View Full Version : Gleason's theorem and N=2
vonnyn@hotmail.com
Jul17-05, 01:27 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Since Gleason\'s Theorem only holds in Hilbert Spaces of dimension\ngreater than or equal to 3, an interesting question raises itself in\nthe 2-dimensional case. Namely, are the Gleason Measures which *can\'t*\nbe represented by density operators to be considered as valid (i.e.\nrealisable in principle) states?\n\nWhatever the answer to this question I would be very interested in its\njustification.\n\nVonny N\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Since Gleason's Theorem only holds in Hilbert Spaces of dimension
greater than or equal to 3, an interesting question raises itself in
the 2-dimensional case. Namely, are the Gleason Measures which *can't*
be represented by density operators to be considered as valid (i.e.
realisable in principle) states?
Whatever the answer to this question I would be very interested in its
justification.
Vonny N
Eugene Stefanovich
Jul18-05, 01:00 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nvonnyn@hotmail.com wrote:\n> Since Gleason\'s Theorem only holds in Hilbert Spaces of dimension\n> greater than or equal to 3, an interesting question raises itself in\n> the 2-dimensional case. Namely, are the Gleason Measures which *can\'t*\n> be represented by density operators to be considered as valid (i.e.\n> realisable in principle) states?\n\n\nStrictly speaking, there are no 2D Hilbert spaces in quantum\nmechanics. Even single particle is described by an infinite-dimensional\nHilbert space.\n\nTrue, people often consider 2D spaces (e.g., spin states of the\nelectron). But these are just subspaces in the total Hilbert space, and\nrigorous definition of density operators or state vectors demands\nconsideration of the full infinite-dimensional space.\n\nEugene.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>vonnyn@hotmail.com wrote:
> Since Gleason's Theorem only holds in Hilbert Spaces of dimension
> greater than or equal to 3, an interesting question raises itself in
> the 2-dimensional case. Namely, are the Gleason Measures which *can't*
> be represented by density operators to be considered as valid (i.e.
> realisable in principle) states?
Strictly speaking, there are no 2D Hilbert spaces in quantum
mechanics. Even single particle is described by an infinite-dimensional
Hilbert space.
True, people often consider 2D spaces (e.g., spin states of the
electron). But these are just subspaces in the total Hilbert space, and
rigorous definition of density operators or state vectors demands
consideration of the full infinite-dimensional space.
Eugene.
vonnyn@hotmail.com
Jul19-05, 02:50 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>> Strictly speaking, there are no 2D Hilbert spaces in quantum\n> mechanics. Even single particle is described by an infinite-dimensional\n> Hilbert space.\n>\n> True, people often consider 2D spaces (e.g., spin states of the\n> electron). But these are just subspaces in the total Hilbert space, and\n> rigorous definition of density operators or state vectors demands\n> consideration of the full infinite-dimensional space.\n>\n\nEvery Hilbert Space we ever use to model a system is only an\napproximation of a much larger domain of phenomena - as in all branches\nof physics. If I am only interested in the spin of a spin-1/2 system\n(i.e. I am not interested in observables like position and momentum) I\nam at liberty to model the problem using the smallest Hilbert Space\nwhich is capable of answering my questions. In fact, if you think about\nit you\'ll find it is impossible *not* to do this.\n\nSo my question remains - when working in the 2-dimensional Hilbert\nSpace, which entities qualify as states? If the answer is still density\noperators only, then I would like to know why the other Gleason\nmeasures should be disallowed.\n\nVonny N.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>> Strictly speaking, there are no 2D Hilbert spaces in quantum
> mechanics. Even single particle is described by an infinite-dimensional
> Hilbert space.
>
> True, people often consider 2D spaces (e.g., spin states of the
> electron). But these are just subspaces in the total Hilbert space, and
> rigorous definition of density operators or state vectors demands
> consideration of the full infinite-dimensional space.
>
Every Hilbert Space we ever use to model a system is only an
approximation of a much larger domain of phenomena - as in all branches
of physics. If I am only interested in the spin of a spin-1/2 system
(i.e. I am not interested in observables like position and momentum) I
am at liberty to model the problem using the smallest Hilbert Space
which is capable of answering my questions. In fact, if you think about
it you'll find it is impossible *not* to do this.
