We introduce a class of bipartite operators acting on $\mathcal{H} \otimes \mathcal{H}$ ( $\mathcal{H}$ being an $n$ -dimensional Hilbert space) defined by a set of $n$ Completely Different Permutations CDP. Bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems. It turns out that any set of CDPs gives rise to a certain direct sum decomposition of the total Hilbert space which enables one simple construction of the corresponding bipartite operator. Interestingly, if set of CDPs defines an abelian group then the corresponding bipartite operator displays an additional property -- the partially transposed operator again corresponds to (in general different) set of CDPs. Therefore, our technique may be used to construct new classes of so called PPT states which are of great importance for quantum information. Using well known relation between bipartite operators and linear maps one use also construct linear maps related to CDPs.We introduce a class of bipartite operators acting on $\mathcal{H} \otimes \mathcal{H}$ ($\mathcal{H}$ being an $n$-dimensional Hilbert space) defined by a set of $n$ Completely Different Permutations CDP. Bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems. It turns out that any set of CDPs gives rise to a certain direct sum decomposition of the total Hilbert space which enables one simple construction of the corresponding bipartite operator. Interestingly, if set of CDPs defines an abelian group then the corresponding bipartite operator displays an additional property -- the partially transposed operator again corresponds to (in general different) set of CDPs. Therefore, our technique may be used to construct new classes of so called PPT states which are of great importance for quantum information. Using well known relation between bipartite operators and linear maps one use also construct linear maps related to CDPs.

arXiv:1712.03277Dec 8, 20170 citations

The Operational Approach to Quantum Mechanics

Charles H. Randall, David J. Foulis

Springer eBooksJan 1, 197853 citations

QUANTUM THEORY BASED ON FUZZY SETS

Ferdinand Chovanec, Mária Jurečkova

A new probability model based on the theory of fuzzy sets is presented. In thismodel, a difference of comparable fuzzy sets is the primary operation. The idea ofa difference of fuzzy sets (fuzzy events) is simple: If we have two comparable eventsa and b ( a ≤ b ), then our knowledge on a and b entails the complete knowledge ofthe complement of a in b , i. e., b ⊖ a .The new algebraic structure of fuzzy sets is called a difference poset (a D-poset) offuzzy sets. Some properties of a lattice ordered D-poset of fuzzy sets (a D-lattice offuzzy sets) are studied. An MV-algebra of fuzzy sets (a Bold algebra) is characterizedin the D-poset of fuzzy sets set-up. The sufficient and necessary conditions for a Dlatticeof fuzzy sets to be a Bold algebra are given. The basic notions of the quantumlogic theory - a state and an observable are defined in D-posets of fuzzy sets.A new probability model based on the theory of fuzzy sets is presented. In thismodel, a difference of comparable fuzzy sets is the primary operation. The idea ofa difference of fuzzy sets (fuzzy events) is simple: If we have two comparable eventsa and b ( a ≤ b ), then our knowledge on a and b entails the complete knowledge ofthe complement of a in b , i. e., b ⊖ a .The new algebraic structure of fuzzy sets is called a difference poset (a D-poset) offuzzy sets. Some properties of a lattice ordered D-poset of fuzzy sets (a D-lattice offuzzy sets) are studied. An MV-algebra of fuzzy sets (a Bold algebra) is characterizedin the D-poset of fuzzy sets set-up. The sufficient and necessary conditions for a Dlatticeof fuzzy sets to be a Bold algebra are given. The basic notions of the quantumlogic theory - a state and an observable are defined in D-posets of fuzzy sets.

The International Conference on Mathematics and Engineering Physics/The International Conference on Mathematics and Engineering PhysicsMay 1, 20060 citations

What is electric charge?: Charge as a local observable in relativistic quantum field theory

Hideyasu Yamashita

It is rather surprising that modern quantum physics does not appear to have provided any clear answer to the simple question ``what is electric charge?''. Even when the total charge operator $Q$ is well-defined, the non-locality of $Q$ implies that it is not an observable in the usual sense, which can be measured by a (local) experimental apparatus. A candidate for the ``local version'' of charge operator is the 4-current operator $j=(j^μ)_{μ=0,1,2,3}$ . However, it is known that the rigorous definition of $j$ is difficult in $(3+1)$ -dimensional Minkowski space. Although it was found that the current can be defined in $(1+1)$ -dimensions (Carey et al.), I argue that even when $j$ can be suitably defined, the interpretability of $j$ as the ``local charge operator'' is dubious. Instead I return to Araki and Wyss (1964), and propose the concept of ``scope-local charge'' $Q_ζ(P)$ for a ``scope'' $P$ , expressed by a finite-dimensional projection. I work in an abstract $C^{*}$ -algebraic setting.It is rather surprising that modern quantum physics does not appear to have provided any clear answer to the simple question ``what is electric charge?''. Even when the total charge operator $Q$ is well-defined, the non-locality of $Q$ implies that it is not an observable in the usual sense, which can be measured by a (local) experimental apparatus. A candidate for the ``local version'' of charge operator is the 4-current operator $j=(j^μ)_{μ=0,1,2,3}$. However, it is known that the rigorous definition of $j$ is difficult in $(3+1)$-dimensional Minkowski space. Although it was found that the current can be defined in $(1+1)$-dimensions (Carey et al.), I argue that even when $j$ can be suitably defined, the interpretability of $j$ as the ``local charge operator'' is dubious. Instead I return to Araki and Wyss (1964), and propose the concept of ``scope-local charge'' $Q_ζ(P)$ for a ``scope'' $P$, expressed by a finite-dimensional projection. I work in an abstract $C^{*}$-algebraic setting.

quant-pharXiv:2608.00468Aug 4, 20260 citations

On the relation between classical and quantum observables

Abhay Ashtekar

Communications in Mathematical PhysicsFeb 1, 198033 citations

We introduce a class of bipartite operators acting on

\mathcal{H} \otimes \mathcal{H}

(

\mathcal{H}

being an

n

-dimensional Hilbert space) defined by a set of

n

Completely Different Permutations CDP. Bipartite operators are of particular importance in quantum information theory to represent states and observables of composite quantum systems. It turns out that any set of CDPs gives rise to a certain direct sum decomposition of the total Hilbert space which enables one simple construction of the corresponding bipartite operator. Interestingly, if set of CDPs defines an abelian group then the corresponding bipartite operator displays an additional property -- the partially transposed operator again corresponds to (in general different) set of CDPs. Therefore, our technique may be used to construct new classes of so called PPT states which are of great importance for quantum information. Using well known relation between bipartite operators and linear maps one use also construct linear maps related to CDPs.

It is rather surprising that modern quantum physics does not appear to have provided any clear answer to the simple question ``what is electric charge?''. Even when the total charge operator

Q

is well-defined, the non-locality of

Q

implies that it is not an observable in the usual sense, which can be measured by a (local) experimental apparatus. A candidate for the ``local version'' of charge operator is the 4-current operator

j=(j^μ)_{μ=0,1,2,3}

. However, it is known that the rigorous definition of

j

is difficult in

(3+1)

-dimensional Minkowski space. Although it was found that the current can be defined in

(1+1)

-dimensions (Carey et al.), I argue that even when

j

can be suitably defined, the interpretability of

j

as the ``local charge operator'' is dubious. Instead I return to Araki and Wyss (1964), and propose the concept of ``scope-local charge''

Q_ζ(P)

for a ``scope''

P

, expressed by a finite-dimensional projection. I work in an abstract

C^{*}

-algebraic setting.