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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Quantum process tomography, non-trace preserving
Consider an unknown quantum process, i.e., a black box, acting on a physical quantum system described by a density matrix $\rho$ associated with a d-dimensional Hilbert space $\mathcal{H}$.
A complete ...
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Find supremum in an expression containing trace
I am working on a problem in which I need to find the supremum of an expression. Namely, the expression below:
$$
\sup_{w > 0}\Big\{\operatorname{tr}[ \rho \log w] - \log\operatorname{tr}[\sigma w ...
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What traces can be estimated in DQC1 (One clean qubit model), and how?
In particular I'm hoping to understand what is written in this paper better: https://arxiv.org/abs/quant-ph/9802037 (On the Power of One Bit of Quantum Information, Knill and Laflamme 1998)
In the ...
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Equality condition on Holder's inequality for matrix for infinity norm
The equality condition for Holder's inequality, $\text{Tr}A^*B \leq ||A||_p||B||_q $ is $|A|^p = \lambda |B|^q$ for scaler $\lambda > 0$. What happens when $p$ or $q$ is $\infty$? I found out that ...
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Decompose into completely stabilizer preserving channel in surface codes
In the article "Sampling-based quasiprobability simulation for fault-tolerant quantum error correction on the surface codes under coherent noise" they are talking about decomposing (possibly ...
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Does ${\rm tr}(\Pi \rho) = 1$ imply $\Pi\rho\Pi=\rho$?
Suppose I have a density matrix $\rho$ and an orthogonal projector $\Pi$. Is it true that, if $tr(\Pi \rho) = 1$ then it must hold that
$$\Pi \rho \Pi = \rho$$?
If yes, how can I prove it?
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How to compute derivatives of partial traces of the form $\frac{\partial \operatorname{Tr}_B(F(\mathbf{X}))}{\partial \mathbf{X}}$?
The Matrix Cookbook says that for any differentiable matrix function $F(\cdot)$, it holds that
$$\frac{\partial \operatorname{Tr}(F(\mathbf{X}))}{\partial \mathbf{X}}=f(\mathbf{X})^{T},$$
where $f(\...
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What do normalization term and partial measurement represent when tracing out ancillary qubits?
I am reading a paper and I am having trouble following some equations.
The system in this paper has $N$ qubits, with $N_A$ ancillary and the rest ($N - N_A$) as data qubits. For the purpose of this ...
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Show that $\langle v,O(v)\rangle= \mathrm{tr}(O|v\rangle\langle v|)$ for $v \in V$
I have a question regarding this exercise:
Let O be an observable on V. Show that $\langle v,O(v)\rangle= \mathrm{tr}(O|v\rangle\langle v|)$ for $v \in V$.
I thought that this exercise is quite easy ...
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Trace calculation from the basis to |+> and |-> states
I was reading the paper; https://arxiv.org/abs/2002.00055
and going through some of the formulas below and I am a bit stuck between (1) and (2).
How the equation (1) turns into (2) is not clear to me.....
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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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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$ ...
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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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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 ...
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