So my question remains - when working in the 2-dimensional Hilbert
Space, which entities qualify as states? If the answer is still density
operators only, then I would like to know why the other Gleason
measures should be disallowed.
Vonny N.
a student
Jul19-05, 02:50 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>vonnyn@hotmail.com wrote:\n> Since Gleason\'s Theorem only holds in Hilbert Spaces of dimension\n> greater than or equal to 3, an interesting question raises itself in\n> the 2-dimensional case. Namely, are the Gleason Measures which *can\'t*\n> be represented by density operators to be considered as valid (i.e.\n> realisable in principle) states?\n>\n> Whatever the answer to this question I would be very interested in its\n> justification.\n\nThe short answer is no, not for physical systems satisfying standard\nquantum mechanics, because states of the latter are represented by\ndensity operators, as a basic postulate.\n\nHowever, one could still ask if there are \'non-standard\' quantum\nsystems out there, corresponding to nonstandard Gleason measures for\nn=2. A problem here, as far as I can see, is that such systems cannot\nexist together with any other system (even of the same sort) - for the\ncomposite system would have to live in a higher-dimensional Hilbert\nspace (n>2), and hence be described by a density operator according to\nGleason\'s theorem - even if the two systems are independent and\nnon-interacting!\n\nAs far as I recall, Gleason\'s theorem shows that for n>2, any\nnon-negative real function on the set of projection operators, which is\nadditive for orthogonal projection operators, must have the form\nf(P) = tr [rho P]\nfor some density operator rho. This is nice for axiomatic QM, where if\none can argue that experimental propositions form a logic isomorphic to\nthe lattice of Hilbert space projections, then one automatically gets\nthe density operator representation for probabilities.\n\nFor n=2, the only projections are 0, 1, and the set of pure states.\nSince the latter projections can be represented on the (Bloch) sphere\nof unit vectors, as\nP(n) = (1 + n.sigma)/2,\nfor all n.n=1 (where sigma is the vector of Pauli spin matrices), it\nfollows that any function satisfying the requirements of Gleason\'s\ntheorem reduces to finding a function f on the unit sphere that\nsatisfies the conditions\nf(n) >=0, f(n) + f(-n) = 1.\n\nOne set of solutions is given by\nf(n) = (1 + a.n)/2 = tr[ rho(a) P(n)],\nfor any vector a in the unit ball, corresponding to the density\noperator\nrho(a) = (1 + a.sigma)/2.\n\nBut there are many other solutions, eg,\nf(n) = 1/2 + (a.n)^3 (|a| <= 1),\nand one can ask whether these correspond to "realisable in principle"\nstates in some consistent theory. However, as pointed out above, such\nsystems would not appear to be able to exist in the same universe as\nany other (even totally independent) system having a Hilbert space\nlogic of propositions, as Gleason\'s theorem implies that the composite\nHilbert space logic must have probabilities given by a density\noperator.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>vonnyn@hotmail.com wrote:
> Since Gleason's Theorem only holds in Hilbert Spaces of dimension
> greater than or equal to 3, an interesting question raises itself in
> the 2-dimensional case. Namely, are the Gleason Measures which *can't*
> be represented by density operators to be considered as valid (i.e.
> realisable in principle) states?
>
> Whatever the answer to this question I would be very interested in its
> justification.
The short answer is no, not for physical systems satisfying standard
quantum mechanics, because states of the latter are represented by
density operators, as a basic postulate.
However, one could still ask if there are 'non-standard' quantum
systems out there, corresponding to nonstandard Gleason measures for
n=2. A problem here, as far as I can see, is that such systems cannot
exist together with any other system (even of the same sort) - for the
composite system would have to live in a higher-dimensional Hilbert
space (n>2), and hence be described by a density operator according to
Gleason's theorem - even if the two systems are independent and
non-interacting!
