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

Please welcome our new moderators!

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

122 submissions , 103 unreviewed
3,497 questions , 1,172 unanswered
4,545 answers , 19,342 comments
1,470 users with positive rep
408 active unimported users
More ...

Experimental test of the non-statisticality theorem?

+ 13 like - 0 dislike
784 views

Context: The recent paper The quantum state cannot be interpreted statistically by Pusey, Barrett and Rudolph shows under suitable assumptions that the quantum state cannot be interpreted as a probability distribution over hidden variables (go read it now).

In the abstract they claim: "This result holds even in the presence of small amounts of experimental noise, and is therefore amenable to experimental test using present or near-future technology."

The claim is supported on page 3 starting with: "In a real experiment, it will be possible to establish with high condence that the probability for each measurement outcome is within $\epsilon$ of the predicted quantum probability for some small $\epsilon> 0$."

Something felt like it was missing so I tried to fill in the details. Here is my attempt:

First, without even considering experimental noise, any reasonable measure of error (e.g. standard squared error) on the estimation of the probabilities is going to have worst case bounded by $$ \epsilon\geq\frac{2^n}{N}, $$ (tight for the maximum likelihood estimator, in this case) where $n$ is the number of copies of the system required for the proof and $N$ is the number of measurements (we are trying to estimate a multinomial distribution with $2^n$ outcomes).

Now, they show that some distance measure on epistemic states (I'm not sure if it matters what it is) satisfies $$ D \ge 1 - 2\epsilon^{1/n}. $$ The point is we want $D=1$. So if we can tolerate an error in this metric of $\delta=1-D$ (what is the operational interpretation of this?), then the number of measurements we must make is $$ N \ge \left(\frac4\delta\right)^n. $$ This looks bad. But how many copies do we really need? Note that the proof requires two non-orthogonal qubit states with overlap $|\langle \phi_0 |\phi_1\rangle|^2 = \cos^2\theta$. The number of copies required is implicitly given by $$ 2\arctan(2^{1/n}-1)\leq \theta. $$ Some back-of-the-Mathematica calculations seems to show that $n$ scales at least quadratically with the overlap of the state.

Is this right? Does it require (sub?)exponentially many measurements in the system size (not suprising, I suppose) and the error tolerance (bad, right?).

This post has been migrated from (A51.SE)
asked Nov 17, 2011 in Theoretical Physics by Chris Ferrie (645 points) [ no revision ]
retagged Mar 7, 2014 by dimension10
Of course, this is not the only distance with this property (consider fidelity-derived distances). Probably it was just the more convenient to prove their bound. But it is a mystery to me why they called it "total variation distance" instead of the more common "trace distance".

This post has been migrated from (A51.SE)
I think the distance chosen does matter as $D(\mu_0,\mu_1) = 1$ iff $\mu_0$ and $\mu_1$ have disjoint support, that is, if your ontological model is actually $\psi$-ontic. So $1-D$ is just a measure of how $\psi$-epistemic is your ontological model.

This post has been migrated from (A51.SE)
Removed some comments (some offensive and some not) which combined to form a debate which was peripheral to this technical and concrete question. There are plenty of places to have debates on this paper and its deep meaning, an issue which attracted some attention. Here, I'd hope, is a place where people can engage with the specifics of the question asked.

This post has been migrated from (A51.SE)

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$\hbar$ysicsOver$\varnothing$low
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
...