64,180 research outputs found
New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic
Intuitionistic logic, in which the double negation law not-not-P = P fails,
is dominant in categorical logic, notably in topos theory. This paper follows a
different direction in which double negation does hold. The algebraic notions
of effect algebra/module that emerged in theoretical physics form the
cornerstone. It is shown that under mild conditions on a category, its maps of
the form X -> 1+1 carry such effect module structure, and can be used as
predicates. Predicates are identified in many different situations, and capture
for instance ordinary subsets, fuzzy predicates in a probabilistic setting,
idempotents in a ring, and effects (positive elements below the unit) in a
C*-algebra or Hilbert space. In quantum foundations the duality between states
and effects plays an important role. It appears here in the form of an
adjunction, where we use maps 1 -> X as states. For such a state s and a
predicate p, the validity probability s |= p is defined, as an abstract Born
rule. It captures many forms of (Boolean or probabilistic) validity known from
the literature. Measurement from quantum mechanics is formalised categorically
in terms of `instruments', using L\"uders rule in the quantum case. These
instruments are special maps associated with predicates (more generally, with
tests), which perform the act of measurement and may have a side-effect that
disturbs the system under observation. This abstract description of
side-effects is one of the main achievements of the current approach. It is
shown that in the special case of C*-algebras, side-effect appear exclusively
in the non-commutative case. Also, these instruments are used for test
operators in a dynamic logic that can be used for reasoning about quantum
programs/protocols. The paper describes four successive assumptions, towards a
categorical axiomatisation of quantitative logic for probabilistic and quantum
systems
Maximum efficiency of a linear-optical Bell-state analyzer
In a photonic realization of qubits the implementation of quantum logic is
rather difficult due the extremely weak interaction on the few photon level. On
the other hand, in these systems interference is available to process the
quantum states. We formalize the use of interference by the definition of a
simple class of operations which include linear optical elements, auxiliary
states and conditional operations.
We investigate an important subclass of these tools, namely linear optical
elements and auxiliary modes in the vacuum state. For this tools, we are able
to extend a previous quantitative result, a no-go theorem for perfect Bell
state analyzer on two qubits in polarization entanglement, by a quantitative
statement. We show, that within this subclass it is not possible to
discriminate unambiguously four equiprobable Bell states with a probability
higher than 50 %.Comment: 6 pages, 2 fig
Alternative fast quantum logic gates using nonadiabatic Landau-Zener-St\"{u}ckelberg-Majorana transitions
A conventional realization of quantum logic gates and control is based on
resonant Rabi oscillations of the occupation probability of the system. This
approach has certain limitations and complications, like counter-rotating
terms. We study an alternative paradigm for implementing quantum logic gates
based on Landau-Zener-St\"{u}ckelberg-Majorana (LZSM) interferometry with
non-resonant driving and the alternation of adiabatic evolution and
non-adiabatic transitions. Compared to Rabi oscillations, the main differences
are a non-resonant driving frequency and a small number of periods in the
external driving. We explore the dynamics of a multilevel quantum system under
LZSM drives and optimize the parameters for increasing single- and two-qubit
gates speed. We define the parameters of the external driving required for
implementing some specific gates using the adiabatic-impulse model. The LZSM
approach can be applied to a large variety of multi-level quantum systems and
external driving, providing a method for implementing quantum logic gates on
them.Comment: 15 pages, 12 figure
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