As far as I recall, Gleason's theorem shows that for n>2, any
non-negative real function on the set of projection operators, which is
additive for orthogonal projection operators, must have the form
f(P) = tr [\rho P]
for some density operator \rho. This is nice for axiomatic QM, where if
one can argue that experimental propositions form a logic isomorphic to
the lattice of Hilbert space projections, then one automatically gets
the density operator representation for probabilities.
For n=2, the only projections are 0, 1, and the set of pure states.
Since the latter projections can be represented on the (Bloch) sphere
of unit vectors, as
P(n) = (1 + n.\sigma)/2,
for all n.n=1 (where \sigma is the vector of Pauli spin matrices), it
follows that any function satisfying the requirements of Gleason's
theorem reduces to finding a function f on the unit sphere that
satisfies the conditions
f(n) >=0, f(n) + f(-n) = 1.
One set of solutions is given by
f(n) = (1 + a.n)/2 = tr[ \rho(a) P(n)],
for any vector a in the unit ball, corresponding to the density
operator
\rho(a) = (1 + a.\sigma)/2.
But there are many other solutions, eg,
f(n) = 1/2 + (a.n)^3 (|a| <= 1),
and one can ask whether these correspond to "realisable in principle"
states in some consistent theory. However, as pointed out above, such
systems would not appear to be able to exist in the same universe as
any other (even totally independent) system having a Hilbert space
logic of propositions, as Gleason's theorem implies that the composite
Hilbert space logic must have probabilities given by a density
operator.
vonnyn@hotmail.com
Jul19-05, 01:13 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>> The short answer is no, not for physical systems satisfying standard\n> quantum mechanics, because states of the latter are represented by\n> density operators, as a basic postulate.\n\nThis part of your argument isn\'t very convincing because my question\nreally amounts to: \'who says so?\'. That is, who says that the density\noperators are the \'basic postulate\' rather than the Gleason Measures?\nIt could just be that we stumbled over the density operators before we\nrealised that Gleason Measures are really nature\'s choice.\n\n> However, one could still ask if there are \'non-standard\' quantum\n> systems out there, corresponding to nonstandard Gleason measures for\n> n=2. A problem here, as far as I can see, is that such systems cannot\n> exist together with any other system (even of the same sort) - for the\n> composite system would have to live in a higher-dimensional Hilbert\n> space (n>2), and hence be described by a density operator according to\n> Gleason\'s theorem - even if the two systems are independent and\n> non-interacting!\n\nIs it a theorem that if you project a Gleason Measure on the larger\nspace down to one on the smaller space then you necessarily get a\ndensity operator on the smaller space? If so, it doesn\'t seem obvious\nto me. Remember that a similar thing happens with self-adjoint\noperators. Namely, when we project the PVM of a perfectly decent\nself-adjoint operator onto a smaller space we can end up with a POVM\nwhich isn\'t a PVM. Physicists have now finally started to accept the\nindispensibility of POVMs.\n\nOf course, I\'m not sure if this argument holds water for the case of\nstates, but thanks for your thoughts.\n\nVonny\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>> The short answer is no, not for physical systems satisfying standard
> quantum mechanics, because states of the latter are represented by
> density operators, as a basic postulate.
This part of your argument isn't very convincing because my question
really amounts to: 'who says so?'. That is, who says that the density
operators are the 'basic postulate' rather than the Gleason Measures?
It could just be that we stumbled over the density operators before we
realised that Gleason Measures are really nature's choice.
> However, one could still ask if there are 'non-standard' quantum
> systems out there, corresponding to nonstandard Gleason measures for
> n=2. A problem here, as far as I can see, is that such systems cannot
> exist together with any other system (even of the same sort) - for the
> composite system would have to live in a higher-dimensional Hilbert
> space (n>2), and hence be described by a density operator according to
> Gleason's theorem - even if the two systems are independent and
> non-interacting!
Is it a theorem that if you project a Gleason Measure on the larger
space down to one on the smaller space then you necessarily get a
density operator on the smaller space? If so, it doesn't seem obvious
to me. Remember that a similar thing happens with self-adjoint
operators. Namely, when we project the PVM of a perfectly decent
self-adjoint operator onto a smaller space we can end up with a POVM
which isn't a PVM. Physicists have now finally started to accept the
indispensibility of POVMs.
