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.

Filter by
Sorted by
Tagged with
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 ...
Physics Penguin's user avatar
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 ...
FDGod's user avatar
  • 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} | \...
X0-user-0X's user avatar
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(.) = \...
QuestionEverything's user avatar
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(...
catmousedog's user avatar
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 $L$?
Kochan's user avatar
  • 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 ...
Newuser7's user avatar
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 ...
MonteNero's user avatar
  • 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 ...
M. Al Jumaily's user avatar
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 ...
Nguyễn An's user avatar
0 votes
1 answer
100 views

Why my density matrix trace is over 1?

Suppose this operator $$ \rho=\frac{a^2}{\cosh^2(r)}\sum_{n=0}^{\infty}\tanh^{2n}(r)|0,n\rangle\langle 0,n|+\frac{b^2}{\cosh^4(r)}\sum_{n=0}^{\infty}(n+1)\tanh^{2n}(r)|1,n+1\rangle\langle 1,n+1| $$ ...
reza's user avatar
  • 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}(...
Denis _J's user avatar
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) \...
JRT's user avatar
  • 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 ...
GaussStrife's user avatar
  • 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 $\sum_{i}^{n} A_i^\dagger A_i \ne \mathbf{I}$...
seeker's user avatar
  • 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 ...
Adrien Amour's user avatar
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 ...
username9's user avatar
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 ...
Pegi's user avatar
  • 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 ...
shashvat's user avatar
  • 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 ...
user19468's user avatar
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 ...
Ron Cohen's user avatar
  • 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?
Lorenzo Laneve's user avatar
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(\...
user1936752's user avatar
  • 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 ...
James Ellis's user avatar
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 ...
user18680's user avatar
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.....
John Parker's user avatar
  • 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^- \...
mikanim's user avatar
  • 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?
John Parker's user avatar
  • 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 ...
snickers_stickers's user avatar
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 $|i\rangle$ ...
123's user avatar
  • 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 ...
ZR-'s user avatar
  • 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 ...
QuestionEverything's user avatar