Downloadable PDF
(IN CASE THE TEXT BELOW IS UNREADABLE)

Preprint

6/20/2019 Copyright

Not So Fast  Qbists - - The Frequency Operator Derivation Of Born’s Rule Is OK!

David Carlyle Anacker, PhD Physics

The business of this essay is to refute the main objection^1,2,3, commonly proffered by Qbists, to the frequency operator derivation of Born’s rule in quantum mechanics. Qbism, short for quantum Bayesianism, is an interpretation of quantum mechanics in terms of subjective Bayesian probability. A more general critique of Qbism and associated bibliography may be found by Googling Marian Kupczynski’s Correspondnce to: marian.kupczynski@uqo.ca  entitled  “Greeks were right: critical comments on Qbism”.

Let us assume ïY ñ  =  cï 0ñ + sï 1ñ  is unity normalized  and for convenience c , s both real. (The same conclusions can be reached for complex c, s.) Also let  êY^Nñ =  çYñçYñ...N factors...çYñ, with due apologies for omitting Ä here and throughout. One notes if ïY ñ is a sum of two kets ïY ^N ñ can be expanded as a sum of 2^N terms, e.g., ïY ³ñ =  c³ï 0ñï 0ñï0ñ

 + c²sï 0ñï 0ñï1ñ  + c²sï 0ñï 1ñï0ñ  + cs²ï 0ñï 1ñï1ñ +  c²sï 1ñï 0ñï0ñ  + cs²ï 1ñï 0ñï1ñ 

+ cs²ï 1ñï 1ñï0ñ  + s³ï 1ñï 1ñï1ñ has 2³ = eight terms. ïY ^N ñ  discussed here is the suitable representative wave vector if one sets up N widely separated systemsïY ñ  =  cï 0ñ + sï 1ñ in order to observe what fraction of them are in state ï 0ñ. “Born’s rule” postulated (at least  according to Born and others before Qbism) this fraction would approach c² as N ® ¥. Finkelstein, Graham and Hartle used only the inherent Hilbert space structure of quantum mechanics to derive Born’s rule. Their proof rests on an equation commonly called the FGH theorem or the frequency operator theorem. Many authors accept the FGH theorem as proof of Born’s rule but as indicated at the outset Qbists and others have purported to refute it.

In the remainder we examine the FGH theorem and why if true it immediately implies Born’s rule, next we discuss the main purported refutation of said theorem, and lastly explain why this purported refutation completely fails. One can streamline this agenda by the convention of introducing a set of N +1 projection operators P^N0, P^N1, P^N2,..., P^NN   that project out of  êY^Nñ   all ket products containing only 0ï 0ñ’s, 1ï 0ñ, 2ï 0ñ’s,..., Nï 0ñ ’s respectively. For example if  çYñ =  cï 0ñ + sï 1ñ we determinedïY ³ñ =  c³ï 0ñï 0ñï0ñ  + c²sï 0ñï 0ñï1ñ  + c²sï 0ñï 1ñï0ñ 

+ cs²ï 0ñï 1ñï1ñ +  c²sï 1ñï 0ñï0ñ  + cs²ï 1ñï 0ñï1ñ  + cs²ï 1ñï 1ñï0ñ  + s³ï 1ñï 1ñï1ñ. Accordingly  P ³0ïY ³ñ  =  s³ï 1ñï 1ñï1ñ, P ³1ïY ³ñ  = cs²ï 0ñï 1ñï1ñ + cs²ï 1ñï 0ñï1ñ 

+ cs²ï 1ñï 1ñï0ñ, P ³3ïY ³ñ  = c³ï 0ñï 0ñï0ñ  etc.  Some useful properties of these projection operators are: S^N_n=0P^Nn = 1operator,  P^Nn P^Nm =  P^Nn dn,m  with which it easily follows 

ïY^Nñ =  S^N_n=0P^NnïY^Nñ, S^N_n=0ïï P^NnïY^Nñ ïï² = 1.We will get to the promised objectives after one more definition.The frequency ( of observing ï 0ñ ) operator F^N is defined