Of course, I'm not sure if this argument holds water for the case of
states, but thanks for your thoughts.
Vonny
Eugene Stefanovich
Jul19-05, 01:14 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\nvonnyn@hotmail.com wrote:\n>>Strictly speaking, there are no 2D Hilbert spaces in quantum\n>>mechanics. Even single particle is described by an infinite-dimensional\n>>Hilbert space.\n>>\n>>True, people often consider 2D spaces (e.g., spin states of the\n>>electron). But these are just subspaces in the total Hilbert space, and\n>>rigorous definition of density operators or state vectors demands\n>>consideration of the full infinite-dimensional space.\n>>\n>\n>\n> Every Hilbert Space we ever use to model a system is only an\n> approximation of a much larger domain of phenomena - as in all branches\n> of physics. If I am only interested in the spin of a spin-1/2 system\n> (i.e. I am not interested in observables like position and momentum) I\n> am at liberty to model the problem using the smallest Hilbert Space\n> which is capable of answering my questions. In fact, if you think about\n> it you\'ll find it is impossible *not* to do this.\n>\n> So my question remains - when working in the 2-dimensional Hilbert\n> Space, which entities qualify as states? If the answer is still density\n> operators only, then I would like to know why the other Gleason\n> measures should be disallowed.\n\nSuppose there is a measure in the 2D subspace of electron\'s spin that is\nnot given by any density operator. Then this measure should be a part of\na complete measure on the full infinite-dimensional electron\'s Hilbert\nspace. I think it can be proved that the "bigger" measure cannot be\nrepresented by a density operator if the "smaller" measure has this\nproperty. I don\'t know the proof, but this statement looks reasonable to\nme. If this theorem is true, then we got a contradiction, because\naccording to Gleason the "bigger" measure with such a property does not\nexist.\n\nEugene.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>vonnyn@hotmail.com wrote:
>>Strictly speaking, there are no 2D Hilbert spaces in quantum
>>mechanics. Even single particle is described by an infinite-dimensional
>>Hilbert space.
>>
>>True, people often consider 2D spaces (e.g., spin states of the
>>electron). But these are just subspaces in the total Hilbert space, and
>>rigorous definition of density operators or state vectors demands
>>consideration of the full infinite-dimensional space.
>>
>
>
> Every Hilbert Space we ever use to model a system is only an
> approximation of a much larger domain of phenomena - as in all branches
> of physics. If I am only interested in the spin of a spin-1/2 system
> (i.e. I am not interested in observables like position and momentum) I
> am at liberty to model the problem using the smallest Hilbert Space
> which is capable of answering my questions. In fact, if you think about
> it you'll find it is impossible *not* to do this.
>
> So my question remains - when working in the 2-dimensional Hilbert
> Space, which entities qualify as states? If the answer is still density
> operators only, then I would like to know why the other Gleason
> measures should be disallowed.
Suppose there is a measure in the 2D subspace of electron's spin that is
not given by any density operator. Then this measure should be a part of
a complete measure on the full infinite-dimensional electron's Hilbert
space. I think it can be proved that the "bigger" measure cannot be
represented by a density operator if the "smaller" measure has this
property. I don't know the proof, but this statement looks reasonable to
me. If this theorem is true, then we got a contradiction, because
according to Gleason the "bigger" measure with such a property does not
exist.
Eugene.
Seratend
Jul20-05, 02:39 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>vonnyn@hotmail.com wrote:\n> Since Gleason\'s Theorem only holds in Hilbert Spaces of dimension\n> greater than or equal to 3, an interesting question raises itself in\n> the 2-dimensional case. Namely, are the Gleason Measures which *can\'t*\n> be represented by density operators to be considered as valid (i.e.\n> realisable in principle) states?\n>\n> Whatever the answer to this question I would be very interested in its\n> justification.\n>\n> Vonny N\n\nI recommend the short paper of P.Bush quant-ph/9909073 that should\nanswer your question.\nIn this paper, Bush has a very simple and elegant demonstration of\ngleason theorem where if we enlarge the set of projectors to the set of\nPOVM it also works for the case n=2 (allows to understand better the\nspecificities of n=2).\n\nIt also gives an example that explains why we can find a function on\nthe set of [orhtogonal] projectors that may not be uniquely extended as\na positive linear functional but still satisfies the properties (on the\nset of orthogonal projectors) of the theorem in the case dim=2. We can\nsee easily that such a function does not satisfy the properties of the\ntheorem for all the POVM (hence the result of his demonstration).\n\nI hope this answer your question.\n\nSeratend.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>vonnyn@hotmail.com wrote:
> Since Gleason's Theorem only holds in Hilbert Spaces of dimension
> greater than or equal to 3, an interesting question raises itself in
> the 2-dimensional case. Namely, are the Gleason Measures which *can't*
> be represented by density operators to be considered as valid (i.e.
