocols for exact simulation required the
communication to grow to infinity with the dimension . Our protocol and
analysis are based on a power series method, resembling Krivine's bound on
Grothendieck's constant, and on the computation of volumes of spherical
tetrahedra.Comment: 19 pages, 3 figures, preliminary version in IEEE FOCS 2007; to appear
in SICOM
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