The average value inequality in sequential effect algebras

Jun Shen,Jun De Wu

doi:10.1007/s10114-009-8683-5

Jun Shen, Jun De Wu

Open Access

https://doi.org/10.1007/s10114-009-8683-5

Copy DOI

Abstract

A sequential effect algebra (E, 0, 1, ⊕, ∘) is an effect algebra on which a sequential product ∘ with certain physics properties is defined; in particular, sequential effect algebra is an important model for studying quantum measurement theory. In 2005, Gudder asked the following problem: If a, b ∈, (E, 0, 1, ⊕, ∘) and a⊥b and a ∘ b⊥a ∘ b, is it the case that 2(a ∘ b) ≤ a 2 ⊕ b 2? In this paper, we construct an example to answer the problem negatively.

Highlights

We show that the above average value inequality does hold in the underlying sequential effect algebras under some additional conditions
We construct a sequential effect algebra to show that the above average value inequality does not always hold
We verify (SEA4), we omit the trivial cases about 0,1: an ◦ (am ◦ ak) = (an ◦ am) ◦ ak = 0, an ◦ (am ◦ bk) = bk ◦ (an ◦ am) = am ◦ (an ◦ bk) = 0, an ◦ (am ◦ cr,s,t) = cr,s,t ◦ (an ◦ am) = am ◦ (an ◦ cr,s,t) = 0, an ◦ (am ◦ dr,s,t) = dr,s,t ◦ (an ◦ am) = am ◦ (an ◦ dr,s,t) = 0, an ◦ (bm ◦ bk) = bk ◦ (an ◦ bm) = bm ◦ (an ◦ bk) = an, an ◦ (bm ◦ cr,s,t) = cr,s,t ◦ (an ◦ bm) = bm ◦ (an ◦ cr,s,t) = 0, an ◦ (bm ◦ dr,s,t) = dr,s,t ◦ (an ◦ bm) = bm ◦ (an ◦ dr,s,t) = an, an ◦ (ci,k,m ◦ cr,s,t) = cr,s,t ◦ (an ◦ ci,k,m) = ci,k,m ◦ (an ◦ cr,s,t) = 0, an ◦ (ci,k,m ◦ dr,s,t) = dr,s,t ◦ (an ◦ ci,k,m) = ci,k,m ◦ (an ◦ dr,s,t) = 0, an ◦ (di,k,m ◦ dr,s,t) = dr,s,t ◦ (an ◦ di,k,m) = di,k,m ◦ (an ◦ dr,s,t) = an, bn ◦ (bm ◦ bk) = bk ◦ (bn ◦ bm) = bm+n+k, bn ◦ (bm ◦ cr,s,t) = cr,s,t ◦ (bn ◦ bm) = bm ◦ (bn ◦ cr,s,t) = cr,s,t, bn ◦ (bm ◦ dr,s,t) = dr,s,t ◦ (bn ◦ bm) = bm ◦ (bn ◦ dr,s,t) = dr,s,n+m+t, bn ◦ (ci,k,m ◦ cr,s,t) = cr,s,t ◦ (bn ◦ ci,k,m) = ci,k,m ◦ (bn ◦ cr,s,t) = ais+kr(when is + kr = 0) or 0(when is + kr = 0), bn ◦ (ci,k,m ◦ dr,s,t) = dr,s,t ◦ (bn ◦ ci,k,m) = ci,k,m ◦ (bn ◦ dr,s,t) = ci,k,m−is−kr, bn ◦ (di,k,m ◦ dr,s,t) = dr,s,t ◦ (bn ◦ di,k,m) = di,k,m ◦ (bn ◦ dr,s,t) = di+r,k+s,n+m−t−is−kr, cx,y,z ◦ (ci,k,m ◦ cr,s,t) = cr,s,t ◦ (cx,y,z ◦ ci,k,m) = 0, cx,y,z ◦ (ci,k,m ◦ dr,s,t) = dr,s,t ◦ (cx,y,z ◦ ci,k,m) = ci,k,m ◦ (cx,y,z ◦ dr,s,t) = axk+yi(when xk + yi = 0) or 0(when xk + yi = 0), cx,y,z ◦ (di,k,m ◦ dr,s,t) = dr,s,t ◦ (cx,y,z ◦ di,k,m) = di,k,m ◦ (cx,y,z ◦ dr,s,t) = cx,y,z−x(k+s)−y(i+r), dx,y,z ◦(di,k,m◦dr,s,t) = dr,s,t◦(dx,y,z ◦di,k,m)= dx+i+r,y+k+s,z+m+t−(is+kr+xk+xs+yi+yr)