> realisable in principle) states?
>
> Whatever the answer to this question I would be very interested in its
> justification.
>
> Vonny N
I recommend the short paper of P.Bush http://www.arxiv.org/abs/quant-ph/9909073 that should
answer your question.
In this paper, Bush has a very simple and elegant demonstration of
gleason theorem where if we enlarge the set of projectors to the set of
POVM it also works for the case n=2 (allows to understand better the
specificities of n=2).
It also gives an example that explains why we can find a function on
the set of [orhtogonal] projectors that may not be uniquely extended as
a positive linear functional but still satisfies the properties (on the
set of orthogonal projectors) of the theorem in the case dim=2. We can
see easily that such a function does not satisfy the properties of the
theorem for all the POVM (hence the result of his demonstration).
I hope this answer your question.
Seratend.
a student
Jul20-05, 02:39 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>vonnyn@hotmail.com wrote:\n> Is it a theorem that if you project a Gleason Measure on the larger\n> space down to one on the smaller space then you necessarily get a\n> density operator on the smaller space? If so, it doesn\'t seem obvious\n> to me. Remember that a similar thing happens with self-adjoint\n> operators. Namely, when we project the PVM of a perfectly decent\n> self-adjoint operator onto a smaller space we can end up with a POVM\n> which isn\'t a PVM. Physicists have now finally started to accept the\n> indispensibility of POVMs.\n\nThe issue is not really one of projection onto a smaller space, because\nthere is in fact no well-defined \'smaller space\' to project onto for a\nnon-Gleason measure: the embedded logic of propositions corresponding\nto the 2-dim subsystem cannot correspond to any 2-dim subspace of the\nlarger space.\n\nIn particular, let {A} denote the set of projections on the 2-dim\nHilbert space, corresponding to a system described by a non-Gleason\nmeasure f(A)(i.e., by a probability measure not generated by a density\noperator). Let {B} denote a set of projections on an n-dim Hilbert\nspace (n>=2), corresponding to a second system described by a\nprobability measure g(B) (either a Gleason or non-Gleason measure - of\ncourse, it must be a Gleason measure if n>2).\n\nNow consider a composite system, generated by the experimental\npropositions corresponding to (A) and {B}. We assume that this\ncomposite system has a logic of propositions corresponding to the set\nof projections {C} on some Hilbert space (if we don\'t have Hilbert\nspace, we can\'t even start talking about Gleason measures). Clearly\nthere must be distinct propositions in C corresponding to each\nexperimental proposition of the form "A and B" (eg, consider the case\nwhere the two subsystems are totally independent and non-interacting).\nThe corresponding \'joint\' projection in {C} will be denoted by A*B.\nNote that A and A*1 are physically equivalent, and hence the\nprojections {A} on the 2-dim Hilbert space are isomorphic to the\nprojections {A*1} on the composite Hilbert space.\n\nIt follows that the Hilbert space of the composite system must have\ndimension of at least 2n (as there are at least this many mutually\northogonal propositions of the form A*B), and so by Gleason\'s theorem\nall probability measures for this system must be generated by some\ndensity operator rho. In particular, one must have\nprob(A*B) = tr[rho A*B) .\nNote, by the way, that this implies that\nf(A) = prob(A) = tr[rho A*1],\nand hence the non-Gleason measure CAN always be represented by a\ndensity operator on a LARGER Hilbert space. In fact, choosing n=2, one\nonly ever needs a 4-dim Hilbert space to do so. It would be\ninteresting to know if one can get away with just 3 dimensions -\nprobably not (consider the case where g(B) is also non-Gleason).