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The Uniqueness Problem of Sequence Product on Operator Effect Algebra $varepsilon (H)$ | | Liu Weihua
; Wu Junde
; | | Date: |
3 Dec 2008 | | Abstract: | Let $H$ be a complex Hilbert space. A quantum effect on $H$ is a bounded
linear self-adjoint operator $A$ defined on $H$ that satisfies $0leq Aleq I$.
We denote the set of all quantum effects on $H$ by $varepsilon (H)$. The
sequential product $circ$ defined on $varepsilon (H)$ is an important concept
because it can be used to describe the quantum measurements. A sequential
product defined by $Acirc B=A^{{1/2}}BA^{{1/2}}$ for any two quantum effects
$A, B$ has recently been introduced and studied. In 2005, Professor Gudder
presented an open problem as following: Is $Acirc B=A^{{1/2}}BA^{{1/2}}$ the
only sequential product on $varepsilon (H)$? In this paper, for dimH=2, we
construct a new sequential product on $varepsilon (H)$ which is different from
$Acirc B=A^{{1/2}}BA^{{1/2}}$. Thus, we answer the Gudder’s open problem
negatively. In particular, our result shows the phase-changed phenomena of
quantum measurement $Acirc B$, this is much more important in physics. | | Source: | arXiv, 0812.0630 | | Services: | Forum | Review | PDF | Favorites | |
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