Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New features!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

123 submissions , 104 unreviewed
3,547 questions , 1,198 unanswered
4,549 answers , 19,357 comments
1,470 users with positive rep
410 active unimported users
More ...

Hilbert-Schmidt basis for many qubits - reference

+ 9 like - 0 dislike
2610 views

Every density matrix of $n$ qubits can be written in the following way $$\hat{\rho}=\frac{1}{2^n}\sum_{i_1,i_2,\ldots,i_n=0}^3 t_{i_1i_2\ldots i_n} \hat{\sigma}_{i_1}\otimes\hat{\sigma}_{i_2}\otimes\ldots\otimes\hat{\sigma}_{i_n},$$ where $-1 \leq t_{i_1i_2\ldots i_n} \leq 1$ are real numbers and $\{\hat{\sigma}_0,\hat{\sigma}_1,\hat{\sigma}_2,\hat{\sigma}_3\}$ are the Pauli matrices. In particular for one particle ($n=1$) it is the Bloch representation.

Such representation is used e.g. in a work by Horodecki arXiv:quant-ph/9607007 (they apply $n=2$ to investigate the entanglement of two qubit systems). It is called decomposition in the Hilbert-Schmidt basis.

The question is if there is any good reference for such representation for qubits - either introducing it for quantum applications or a review paper? I am especially interested in the constrains on $t_{i_1i_2\ldots i_n}$.

This post has been migrated from (A51.SE)
asked Nov 22, 2011 in Theoretical Physics by Piotr Migdal (1,250 points) [ no revision ]
Most voted comments show all comments
@MateusAraújo: Thanks. I found it also as discrete Wigner function http://arxiv.org/abs/quant-ph/0410117

This post has been migrated from (A51.SE)
@MateusAraújo: Don't forget to post your links as an answer! :)

This post has been migrated from (A51.SE)
[This](http://arxiv.org/abs/quant-ph/0308019) seems to be what you're looking for.

This post has been migrated from (A51.SE)
Maybe the discrete Wigner function is a bit other story, because they need to use trace on $GF(2^n)$ and exchange components in Hilbert-Schmidt scalar product.

This post has been migrated from (A51.SE)
I guess, but I never heard about simple expressions for constrains

This post has been migrated from (A51.SE)
Most recent comments show all comments
Googling leads me into progressively more mathematical territory. There is a small chance that this is a result well known enough to mathematicians that it jumped into QI sideways as a standard tool used by a mathematical physicist. I've seen the Hilbert-Schmidt basis used without even labelling it. See for example the [answer I gave to my own question](http://theoreticalphysics.stackexchange.com/q/537/569) a couple of days ago. I used it without thinking of the name, as did the paper I was referring to in that post. This unfortunately only compounds the stupidity shown in my first comment...

This post has been migrated from (A51.SE)
I supposed the decomposition itself is rather standard consequence of axioms of quantum mechanics and linear algebra, e.g. section 5.3 in my e-print http://arxiv.org/abs/quant-ph/0104126v1

This post has been migrated from (A51.SE)

3 Answers

+ 3 like - 0 dislike

I use this decomposition all the time, but I have never read a paper solely devoted to the topic. From my experience a complete characterization of the constraints on $t_{i_{1}, t_{2},..t_{n}}$ is tricky, and so if you want to be sure $\rho$ is physical you should calculate the density matrix and its eigenvalues.

However, there are a lot of necessary conditions that have a useful form in this decomposition. For example, for a positive unit-trace Hermitian operator $\rho$ is follows that

$|t_{i_{1}, i_{2},.. i_{n}}| \leq 1$

$tr ( \rho^{2} ) =\frac{1}{2^{n}} \sum_{i_{1}, i_{2},.. i_{n}} t_{i_{1}, i_{2},.. i_{n}}^{2} \leq 1 $

The above condition tells us that if we think of $t$ as a vector in a real vector space, then the physical states live within the unit sphere. This is a bit like the Bloch sphere for 1 qubit but for many qubits we have some other constraints that take the form of hyperplanes. For every $\vert \psi \rangle$ expressed in the same form $\vert \psi \rangle \langle \psi \vert = \frac{1}{2^{n}} \sum_{i_{1},i_{2},... i_{n}} Q_{i_{1},i_{2},... i_{n}}\sigma_{i1} \otimes \sigma_{i2}... \sigma_{in}$ we require that

$\langle \psi \vert \rho \vert \psi \rangle \geq 0 $ and so $\sum Q_{i_{1},i_{2},... i_{n}}t_{i_{1},i_{2},... i_{n}}\geq 0$ which defines a hyperplane.

The problem is you have a hyperplane for every $\psi$ so that requiring $t$ to satisfy every inequality one of the infinite hyperplanes is impossible to check by brute force. If you want sufficient conditions for positivity of $\rho$ I suspect you have to calculate eigenvalues.

This post has been migrated from (A51.SE)
answered Nov 25, 2011 by Earl (405 points) [ no revision ]
+1 from me. I use it a lot too, but couldn't think of anything interesting to say!

This post has been migrated from (A51.SE)
+ 1 like - 0 dislike

A good starting point, I have checked just chapter 4 but there is more, is

R. R. Puri, Mathematical Methods of Quantum Optics, Springer (2001) (see here).

This post has been migrated from (A51.SE)
answered Nov 24, 2011 by JonLester (376 points) [ no revision ]
Thank you, Jon. In the meantime I found papers about discrete phase space, where $\hat{\sigma}_{\vec{i}}$ is a shift operator (Gibbons, Hoffman, Wootters [quant-ph/0401155](http://arxiv.org/abs/quant-ph/0401155), Paz, Roncaglia, Saraceno [quant-ph/0410117](http://arxiv.org/abs/quant-ph/0410117) and Sec 4.2 of Chris Ferrie's [1010.2701](http://arxiv.org/abs/1010.2701)).

This post has been migrated from (A51.SE)
+ 1 like - 0 dislike

Claudio Altafini studies precisely this subject, in Tensor of coherences parameterization of multiqubit density operators for entanglement characterization and some follow-ups.

This post has been migrated from (A51.SE)
answered Nov 25, 2011 by Mateus Araújo (270 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverfl$\varnothing$w
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...