\n\nNow, suppose that the projections {A}, or equivalently {A*1} do in fact\ncorrespond to some 2-dim subspace S of the composite Hilbert space\n(this includes the case where the composite space is be the tensor\nproduct of the two subsystem Hilbert spaces, with A*B being the tensor\nproduct of A and B). But in this case one must then have\nf(A) = prob(A) = prob(A*1) = tr[rho A*1] = tr_S[rho_S A*1]\nwhere rho_S is the projection of rho onto S, and tr_S denotes the trace\nover S. But it was assumed that f(A) was non-Gleason. Contradiction!\nSo the supposition is false, as advertised, i.e., the projections {A}\nCANNOT be embedded in any 2-d subspace of the composite system.\n\nThe above does not rule out non-Gleason systems per se, but it shows\nthat, for example, they cannot be combined as tensor products with each\nother or with standard quantum systems. This implies, I think, that\ntheir observables must be automatically entangled with all other\nsystems.\n\nFurther, they cannot evolve independently of other systems - only\nunitary evolution preserves the logic of the projections (Wigner\'s\ntheorem), and hence, by an argument similar to the one above, a unitary\noperator U for the composite system cannot decompose into the tensor\nproduct of two non-commuting unitary operators U_1and U_2 for the\nsubsystems.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>vonnyn@hotmail.com wrote:
> Is it a theorem that if you project a Gleason Measure on the larger
> space down to one on the smaller space then you necessarily get a
> density operator on the smaller space? If so, it doesn't seem obvious
> to me. Remember that a similar thing happens with self-adjoint
> operators. Namely, when we project the PVM of a perfectly decent
> self-adjoint operator onto a smaller space we can end up with a POVM
> which isn't a PVM. Physicists have now finally started to accept the
> indispensibility of POVMs.
The issue is not really one of projection onto a smaller space, because
there is in fact no well-defined 'smaller space' to project onto for a
non-Gleason measure: the embedded logic of propositions corresponding
to the 2-dim subsystem cannot correspond to any 2-dim subspace of the
larger space.
In particular, let {A} denote the set of projections on the 2-dim
Hilbert space, corresponding to a system described by a non-Gleason
measure f(A)(i.e., by a probability measure not generated by a density
operator). Let {B} denote a set of projections on an n-dim Hilbert
space (n>=2), corresponding to a second system described by a
probability measure g(B) (either a Gleason or non-Gleason measure - of
course, it must be a Gleason measure if n>2).
Now consider a composite system, generated by the experimental
propositions corresponding to (A) and {B}. We assume that this
composite system has a logic of propositions corresponding to the set
of projections {C} on some Hilbert space (if we don't have Hilbert
space, we can't even start talking about Gleason measures). Clearly
there must be distinct propositions in C corresponding to each
experimental proposition of the form "A and B" (eg, consider the case
where the two subsystems are totally independent and non-interacting).
The corresponding 'joint' projection in {C} will be denoted by A*B.
Note that A and A*1 are physically equivalent, and hence the
projections {A} on the 2-dim Hilbert space are isomorphic to the
projections {A*1} on the composite Hilbert space.
It follows that the Hilbert space of the composite system must have
dimension of at least 2n (as there are at least this many mutually
orthogonal propositions of the form A*B), and so by Gleason's theorem
all probability measures for this system must be generated by some
density operator \rho. In particular, one must have
prob(A*B) = tr[\rho A*B) .
Note, by the way, that this implies that
f(A) = prob(A) = tr[\rho A*1],
and hence the non-Gleason measure CAN always be represented by a
density operator on a LARGER Hilbert space. In fact, choosing n=2, one
only ever needs a 4-dim Hilbert space to do so. It would be
interesting to know if one can get away with just 3 dimensions -
probably not (consider the case where g(B) is also non-Gleason).
