58,173 research outputs found
The Representation of Natural Numbers in Quantum Mechanics
This paper represents one approach to making explicit some of the assumptions
and conditions implied in the widespread representation of numbers by composite
quantum systems. Any nonempty set and associated operations is a set of natural
numbers or a model of arithmetic if the set and operations satisfy the axioms
of number theory or arithmetic. This work is limited to k-ary representations
of length L and to the axioms for arithmetic modulo k^{L}. A model of the
axioms is described based on states in and operators on an abstract L fold
tensor product Hilbert space H^{arith}. Unitary maps of this space onto a
physical parameter based product space H^{phy} are then described. Each of
these maps makes states in H^{phy}, and the induced operators, a model of the
axioms. Consequences of the existence of many of these maps are discussed along
with the dependence of Grover's and Shor's Algorithms on these maps. The
importance of the main physical requirement, that the basic arithmetic
operations are efficiently implementable, is discussed. This conditions states
that there exist physically realizable Hamiltonians that can implement the
basic arithmetic operations and that the space-time and thermodynamic resources
required are polynomial in L.Comment: Much rewrite, including response to comments. To Appear in Phys. Rev.
The Physics and Mathematics of the Second Law of Thermodynamics
The essential postulates of classical thermodynamics are formulated, from
which the second law is deduced as the principle of increase of entropy in
irreversible adiabatic processes that take one equilibrium state to another.
The entropy constructed here is defined only for equilibrium states and no
attempt is made to define it otherwise. Statistical mechanics does not enter
these considerations. One of the main concepts that makes everything work is
the comparison principle (which, in essence, states that given any two states
of the same chemical composition at least one is adiabatically accessible from
the other) and we show that it can be derived from some assumptions about the
pressure and thermal equilibrium. Temperature is derived from entropy, but at
the start not even the concept of `hotness' is assumed. Our formulation offers
a certain clarity and rigor that goes beyond most textbook discussions of the
second law.Comment: 93 pages, TeX, 8 eps figures. Updated, published version. A summary
appears in Notices of the Amer. Math. Soc. 45 (1998) 571-581, math-ph/980500
The Representation of Numbers by States in Quantum Mechanics
The representation of numbers by tensor product states of composite quantum
systems is examined. Consideration is limited to k-ary representations of
length L and arithmetic modulo k^{L}. An abstract representation on an L fold
tensor product Hilbert space H^{arith} of number states and operators for the
basic arithmetic operations is described. Unitary maps onto a physical
parameter based tensor product space H^{phy} are defined and the relations
between these two spaces and the dependence of algorithm dynamics on the
unitary maps is discussed. The important condition of efficient implementation
by physically realizable Hamiltonians of the basic arithmetic operations is
also discussed.Comment: Paper, 8 pages, for Proceedings, QCM&C 3, O Hirota and P. Tombesi,
Editors, Kluver/Plenum, publisher
Efficient Implementation and the Product State Representation of Numbers
The relation between the requirement of efficient implementability and the
product state representation of numbers is examined. Numbers are defined to be
any model of the axioms of number theory or arithmetic. Efficient
implementability (EI) means that the basic arithmetic operations are physically
implementable and the space-time and thermodynamic resources needed to carry
out the implementations are polynomial in the range of numbers considered.
Different models of numbers are described to show the independence of both EI
and the product state representation from the axioms. The relation between EI
and the product state representation is examined. It is seen that the condition
of a product state representation does not imply EI. Arguments used to refute
the converse implication, EI implies a product state representation, seem
reasonable; but they are not conclusive. Thus this implication remains an open
question.Comment: Paragraph in page proof for Phys. Rev. A revise
A note on information theoretic characterizations of physical theories
Clifton, Bub, and Halvorson [Foundations of Physics 33, 1561 (2003)] have
recently argued that quantum theory is characterized by its satisfaction of
three information-theoretic axioms. However, it is not difficult to construct
apparent counterexamples to the CBH characterization theorem. In this paper, we
discuss the limits of the characterization theorem, and we provide some
technical tools for checking whether a theory (specified in terms of the convex
structure of its state space) falls within these limits.Comment: 16 pages, LaTeX, Contribution to Rob Clifton memorial conferenc
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