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.
32
questions
4
votes
1
answer
131
views
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 ...
2
votes
1
answer
76
views
Given that for every valid density matrix $\rho$, $\text{Tr}(M\rho) = 1$; what can we conclude about matrix $M$?
My intuition says that $M$ has to be the identity matrix, but I am not able to show it rigorously. I tried playing around using spectral decomposition. If
$$
\rho = \sum_i \lambda_i |\lambda_i \rangle ...
- 571
2
votes
2
answers
88
views
Prove the fidelity equals $F( \rho , \sigma) = |\langle \psi_{\rho} | \psi_{\sigma}\rangle|^2$ for pure states
I am trying to learn by myself quantum computing and information and I have a very simple question concerning the demonstration of the following equality: $F( \rho , \sigma) = |\langle \psi_{\rho} | \...
- 155
1
vote
1
answer
26
views
Prove that $Tr(\chi(\rho_A) \log(\chi(\rho_A)) = Tr(\rho_A \log(\chi(\rho_A))$
I am trying to see how the following statement about trace $Tr$ is true.
$$
Tr(\chi(\rho_A) \log(\chi(\rho_A)) = Tr(\rho_A \log(\chi(\rho_A)),
$$
for some quantum state $\rho_A$, Where,
$$
\chi(.) = \...
- 1,776
3
votes
1
answer
43
views
What trace properties are used in the identity ${\rm tr}_A{\rm tr}_B(\rho\Pi)={\rm tr}_A(\rho_A{\rm tr}_B(\rho_B\Pi))$?
To turn the probability of the projection over the Hilbert space $\mathcal H_A \otimes \mathcal H_B$ into the POVM probabilty over $\mathcal H_A$ we we use this equality:
$$tr_Atr_B(ρΠ_i)=tr_A(ρ_Atr_B(...
- 133
2
votes
2
answers
88
views
Does the inequality $\mathrm{tr}(L^\dagger L \rho^2)-\mathrm{tr}(L^\dagger \rho L\rho )\geq 0$ hold generally?
Does the inequality
$$\mathrm{tr}(L^\dagger L \rho^2)-\mathrm{tr}(L^\dagger \rho L\rho )\geq 0$$
hold for any density matrix $\rho$ and any non-Hermitian Lindblad operator $L$?
- 31
1
vote
1
answer
57
views
Quantum Relative entropy- the math and intuition
I am new to quantum information theory and have been reading Mark Wilde's notes on quantum relative entropy.
http://www.markwilde.com/teaching/2015-fall-qit/lectures/lecture-19.pdf
I have three basic ...
- 13
1
vote
0
answers
31
views
List of inequalities for purity of a traced out bipartite system
I would like to know if there are inequalities related to the purity of the partial trace of a bipartite system.
The purity $P$ of a density matrix $\rho$ is given by
$$P(\rho) = Tr(\rho^2).$$
The ...
- 2,339
0
votes
1
answer
195
views
How to prove that the trace of a density matrix is $1$?
Equation 2 gives the following proof:
$$
\text{Tr}[\rho] = \sum_x \langle x\vert \rho\vert x\rangle = \sum_x \langle x\vert
\sum_i p_i\vert \psi_i\rangle \langle \psi_i\vert\vert x\rangle = \sum_i ...
- 800
1
vote
1
answer
54
views
Minimum and maximum of $Tr(\rho\sigma)+\sqrt{1-Tr(\rho^2)}\sqrt{1-Tr(\sigma^2)}$
How do I find the maximum and minimum of the following expression?
$$F_N(\rho,\sigma)=Tr(\rho\sigma)+\sqrt{1-Tr(\rho^2)}\sqrt{1-Tr(\sigma^2)}.$$
I think of using this inequality
$$Tr(\rho\sigma)\leq ...
- 11
0
votes
1
answer
100
views
Why my density matrix trace is over 1?
Suppose this operator
$$
\rho=\frac{a^2}{\cosh^2(r)}\sum_{n=0}^{\infty}\tanh^{2n}(r)|0,n\rangle\langle 0,n|+\frac{b^2}{\cosh^4(r)}\sum_{n=0}^{\infty}(n+1)\tanh^{2n}(r)|1,n+1\rangle\langle 1,n+1|
$$
...
- 389
3
votes
1
answer
169
views
Haar measure : trace of an operator squared and square of the trace of an operator
From doing numerical simulations, I seem to have the following results :
$$ \int d \rho \,\, \text{Tr}(\rho M^\dagger M) = \frac{1}{d} \text{Tr}(M^\dagger M) $$
and
$$ \int d \rho \,\, \left|\text{Tr}(...
- 33
4
votes
1
answer
144
views
Does closeness in trace distance imply close measurement outcomes?
Suppose we have two density matrices $\rho$ and $\rho'$ such that $\|\rho - \rho'\|_1 \leq \varepsilon$. Let $\{\Lambda, I - \Lambda\}$ be elements of some POVM. If it holds that
$$Tr(\Lambda\rho) \...
- 457
2
votes
2
answers
399
views
Prove the triangle inequality for the trace norm: $\|M+N\|_1\le \|M\|_1+\|N\|_1$
I have been trying to show that $$||M+N|| \le ||M|| + ||N||$$ However, I seem to be missing some fundamental property of either how the trace or square root acts on these sums of matrices, or how the ...
- 1,107
1
vote
1
answer
133
views
Can one turn a non-Trace Preserving map into one that is Trace Preserving?
A trace non-preserving quantum channel $\mathcal{A}$ takes a state $\rho$ to $\rho^\prime$, i.e., $\sum_{i=1}^{n} A_i \rho A_i^\dagger = \rho^\prime$, with $\sum_{i}^{n} A_i^\dagger A_i \ne \mathbf{I}$...
- 149
0
votes
1
answer
62
views
measurement probability from density operator?
I've been through this before but I can't fully get my head round this upon review.
So the density operator $\hat{\rho}=\sum_j p_j|\psi_j\rangle\!\langle \psi_{j}|$ for pure states $|\psi_{j}>$ at ...
- 329
1
vote
1
answer
119
views
Quantum process tomography, non-trace preserving
Consider an unknown quantum process, i.e., a black box, acting on a physical quantum system described by a density matrix $\rho$ associated with a d-dimensional Hilbert space $\mathcal{H}$.
A complete ...
- 67
2
votes
0
answers
55
views
Find supremum in an expression containing trace
I am working on a problem in which I need to find the supremum of an expression. Namely, the expression below:
$$
\sup_{w > 0}\Big\{\operatorname{tr}[ \rho \log w] - \log\operatorname{tr}[\sigma w ...
- 165
2
votes
1
answer
199
views
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 ...
- 665
2
votes
0
answers
173
views
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 ...
- 55
0
votes
1
answer
66
views
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 ...
- 1,144
5
votes
2
answers
86
views
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?
- 327
3
votes
1
answer
136
views
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(\...
- 2,439
4
votes
1
answer
173
views
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 ...
- 103
0
votes
1
answer
43
views
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 ...
0
votes
1
answer
111
views
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.....
- 1,011
0
votes
1
answer
138
views
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^- \...
- 277
2
votes
3
answers
2k
views
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?
- 1,011
7
votes
2
answers
267
views
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 ...
2
votes
1
answer
68
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 $|i\rangle$ ...
- 61
4
votes
2
answers
1k
views
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 ...
- 2,368
1
vote
1
answer
73
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 ...
- 1,776