tag:blogger.com,1999:blog-1938680703532237797.post2202473591592168132..comments2016-06-02T00:14:23.402-07:00Comments on Spherical Harmonics: On quantum measurement (Part 4: Born's rule)Chris Adamihttps://plus.google.com/109210086614267908715noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-1938680703532237797.post-48905675306343473062014-09-13T14:52:39.195-07:002014-09-13T14:52:39.195-07:00^ Thanks!^ Thanks!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1938680703532237797.post-13885998189399617592014-08-27T20:57:47.484-07:002014-08-27T20:57:47.484-07:00Regarding the orthogonality argument - I think a f...Regarding the orthogonality argument - I think a formal version might look something like this:<br /><br />Imagine that the quantum state-space is n-dimensional. So when you describe some arbitrary state, instead of:<br />|Q>=a |0> + b |1><br />It's<br />|Q>=a_0 |0> + a_1 |1> + a_2 |2> ... a_n |n><br /><br />Subject to the typical normalization conditions. Now let's say that |q> and |r> are two states that are chosen from an even random distribution.<br /><br />Then there is some basis set that has |q_0>=|q> as it's first element so that we can write |r> in that basis:<br />|r>=a_0| |q_0> + a_1 |q_1> + a_2 |q_2> ... a_n | q_n><br /><br />Because of the normalization conditions, we have that a_0 goes as the 1 divided by the square root of n, so it goes to 0 as n gets big. But that means that we expect |q> and |r> to get 'more orthogonal' as n gets bigger.<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1938680703532237797.post-16091337451776599742014-08-23T22:28:57.128-07:002014-08-23T22:28:57.128-07:00An arbitrary state |a> cannot be entangled with...An arbitrary state |a> cannot be entangled with more than one system. So |a>|b1>=|a>|b2> is not possible. This is called the "monogamy of entanglement", see http://www.quantiki.org/wiki/Monogamy_of_entanglementChris Adamihttp://www.blogger.com/profile/02447043823985095127noreply@blogger.comtag:blogger.com,1999:blog-1938680703532237797.post-25743039091398737812014-08-09T12:02:14.144-07:002014-08-09T12:02:14.144-07:00Your blog is fascinating, I've read all of ent...Your blog is fascinating, I've read all of entropy and quantum measurement articles in one session! That said, I'm now haunted by this passage:<br /><br />"Oh, it just so happens that a classical system, because it has so many entangled particles, must be described in terms of a basis that is so high-dimensional that it will appear orthogonal to any other high-dimensional system (simply because almost all vectors in a high-dimensional space are orthogonal)."<br /><br />Could you reformulate this? I feel like I'm missing the obvious...<br /><br />If I have a system which is one qubit |a> entangled with many many particles b. <br />ie. S= { |a>|b1=a>|b2=a>|b3=a>...|b(n-1)=a>.} with n huge. Then I measure in the 2^n dimensional orthonormal basis B= { |m>, -1<m<2^n }.<br /><br />I still get a non-classical result no?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1938680703532237797.post-2852745833771600482014-08-05T21:07:06.432-07:002014-08-05T21:07:06.432-07:00It's the reaction I was going for. Glad to see...It's the reaction I was going for. Glad to see that it occurred in at least one individual!Chris Adamihttp://www.blogger.com/profile/02447043823985095127noreply@blogger.comtag:blogger.com,1999:blog-1938680703532237797.post-77305825324490214952014-08-04T07:27:10.273-07:002014-08-04T07:27:10.273-07:00"simply because almost all vectors in a high-..."simply because almost all vectors in a high-dimensional space are orthogonal"<br /><br />reading this, i literally smacked my forehead in shock and recognition of it's plain obviousness and correctness, a reaction i have formerly only seen in cartoons!Anonymousnoreply@blogger.com