# 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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### Efficient way to calculate trace of product of Pauli string and matrix?

Basically the title, but more formally: is there a way to efficiently calculate the trace of the product of a Pauli string $P$ and a $2^n \times 2^n$ matrix $M$? That is, is there a way to calculate ...
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### What traces can be estimated in DQC1 (One clean qubit model), and how?

In particular I'm hoping to understand what is written in this paper better: https://arxiv.org/abs/quant-ph/9802037 (On the Power of One Bit of Quantum Information, Knill and Laflamme 1998) In the ...
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### Equality condition on Holder's inequality for matrix for infinity norm

The equality condition for Holder's inequality, $\text{Tr}A^*B \leq ||A||_p||B||_q$ is $|A|^p = \lambda |B|^q$ for scaler $\lambda > 0$. What happens when $p$ or $q$ is $\infty$? I found out that ...
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### Decompose into completely stabilizer preserving channel in surface codes

In the article "Sampling-based quasiprobability simulation for fault-tolerant quantum error correction on the surface codes under coherent noise" they are talking about decomposing (possibly ...
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### Does ${\rm tr}(\Pi \rho) = 1$ imply $\Pi\rho\Pi=\rho$?

Suppose I have a density matrix $\rho$ and an orthogonal projector $\Pi$. Is it true that, if $tr(\Pi \rho) = 1$ then it must hold that $$\Pi \rho \Pi = \rho$$? If yes, how can I prove it?
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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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### What is the physical intuition behind taking the partial trace of a state?

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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### 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 $|i\rangle$ ...
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 ...
### 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 ...