Now, suppose that the projections {A}, or equivalently {A*1} do in fact
correspond to some 2-dim subspace S of the composite Hilbert space
(this includes the case where the composite space is be the tensor
product of the two subsystem Hilbert spaces, with A*B being the tensor
product of A and B). But in this case one must then have
f(A) = prob(A) = prob(A*1) = tr[\rho A*1] = tr_S[\rho_S A*1]
where \rho_S is the projection of \rho onto S, and tr_S denotes the trace
over S. But it was assumed that f(A) was non-Gleason. Contradiction!
So the supposition is false, as advertised, i.e., the projections {A}
CANNOT be embedded in any 2-d subspace of the composite system.
The above does not rule out non-Gleason systems per se, but it shows
that, for example, they cannot be combined as tensor products with each
other or with standard quantum systems. This implies, I think, that
their observables must be automatically entangled with all other
systems.
Further, they cannot evolve independently of other systems - only
unitary evolution preserves the logic of the projections (Wigner's
theorem), and hence, by an argument similar to the one above, a unitary
operator U for the composite system cannot decompose into the tensor
product of two non-commuting unitary operators U_{1and} U_2 for the
subsystems.
a student
Jul21-05, 12:26 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I guess that Busch\'s result shows that one cannot choose both of POVMs\nand non-Gleason measures to be fundamental. The result doesn\'t\nactually rule out one or the other though.\n\nIn particular, Gleason\'s theorem (as stated by Gleason) is important in\nquantum-logic-axiomatic approaches: it is only of fundamental interest\nif the quantum logic of projections is of fundamental significance (and\nPOVMs do not form an orthomodular poset). In contrast, Busch\'s\nderivation is important in convex-set axiomatic approaches: it is only\nof fundamental interest if probabilities are (convex-linearly)\ngenerated by a positive cone of operators.\n\nIn other approaches, eg, C*-algebra approaches, Gleason\'s theorem is\nhardly relevant at all (one can derive density operators plus a trivial\nset of \'non-normal\' states without considering projections or POVMs at\nall).\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I guess that Busch's result shows that one cannot choose both of POVMs
and non-Gleason measures to be fundamental. The result doesn't
actually rule out one or the other though.
In particular, Gleason's theorem (as stated by Gleason) is important in
quantum-logic-axiomatic approaches: it is only of fundamental interest
if the quantum logic of projections is of fundamental significance (and
POVMs do not form an orthomodular poset). In contrast, Busch's
derivation is important in convex-set axiomatic approaches: it is only
of fundamental interest if probabilities are (convex-linearly)
generated by a positive cone of operators.
In other approaches, eg, C*-algebra approaches, Gleason's theorem is
hardly relevant at all (one can derive density operators plus a trivial
set of 'non-normal' states without considering projections or POVMs at
all).
Seratend
Jul21-05, 01:33 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>a student wrote:\n> I guess that Busch\'s result shows that one cannot choose both of POVMs\n> and non-Gleason measures to be fundamental. The result doesn\'t\n> actually rule out one or the other though.\n>\nI am not sure I follow you (note: the result and demo of bush paper is\ngiven in the first page). We have two sets P(H) (the set of\n[orthogonal] projectors ) and E(H) (the set of the positive operators)\nwhere we define the "gleason/non gleason" measure. We have P(H)\nincluded in E(H) => we restrict the set of possible measures\n(satisfying the gleason properties) when we enlarge the set P(H) into\nE(H).\n\n> In other approaches, eg, C*-algebra approaches, Gleason\'s theorem is\n> hardly relevant at all (one can derive density operators plus a trivial\n> set of \'non-normal\' states without considering projections or POVMs at\n> all).\n\nI do not understand this statement. Any C*-algebras is isomorphic to\nthe usual norm closed self-adjoint sub algebra B(H) for some given\nHilbert space H, hence we may use the results of gleason theorem (~ we\nare playing with words).\n\nSeratend.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>a student wrote:
> I guess that Busch's result shows that one cannot choose both of POVMs
> and non-Gleason measures to be fundamental. The result doesn't
> actually rule out one or the other though.
