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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3 votes
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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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2 votes
3 answers
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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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6 votes
2 answers
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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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