[Community Question] Linear-algebra: partial trace of a subsystem

One of our user asked:

Let there exists a pure composite system of 2 subsystems; namely, "1" and "2". Suppose $\hat{\rho} = \frac{|00\rangle \langle 00|+|00\rangle\langle 11| + |11\rangle\langle 00| + |11\rangle \langle 11|}{2}$ is a pure state operator for the this pure composite system.

The reduce state operator for subsystem "1" is

$\hat{\rho}^{1} = Tr_{2}\left(\hat{\rho}\right)$

= $\frac{1}{2} Tr_{2} \left(|0\rangle\langle 0|\otimes|0\rangle \langle 0| + |1\rangle\langle0|+|0\rangle\langle1| \otimes|0\rangle\langle 1| \otimes |0\rangle\langle1|+|1\rangle\langle1|\otimes|1\rangle\langle1| \right)$

= $\frac{1}{2}\left(|0\rangle \langle0|\cdot Tr\left(|0\rangle \langle 0|\right) + \cdot \cdot \cdot \right)$

I am unable to understand how the third inequality comes about. Any help is appreciated.

Comments

[Community Question] Linear-algebra: non-negative matrix satisfying two conditions

One of our user asked: A real matrix $B$ is called non-negative if every entry is non-negative. We will denote this by $B\ge 0$ . I want to find a non-negative matrix $B$ satisfying the following two conditions: (1) $(I-B)^{-1}$ exists but not non-negative. Here $I$ is the identity matrix. (2) There is a non-zero and non-negative vector $\vec{d}$ such that $(I-B)^{-1}\vec{d}\ge 0$ . I tried all the $2\times 2$ matrices, but it did not work. I conjecture that such a $B$ does not exist, but don't know how to prove it.

[Community Question] Geometry: The limit about the line connecting the intersection of a circle and the $y$-axis and the intersection of the shrinking circle and a fixed circle

One of our user asked: There is a fixed circle $C_1$ with equation $(x - 1)^2 + y^2 = 1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0, r)$ , $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$ -axis. What happens to $R$ as $C_2$ shrinks, that is, as $r \to 0^+$ ? (The figure is made with GeoGebra ) In order to solve this problem, I made a script using GeoGebra in which the circle $C_2$ is a dynamic one whose radius $r$ can be adjusted with a slider. As I set $r \to 0^+$ , the figure seems to suggest that $R \to (4,0)$ . In particular, this is the state with $r = 0.001$ , in which $R$ is reported to be $(3.9999997523053,0)$ : However, I would like to find out a way to prove (or disprove, though unlikely) my guess that $$\lim_{r \to 0} R = (4,0).$$ But I have little idea. Any help would be appreciated.

Order of elements of the Prüfer groups $\mathbb{Z}(p^{\infty})$

Let $\mathbb{Z}(p^{\infty})$ be defined by $\mathbb{Z}(p^{\infty}) = \{ \overline{a/b} \in \mathbb{Q}/ \mathbb{Z} / a,b \in \mathbb{Z}, b=p^i$ $ with$ $ i \in \mathbb{N} \}$ , I wish show that any element in $\mathbb{Z}(p^{\infty})$ has order $p^n$ with $n \in \mathbb{N}$ . i try several ways but I have not been successful, some help ?? thank you

Win At Mathematics

Search This Blog