F^NïY^Nñ = S^N_n=0(n/N)P^NnïY^Nñ. According to the previous paragraph’s statement of Born’s rule, sufficient to proving it is to prove  F^NïY^Nñ ® c² ïY^Nñ as N ® ¥. Obviously this is the same as

proving F^NïY^Nñ - c² ïY^Nñ  ® 0 as N ® ¥. Quantum mechanics is a Hilbert space structure

in which by definition ïuñ = 0 implies á u êuñ = 0 and vice versa , i.e. ïuñ  = 0 implies

ïïu ïï² = 0 and vice versa. FGH (and independently others) consequently set about proving instead the so called frequency operator or  FGH theorem: 

             [1]     (For çYñ = cï 0ñ + ....,)  ïïF^NïY^Nñ - c² ïY^Nñ ïï² ® 0 as N ® ¥  

The literature shows a consensus of authors agreeing equation [1], the FGH theorem, has been completely proven as a matter of pure mathematics. An immediate direct implication of the FGH theorem is that unit magnitude ïY^Nñ is comprised of a sole eigenvector of observable F^N

as N ® ¥, corresponding to eigenvalue c², and consequently the measured frequency of observing 

ï 0ñ must be this eigenvalue c² in the N ® ¥  limit. The FGH theorem is very significant since it obviously implies the fundamental statistical prediction regarding experiment of quantum mechanics, Born’s rule, from nothing but the underlying Hilbert space structure of quantum mechanics. 

In 1990 E.J. Squires¹ purported to refute by contradiction the FGH theorem [1] by contradicting its above noted direct implication that unit magnitude ïY^Nñ is comprised of a sole eigenvector of observable F^N as N ® ¥, corresponding to eigenvalue c². Squires looked at the exampleïY ñ  =  

cï 0ñ + sï 1ñ unity normalized with c , s both real, claiming for such ïY ñ, as N ® ¥ ïY^Nñ in fact becomes orthogonal to all F^N eigenvector subspaces. Squire’s assertion was that

[2] (for ïY ñ = cï 0ñ + sï 1ñ, ) á Y^N êP^NnïY^Nñ = ïï P^NnïY^Nñ ïï² ® 0 for all 0 £ n £ N as N ® ¥.

Curiously, in a 2005 arXive paper² Carlton Cave and Rudiger Shack asserted that FGH theorem [1] had been rigorously verified as a matter of pure mathematics and simultaneously quoted and supported Squires’ 1990 purported refutation of it. This author does not know what other to make of Cave and Shacks 2005 arXive paper except that thus they actually say the FGH theorem is simultaneously right and wrong!!

Firstly, as regards Squires’ purported 1990 refutation of the FGH theorem [1], his argument can be expected to have a flaw since [1] has been exhaustively and repeatedly verified as a matter of pure mathematics and [1] clearly implies that unit magnitude ïY^Nñ is an eigenvector of observable F^N with eigenvalue c² as N® ¥.

Secondly, if one adopts Squires’ test case ï Y ñ  =  cï 0ñ + sï 1ñ unity normalized with c , s both real, it is possible to show Squires was mistaken because it turns out if one selects n = Nc²  exactly an integer and makes N ® ¥, ïï P^Nn = Nc²ïY^Nñ ïï² ® 1, not 0 as Squires’ alleged, while simultaneously for all other 0 £ n £ N, ïï P^Nn ïY^Nñ ïï² ® 0  since S^N_n=0ïï P^NnïY^Nñ ïï² = 1.

