34,392 research outputs found
Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb
We report the experimental realization and characterization of one 60-mode
copy, and of two 30-mode copies, of a dual-rail quantum-wire cluster state in
the quantum optical frequency comb of a bimodally pumped optical parametric
oscillator. This is the largest entangled system ever created whose subsystems
are all available simultaneously. The entanglement proceeds from the coherent
concatenation of a multitude of EPR pairs by a single beam splitter, a
procedure which is also a building block for the realization of
hypercubic-lattice cluster states for universal quantum computing.Comment: Accepted by PRL. 5 pages, 5 figures + 14 pages, 9 figures of
supplemental material. Ver3: better experimental dat
Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives
Boundary algebra [BA] is a algebra of type , and a simplified notation for Spencer-Brown’s (1969) primary algebra. The syntax of the primary arithmetic [PA] consists of two atoms, () and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting, indifferently, the presence or absence of () into a PA formula yields a BA formula. The BA axioms are A1: ()()= (), and A2: “(()) [abbreviated ‘⊥’] may be written or erased at will,” implying (⊥)=(). The repeated application of A1 and A2 simplifies any PA formula to either () or ⊥. The basis for BA is B1: abc=bca (concatenation commutes & associates); B2, ⊥a=a (BA has a lower bound, ⊥); B3, (a)a=() (BA is a complemented lattice); and B4, (ba)a=(b)a (implies that BA is a distributive lattice). BA has two intended models: (1) the Boolean algebra 2 with base set B={(),⊥}, such that () ⇔ 1 [dually 0], (a) ⇔ a′, and ab ⇔ a∪b [a∩b]; and (2) sentential logic, such that () ⇔ true [false], (a) ⇔ ~a, and ab ⇔ a∨b [a∧b]. BA is a self-dual notation, facilitates a calculational style of proof, and simplifies clausal reasoning and Quine’s truth value analysis. BA resembles C.S. Peirce’s graphical logic, the symbolic logics of Leibniz and W.E. Johnson, the 2 notation of Byrne (1946), and the Boolean term schemata of Quine (1982).Boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; G. Spencer-Brown; C.S. Peirce; existential graphs
Metrical properties of the set of bent functions in view of duality
In the paper, we give a review of metrical properties of the entire set of bent functions and its significant subclasses of self-dual and anti-self-dual bent functions. We present results for iterative construction of bent functions in n + 2 variables based on the concatenation of four bent functions and consider related open problem proposed by one of the authors. Criterion of self-duality of such functions is discussed. It is explored that the pair of sets of bent functions and affine functions as well as a pair of sets of self-dual and anti-self-dual bent functions in n > 4 variables is a pair of mutually maximally distant sets that implies metrical duality. Groups of automorphisms of the sets of bent functions and (anti-)self-dual bent functions are discussed. The solution to the problem of preserving bentness and the Hamming distance between bent function and its dual within automorphisms of the set of all Boolean functions in n variables is considered
Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count
We present an infinite family of protocols to distill magic states for
-gates that has a low space overhead and uses an asymptotic number of input
magic states to achieve a given target error that is conjectured to be optimal.
The space overhead, defined as the ratio between the physical qubits to the
number of output magic states, is asymptotically constant, while both the
number of input magic states used per output state and the -gate depth of
the circuit scale linearly in the logarithm of the target error (up to
). Unlike other distillation protocols, this protocol
achieves this performance without concatenation and the input magic states are
injected at various steps in the circuit rather than all at the start of the
circuit. The protocol can be modified to distill magic states for other gates
at the third level of the Clifford hierarchy, with the same asymptotic
performance. The protocol relies on the construction of weakly self-dual CSS
codes with many logical qubits and large distance, allowing us to implement
control-SWAPs on multiple qubits. We call this code the "inner code". The
control-SWAPs are then used to measure properties of the magic state and detect
errors, using another code that we call the "outer code". Alternatively, we use
weakly-self dual CSS codes which implement controlled Hadamards for the inner
code, reducing circuit depth. We present several specific small examples of
this protocol.Comment: 39 pages, (v2) renamed "odd" and "even" weakly self-dual CSS codes of
(v1) to "normal" and "hyperbolic" codes, respectively. (v3) published in
Quantu
Uniqueness and Homogeneity of Ordered Relational Structures
There are four major results in the paper. (1) In a general ordered relational structure that is order dense, Dedekind complete, and whose dilations (automorphisms with fixed points) are Archimedean, various consequences of finite uniqueness are developed (Theorem 2.6). (2) Replacing the Archimedean assumption by the assumption that there is a homogeneous subgroup of automorphisms that is Archimedean ordered is sufficient to show that the structure can be represented numberically as a generalized unit structure in the sense that the defining real relations satisfy the usual numerical property of homogeneity (Theorem 3.4). The last two results pertain just to idempotent concatenation structures. (3) In a closed, idempotent, solvable, and Dedekind complete concatenation structure, homogeneity is equivalent to the structure satisfying an inductive property analogous to the condition for homogeneity in a positive concatenation structure (Theorem 4.3). Finally, (4) an axiomatization is given for an idempotent structure to be of scale type (2, 2), which has previously been shown to be equivalent to a dual bilinear representation. Basically two operations are defined in terms of the given one, and the conditions are that each must be right autodistributive and together they satisfy a generalized bisymmetry property. The paper ends listing several unsolved problems.Psycholog
Recent progress in random metric theory and its applications to conditional risk measures
The purpose of this paper is to give a selective survey on recent progress in
random metric theory and its applications to conditional risk measures. This
paper includes eight sections. Section 1 is a longer introduction, which gives
a brief introduction to random metric theory, risk measures and conditional
risk measures. Section 2 gives the central framework in random metric theory,
topological structures, important examples, the notions of a random conjugate
space and the Hahn-Banach theorems for random linear functionals. Section 3
gives several important representation theorems for random conjugate spaces.
Section 4 gives characterizations for a complete random normed module to be
random reflexive. Section 5 gives hyperplane separation theorems currently
available in random locally convex modules. Section 6 gives the theory of
random duality with respect to the locally convex topology and in
particular a characterization for a locally convex module to be
prebarreled. Section 7 gives some basic results on convex
analysis together with some applications to conditional risk measures. Finally,
Section 8 is devoted to extensions of conditional convex risk measures, which
shows that every representable type of conditional convex risk
measure and every continuous type of convex conditional risk measure
() can be extended to an type
of lower semicontinuous conditional convex risk measure and an
type of continuous
conditional convex risk measure (), respectively.Comment: 37 page
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