partial trace

Suppose VV, WW are finite-dimensional vector spaces over a field, with dimensions mm and nn, respectively. For any space AA let L(A)L(A) denote the space of linear operators on AA. The partial trace over WW, TrW_{W}, is a mapping

TL(VW)Tr W(T)L(V). T \in L(V \otimes W) \mapsto Tr_{W}(T) \in L(V).

Let e 1,,e me_{1}, \ldots, e_{m} and f 1,,f nf_{1}, \ldots, f_{n} be bases for VV and WW respectively. Then TT has a matrix representation {a kl,ij}\{a_{kl,ij}\} where 1k,im1 \le k,i \le m and 1l,jn1 \le l,j \le n relative to the basis of the space VWV \otimes W given by e kf le_{k} \otimes f_{l}. Consider the sum

b k,i= j=1 na kj,ij b_{k,i} = \sum_{j=1}^{n}a_{kj,ij}

for k,ik,i over 1,,m1, \ldots, m. This gives the matrix b k,ib_{k,i}. The associated linear operator on VV is independent of the choice of bases and is defined as the partial trace.


Consider a quantum system, ρ\rho, in the presence of an environment, ρ env\rho_{env}. Consider what is known in quantum information theory as the CNOT gate:

U=|0000|+|0101|+|1110|+|1011|. U={|00\rangle}{\langle 00|} + {|01\rangle}{\langle 01|} + {|11\rangle}{\langle 10|} + {|10\rangle}{\langle 11|}.

Suppose our system has the simple state |11|{|1\rangle}{\langle 1|} and the environment has the simple state |00|{|0\rangle}{\langle 0|}. Then ρρ env=|1010|\rho \otimes \rho_{env} = {|10\rangle}{\langle 10|}. In the quantum operation formalism we have

T(ρ)=12Tr envU(ρρ env)U =12Tr env(|1010|+|1111|)=|11|0|0+|11|1|12=|11| T(\rho) = \frac{1}{2}Tr_{env}U(\rho \otimes \rho_{env})U^{\dagger} = \frac{1}{2}Tr_{env}({|10\rangle}{\langle 10|} + {|11\rangle}{\langle 11|}) = \frac{{|1\rangle}{\langle 1|}{\langle 0|0\rangle} + {|1\rangle}{\langle 1|}{\langle 1|1\rangle}}{2} = {|1\rangle}{\langle 1|}

where we inserted the normalization factor 12\frac{1}{2}.

Last revised on March 6, 2010 at 23:40:54. See the history of this page for a list of all contributions to it.