# How to get the matrices in partial trace

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1. Jun 28, 2016

### munirah

1. The problem statement, all variables and given/known data
Good day,

I want to ask the matrix that obtained from below formula and example.

$$tr_A(L_{AB})=\sum_i [(\langle i|\otimes id)L_{AB}(|i\rangle\otimes id)]$$

this formula above can be represented as in matrix form below,

$$tr_A(L_{AB})= \left(\array{1&0&0&0\\0&1&0&0}\right)\cdot \left(\array{0&0&1&0\\1&0&0&0\\0&0&0&0\\0&0&0&0} \right)\cdot \left( \array{1&0\\0&1\\0&0\\0&0}\right)=\left(\array{0&0\\1&0} \right)$$

My question are:

1.what its mean by d in this formula? How can I get this d?

2.How he get the first 2X4 matrix? I already calculate but I just get first row only. How it compute from the formula exactly?

Thank you

2. Jul 1, 2016

### dpopchev

Can you give more information from where you took this example as it seems a little out of context.

From what I see the i stands for the identity matrix, and id for the 0 matrix, the O with the cross stands for tensor product, and take into account that < | and | > come from bra - ket notation of vectors and covectors.

This appears to be a formula for the trace of a matrix L_{AB} which I have no idea from where comes.