Questions tagged [trace]

For questions about trace, the sum of elements on the main diagonal of a square matrix, which can concern matrices, operators, or functions.

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Does $\mathrm{Tr}(\rho\sigma) > 0$ prove that a state $\sigma$ is separable?

As an example I have the density matrix: $\rho = \frac{1}{3}(| \phi^+ \rangle\langle\phi^+| + | 00 \rangle\langle 00|+| 11 \rangle\langle11| )$ And the two-qubit state is: $\frac{1}{3}(| \phi^- \...
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3answers
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Why does the trace of density operators need to be one?

Usually, the textbook starts with a few assumptions of what density operator $\rho$ has. One of them is $Tr(\rho) = 1$. Why is that?
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Understanding partial trace

I want to confirm my understanding of a partial trace. Essentially, we have a system that $H_a \otimes H_b$. When we trace out system $b$, what we are doing is basically reducing the system down to as ...
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52 views

How is $\sum_i\langle i|M|i\rangle$ correlated to $\mathrm{tr}(M)$?

In the book Quantum computation and quantum information, it says to evaluate $tr(A|\psi\rangle\langle\psi|)$ using Gram-Schmidt procedure to extend $|\psi\rangle$ to an orthonormal basis $|i\rangle$ ...
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How does the term $|\psi\rangle\langle\psi|$ come from while calculating the expectation value? [duplicate]

Suppose there're two systems $A$ and $B$, if we're in a pure state $|\psi\rangle\in\mathbb{H}_a\otimes\mathbb{H}_b$, let $\hat A$ be an operator acts on $\mathbb{H}_a$, and $|\psi\rangle=\sum_{i,j}\...
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2answers
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How do I trace out the second qubit to find the reduced density operator? [duplicate]

I'm doing an exercise to trace out the second qubit to find the reduced density operator for the first qubit: $tr_2|11\rangle\langle00| = |1\rangle\langle0|\langle0|1\rangle$ I'm just wondering if I ...
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1answer
48 views

Prove that for a general tri-partite state $\rho_{ABE}$, $H(AB) = H(E)$

How do I prove that for a general tri-partite state $\rho_{ABE}$, the following holds: $$ H(\rho_{AB}) = H(\rho_{E}), H(\rho_{AE}) = H(\rho_{B}), $$ where, $H$ is the Von Neumann entropy. Would ...