>
I am not sure I follow you (note: the result and demo of bush paper is
given in the first page). We have two sets P(H) (the set of
[orthogonal] projectors ) and E(H) (the set of the positive operators)
where we define the "gleason/non gleason" measure. We have P(H)
included in E(H) => we restrict the set of possible measures
(satisfying the gleason properties) when we enlarge the set P(H) into
E(H).
> In other approaches, eg, C*-algebra approaches, Gleason's theorem is
> hardly relevant at all (one can derive density operators plus a trivial
> set of 'non-normal' states without considering projections or POVMs at
> all).
I do not understand this statement. Any C*-algebras is isomorphic to
the usual norm closed self-adjoint sub algebra B(H) for some given
Hilbert space H, hence we may use the results of gleason theorem (~ we
are playing with words).
Seratend.
a student
Jul22-05, 01:07 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>I am not saying anything about the mathematics of Gleason\'s theorem and\nBusch\'s theorem. I am pointing out that their significance is\ndifferent depending on the axiomatic approach one takes (which is\nprobably more than playing with words, but that may be a matter of\nopinion!).\n\nThe only point of having a Gleason-type theorem is to "derive" (rather\nthan postulate, as in standard QM), the density operator representation\nfor physical states. This derivation must be made on the basis of an\nassumed structure of interest, obtained by more fundamental\nconsiderations, and hence is axiomatic in nature. But there are\ndifferent types of axiomatic approaches, and corresponding Gleason-type\ntheorems. In particular, Gleason and Busch are deriving states\nrepresentations from rather different (conceptually distinct)\nassumptions, rather than trying to outdo one another.\n\nGleason was interested in the quantum logic approach, where the\nfundamental structure is the orthomodular poset of projection\noperators. In this approach states are defined as positive additive\nfunctions on the logical propositions, which are additive for mutually\nexclusive propositions, and Gleason showed this meant that if enough\naxioms were added to get a Hilbert space logic, then one ended up with\ndensity operators.\n\nIn contrast the typical quantum logician is not very interested in\nPOVMs(!) , and so is not interested in Busch\'s derivation. POVM type\napproaches arise naturally when one characterises positive observables\nby a cone, and states by positive linearly-convex measures on the cone.\nBusch\'s results are of (axiomatic) interest only in this context.\n\nIn the algebraic approach, states are defined as continuous linear\nfunctionals on the set of bounded operators, which are positive for\npositive operators. The corresponding "Gleason theorem" does not\nactually lead to a density operator representation (normal states) -\nit leads also to other types of states (singular states, and mixtures\nof these with normal states). No projections or POVMs are necessary or\nconceptually relevant to this theorem.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>I am not saying anything about the mathematics of Gleason's theorem and
Busch's theorem. I am pointing out that their significance is
different depending on the axiomatic approach one takes (which is
probably more than playing with words, but that may be a matter of
opinion!).
The only point of having a Gleason-type theorem is to "derive" (rather
than postulate, as in standard QM), the density operator representation
for physical states. This derivation must be made on the basis of an
assumed structure of interest, obtained by more fundamental
considerations, and hence is axiomatic in nature. But there are
different types of axiomatic approaches, and corresponding Gleason-type
theorems. In particular, Gleason and Busch are deriving states
representations from rather different (conceptually distinct)
assumptions, rather than trying to outdo one another.
Gleason was interested in the quantum logic approach, where the
fundamental structure is the orthomodular poset of projection
operators. In this approach states are defined as positive additive
functions on the logical propositions, which are additive for mutually
exclusive propositions, and Gleason showed this meant that if enough
axioms were added to get a Hilbert space logic, then one ended up with
density operators.
In contrast the typical quantum logician is not very interested in
POVMs(!) , and so is not interested in Busch's derivation. POVM type
approaches arise naturally when one characterises positive observables
by a cone, and states by positive linearly-convex measures on the cone.
Busch's results are of (axiomatic) interest only in this context.
In the algebraic approach, states are defined as continuous linear
functionals on the set of bounded operators, which are positive for
positive operators. The corresponding "Gleason theorem" does not
actually lead to a density operator representation (normal states) -
it leads also to other types of states (singular states, and mixtures
of these with normal states). No projections or POVMs are necessary or
conceptually relevant to this theorem.
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