live! from Soda Hall!

[nibot]

Meh. I was all set to get an MRI brain-scan tomorrow morning in return for taking some quantum superposition of a sugar pill and this substance:

But it seems that "because of their cholinomimetic actions, cholinesterase inhibitors should be prescribed with care to patients with a history of asthma or obstructive pulmonary disease" I'm not eligible for the experiment. )-:

The other funny thing that happened today is that I called Diane and Christos Papadimitriou answered the phone!

Comments

nitpicking....

[anemone.livejournal.com]

A superposition isn't the same as a mixed state...

Re: nitpicking....

[nibot.livejournal.com]

What's the difference exactly?

Re: nitpicking....

[anemone.livejournal.com]

So, now I'm about to betray the fact that I know very little about quantum computing, and I may even have gotten my terms right.

A mixed state, if I remember correctly, is one where you have some probability of one thing being true, and some probably of another thing being true. Like, with probability 1/2, the two bits are 01, and with probability 1/2, they are 10.

But if you have a superposition, then it's a state like |10> - |01> (there's a normalizing constant there).

If you perform a measurement in the right basis, you can tell the difference between these two things. I am not at a postscript-capable computer at the moment, but you might try looking at:
http://www.cs.berkeley.edu/~vazirani/qc.html

It's probably in one of the first few lectures.

Re: nitpicking....

[nibot.livejournal.com]

In my mind those are the same thing. If you have the state (|10>-|01>)/sqrt(2), then you have equal probability of measuring 10 as you have as measuring 01, and after you measure the first bit, you know what the second will be. I'm fairly confident 'superposition' and 'mixed state' are synonymous (modulo the little convention that you normalize states so that

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In my mind those are the same thing. If you have the state (|10>-|01>)/sqrt(2), then you have equal probability of measuring 10 as you have as measuring 01, and after you measure the first bit, you know what the second will be. I'm fairly confident 'superposition' and 'mixed state' are synonymous (modulo the little convention that you normalize states so that <x|x>=1.)

Re: nitpicking....

[anemone.livejournal.com]

The superposition, |00> + |11> is not the same as
with probability 1/2, 00
1/2 11.

That I am pretty sure about. I think the second setup is called a mixed s state. (this I am not so sure about).

I can't remember the experiement that distiguishes the two, and none of my quantum friends are nearby, and it was not, alas, in the first three of Umesh's notes.

Re: nitpicking....

[nibot.livejournal.com]

kennyjensen says you're right about mixed states and superpositions being different:

i'm going to have to go with greenanemone on this one.  a mixed state is                                                                     
different than a superposition.  a mixed state describes an ensemble of                                                                      
states.  maybe 1/2 of the states in the ensemble are in the state |00>+|01>                                                                  
and maybe the other 1/2 are in the state |00>+|11>.  a superposition                                                                         
describes one state which could have components along a number of basis                                                                      
states.                                                                                                                                      
                                                                                                                                             
if you want to learn more about mixed states look up stuff on the "density                                                                   
matrix."  sakurai has info about it.  but actually, nielson and chuang, a                                                                    
quantum computing text, might be the best resource.             

But I think that 'superposition' does have the meaning I intended.

Re: nitpicking....

[nibot.livejournal.com]

And I think (|00> + |11>)/sqrt(2) is the same as "with probability 1/2 |00>, with prob 1/2 |11>". i.e., Kenny describes a mixed state as a collection of superpositions in that form.

[anemone.livejournal.com]

So, I walked to my quantum buddy and asked.

Consider case A, a quantum bit in
|0> + |1> (I'm ignoring normalizing constants)

and the mixed state
with probability 1/2, |0>
with probability 1/2, |1>

If you measure in the standard 0,1 basis, they appear the same. But suppose you measure in a rotated basis, with your basis vectors |0> + |1> and |0> - |1> (again, normalized). To measure a quantum state in a basis, you take the dot product of state with each of the basis vectors, square them, and that gives you a probability distribution over the basis vectors.

In case A, you'll always get back the 1 on the first vector and zero on the second.

In case B, half the time you'll get 1 on the first vector and zero on the second.

Then, in case A, you always get back the first vector.

In case B, you get back the first vector half the time, and the second vector the other half the time.