To verify this assume ï Y ñ = cï 0ñ + sï 1ñ unity normalized with c , s both real, and consider that the direct expansion of ïY^Nñ contains 2^N terms each being an N ket product of the form

cⁿç0ñç0ñ...n factors...ç0ñs^{N - n}ç1ñç1ñ...(N - n) factors...ç1ñ although of course the ç0ñ and ç1ñ kets are generally intermingled. The multiplicity of different terms with n out of N kets a ç0ñ is N!/[(N - n)! n!] and such terms are all mutually orthogonal having individual

norm² = c²ⁿs^{2(N - n)} . It directly follows

[3]        ïï P^Nn ïY^Nñ ïï²  =  c²ⁿs^{2(N - n)}  N!/[(N - n)! n!]

To finish refuting Squires (1990) it only remains to show that if n = Nc² happens to be an integer, the right hand side of  formula [3] ® 1 as  N ® ¥. There are at least three ways to accomplish this: 

(l) Recast the r.h.s. of [3] algebraically, presumably with the aid of Sterling’s formulae for the log of factorials, so that the above objective can be analytically demonstrated. Perhaps this option is better left to those with extensive mathematical handbooks.

(ll) One notes that the r.h.s.of [3] is also a formula for computation of the probabilities attendant N tosses of a two sided “coin” which has “0“ and “1“ side probabilities c² and s² respectively for a single throw. The r.h.s. of [3] gives the probability of getting n “0“ sides up in N throws and is known to go to 1 as N goes to ¥ for n = c²N an integer and go to zero as N goes to ¥ for n held to be an integer any other set fraction of N. The author believes gambling mavens may know this but perhaps some readers would have to reflect awhile.

(ll1) One can sample formula [3] over a variety of points in ( c, s, N, n ) space - - this is the route this author has taken - - results shown below are already enough to empirically refute Squires.

                                                                                                                           ( Assuming

                                                                                  çYñ =  cï 0ñ + sï 1ñ )

                                                                                  ïï P^NnïY^Nñ ïï² =

Data  Point     c       s                 N             n                    N!/[(N - n)!n!]c²ⁿs^{2(N - n)}  

               

   D₁         0.5    0.75^1/2         1000       0                      0.000000

   D₂       0.5    0.75^1/2        1000       500                  0.000000

   A₁       0.5    0.75^1/2        1000       250 = Nc²       1.000000

   A₂       0.5    0.75^1/2        10000     2500 = Nc²     1.000003

   B₁       1/5    (24/25)^1/2    1000       40 = Nc²         1.000000

   B₂       1/5    (24/25)^1/2    10000     400 = Nc²       1.000004

 

Some care is required to numerically evaluate expression [3] to obtain the right most column. This study used Sterling’s approximation ln (m!) = m ln(m) - m for large m to recast

ïï P^NnïY^Nñ ïï² =  exp[N ln(N) - (N - n) ln (N - n) - n ln(n)] exp[2n ln(c) + 2(N - n)ln(s)]. Then a CASIO fx - 260 SOLAR electronic hand calculator was used to separately compute the two exponential function arguments after which they were added and their sum exponentiated. Of course a desktop digital computer with 16 significant figure accuracy processor could avoid the 3 and 4 seen in the 6th decimal place in the last column. Procedure (lll) refutes Squire’s 1990 argument that FGH theorem [1] is mistaken. (Parenthetically, sampling of [3] also verifies observation (ll) ).

In conclusion, a consensus that the FGH theorem [1] has been proven as a matter of pure mathematics ipso facto already prejudices against E.J. Squires’ purported refutation of it in 1990. Our essay additionally refutes Squire’s attack on [1] and by extension Cave and Shack’s purported 2005 rejection of frequency operator derivation of Born’s rule in quantum mechanics.

Footnotes:

¹ E. J. Squires, Phys. Lett. A 145, 67 (1990).

² “Properties of the frequency operator do not imply the quantum probability postulate”, Carlton

   M. Caves and Rudiger Shack, arXiv: quant - ph/0409144v3 9 Oct 2005.

³ “The Born Rule, the Frequency Operator and the Infinite Limit”, R. Srikanth, Dipanker

   Home, Somesh Kumar, ISCQI 2011: Jan 19 - 22.