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
4,544 answers , 19,342 comments
1,470 users with positive rep
408 active unimported users
More ...

Unknown quantum state with promise of classical data

+ 1 like - 0 dislike
12 views

I am trying to solve a problem in the measurement and identification of quantum states with a promise as to what states it could be. Here is the problem. Imagine a system that produces qubits in one of four states $S = \{a,b,c,d \}$, evenly distributed. In one shot, I can receive $K$ copies of a state in $S$. However, a random unitary, evenly distributed in $SU(2)$ has been applied, so, at the detectors I receive $S^{\prime} = \{Ha,Hb,Hc,Hd \}$. I have a detector system that includes $2M$ detectors and they represent the projection operators onto any pair of basis vectors in any basis (thus we can choose which $M$ bases we want to use the detectors in). All I want to do is decide if the state is $a,b,c,$ or $d$. How many copies of the state do I need (ie what is a minimum for $K$)? What is the smallest $M$?

Another, more general question is this: what should I use to represent the state before I do a measurement? Should it not be a density matrix which is an integral over all states? Since there is a random unitary applied, that means that I can receive, in a fixed basis, any state with equal probability. What would be the update rule for this density matrix after the $j^{th}$ measurement result?

I realize there are pieces missing in my question. I will revise it this post this evening, but hopefully it will give the general idea.

This post has been migrated from (A51.SE)
asked Feb 28, 2012 in Theoretical Physics by user442920 (90 points) [ no revision ]
retagged Mar 7, 2014 by dimension10
What if instead of receiving $K$ copies of a state, you can receive $k$ copies of $N$ states chosen randomly from $S$. That is to say, you receive a string of qubits, and you get $K$ copies of each qubit. Given some setup, you will accumulate 4 different probability distributions because you are also allowed to know when a qubit has been passed. Can you not then use this added information to decide on which distribution corresponds to which original state? There is certainly information embedded in the promise of 4 different distributions and I guess I am trying to squeeze that out.

This post has been migrated from (A51.SE)
In that case, the Holevo quantity bounds the amount of information you can obtain, and hence can be used to upper bound the probability of correctly guessing the correct state. But be warned, it can be an extremely loose bound.

This post has been migrated from (A51.SE)
I guess I should have paid more attention in QInformation class.... :(....let me see if I can make this an interesting question.

This post has been migrated from (A51.SE)
I'm not quite sure I understand: If $H$ is chosen randomly then $Ha$ is uncorrelated with $a$. Averaging over $H$ you get $\rho = \sum_H p(H) (H a)^{\otimes K} = \sum_H p(H) (H b)^{\otimes K} = \sum_H p(H) (H c)^{\otimes K} = \sum_H p(H) (H d)^{\otimes K}$, so no matter how many copies you have you can't distinguish the initial state, since these states are related by unitary operators (and you've just applied a random unitary).

This post has been migrated from (A51.SE)
@PiotrMigdal: Looks like we were typing at the same time about the same thing!

This post has been migrated from (A51.SE)
I don't understand the problem. Once the state is randomly rotated there is no way to tell if it was $a$,$b$,$c$ or $d$.

This post has been migrated from (A51.SE)

1 Answer

+ 0 like - 0 dislike

I think that it may be quite difficult to find the most efficient solution, which would involve collective measurements involving several of the inputs at once (although in your question you seem to rule out this possibility by mentioning measurements of single qubits at a time). Also, one should consider POVMs rather than projective measurements.

Assuming that you measure only one qubit at a time, I can answer your question of how to represent them. The first qubit is as you suspect the fully mixed state. After that you should use Bayes' rule to update your probability distributions for S (if S is not assumed to always be uniformly random) and S'. This is not my expertise, but this part of it is a classical problem not a quantum problem. Use these updated probability distributions to form the density operator of the next state you receive. You then need to decide which type of measurement is best given your current state of knowledge about the probabilities. I can't help you there.

This post has been migrated from (A51.SE)
answered Mar 1, 2012 by Dan Stahlke (70 points) [ no revision ]
Collective measurements tend to do better. But as mentioned in the comments the original question asks about distinguishing states which are indistinguishable: they have the same reduced density matrix.

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$\hbar$ysicsOv$\varnothing$rflow
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
...