A Proof of Vivo-Pato-Oshanin's Conjecture on the Fluctuation of von Neumann Entropy
aa r X i v : . [ m a t h - ph ] J u l A Proof of Vivo-Pato-Oshanin’s Conjectureon the Fluctuation of von Neumann Entropy
Lu Wei ∗ Department of Electrical and Computer EngineeringUniversity of Michigan-Dearborn, MI 48128, USA (Dated: July 10, 2018)It was recently conjectured by Vivo, Pato, and Oshanin [Phys. Rev. E , 052106 (2016)] thatfor a quantum system of Hilbert dimension mn in a pure state, the variance of the von Neumannentropy of a subsystem of dimension m ≤ n is given by − ψ ( mn + 1) + m + nmn + 1 ψ ( n ) − ( m + 1)( m + 2 n + 1)4 n ( mn + 1) , where ψ ( · ) is the trigamma function. We give a proof of this formula. I. BACKGROUND AND THE CONJECTURE
Consider a composite quantum system that consistsof two subsystems A and B of Hilbert space dimensions m and n . The Hilbert space H A + B of the compositesystem is given by the tensor product of the Hilbertspaces of the subsystems, H A + B = H A ⊗ H B . The ran-dom pure state of the composite system is written asa linear combination of the random coefficients x i,j andthe complete basis (cid:8)(cid:12)(cid:12) i A (cid:11)(cid:9) and (cid:8)(cid:12)(cid:12) j B (cid:11)(cid:9) of H A and H B , | ψ i = P mi =1 P nj =1 x i,j (cid:12)(cid:12) i A (cid:11) ⊗ (cid:12)(cid:12) j B (cid:11) . The correspondingdensity matrix ρ = | ψ i h ψ | has the natural constrainttr( ρ ) = 1. This implies that the m × n random coeffi-cient matrix X = ( x i,j ) satisfiestr (cid:16) XX † (cid:17) = 1 . (1)Without loss of generality, it is assumed that m ≤ n .The reduced density matrix ρ A of the smaller sub-system A admits the Schmidt decomposition ρ A = P mi =1 λ i (cid:12)(cid:12) φ Ai (cid:11) (cid:10) φ Ai (cid:12)(cid:12) , where λ i is the i -th largest eigenvalueof XX † . The conservation of probability (1) now impliesthe constraint P mi =1 λ i = 1 . The probability measure ofthe random coefficient matrix X is the Haar measure,where the entries are uniformly distributed over all thepossible values satisfying the constraint (1). The re-sulting eigenvalue density of XX † is well known (see,e.g., [1]), f ( λ ) = Γ( mn ) c δ − m X i =1 λ i ! × Y ≤ i By the construction (1), the random coefficient matrix X has a natural relation with a Wishart matrix YY † as XX † = YY † tr (cid:16) YY † (cid:17) , (9)where Y is an m × n ( m ≤ n ) matrix of independently andidentically distributed complex Gaussian entries. Thedensity of the eigenvalues 0 < θ m < · · · < θ < ∞ of YY † equals [12] g ( θ ) = 1 c Y ≤ i Since T = P mi =1 θ i ln θ i + 2 P ≤ i 1. Calculating I A By the fact that (cf. (29)) X ( x ) = K ( x , x ) , (39)one inserts (37) into (31) to obtain I A = m !( n − A m − ,m − − A m − ,m ) , (40)where for convenience we have further defined (cf. (45)) A s,t = A ( n − m +1 ,n − m +1) s,t ( n − m + 2) . (41)We now use (45), and the contribution to the sum A m − ,m − = m − X k =0 (cid:18) m − − k (cid:19) ( n − m + 2 + k )! k ! × (cid:0) ( ψ ( n − m + 3 + k ) + 2 ψ (2) − ψ (3 − m + k )) + ψ ( n − m + 3 + k ) + 2 ψ (2) − ψ (3 − m + k ) (cid:1) (42)consists of the cases when the binomial terms are zero( k = 0 , . . . , m − 3) with the polygamma functions beinginfinity and are nonzero ( k = m − , m − 1) with thepolygamma functions being finite. Namely, we have A m − ,m − = ( n + 1)!( m − (cid:0) ψ ( n + 2) + ψ ( n + 2) (cid:1) + n !( m − (cid:0) ( ψ ( n + 1) + 2) + ψ ( n + 1) − (cid:1) + m − X k =0 ( n − m + 2 + k )!( m − − k )! k ! 4 ψ (3 − m + k ) − ψ (3 − m + k )Γ (3 − m + k ) , (43) B ( α,β ) s,t ( q ) = Z ∞ x q e − x ln x L ( α ) s ( x ) L ( β ) t ( x ) d x = ( − s + t min( s,t ) X k =0 (cid:18) q − αs − k (cid:19)(cid:18) q − βt − k (cid:19) Γ( q + 1 + k ) k ! × (cid:0) ψ ( q + 1 + k ) + ψ ( q − α + 1) + ψ ( q − β + 1) − ψ ( q − α − s + 1 + k ) − ψ ( q − β − t + 1 + k ) (cid:1) . (44) A ( α,β ) s,t ( q ) = Z ∞ x q e − x ln x L ( α ) s ( x ) L ( β ) t ( x ) d x = ( − s + t min( s,t ) X k =0 (cid:18) q − αs − k (cid:19)(cid:18) q − βt − k (cid:19) Γ( q + 1 + k ) k ! × (cid:16)(cid:0) ψ ( q + 1 + k ) + ψ ( q − α + 1) + ψ ( q − β + 1) − ψ ( q − α − s + 1 + k ) − ψ ( q − β − t + 1 + k ) (cid:1) + ψ ( q + 1 + k ) + ψ ( q − α + 1) + ψ ( q − β + 1) − ψ ( q − α − s + 1 + k ) − ψ ( q − β − t + 1 + k ) (cid:17) . (45)which by interpreting the gamma and polygamma func-tions of negative integer arguments as the limit ǫ → − l + ǫ ) = ( − l l ! ǫ (cid:0) ψ ( l + 1) ǫ + o (cid:0) ǫ (cid:1)(cid:1) , (46a) ψ ( − l + ǫ ) = − ǫ (cid:0) − ψ ( l + 1) ǫ + o (cid:0) ǫ (cid:1)(cid:1) , (46b) ψ ( − l + ǫ ) = 1 ǫ (cid:0) o (cid:0) ǫ (cid:1)(cid:1) , (46c)leads to a well-defined limit1( m − − k )! ψ (3 − m + k ) − ψ (3 − m + k )Γ (3 − m + k ) =2( m − − k ) ( m − − k ) , k = 0 , . . . , m − . (47) In the same manner that has led to A m − ,m − , we obtain A m − ,m = n !( m − ψ ( n + 1) + 1) − ( n − m − ψ ( n ) + 1) + m − X k =0 ( n − m + 2 + k )! k ! × m − − k )( m − − k )( m − − k )( m − k ) . (48)Finally, we insert (43), (47), (48) into (40) and sim-plify the expression by rearranging the sums as well asusing (27) to obtain I A = m (cid:0) n ( m + n ) ψ ( n ) + 3 n ( m + n ) ψ ( n ) + ( m + 9 mn + 3 m + 3 n + 2) ψ ( n ) + m + 3 mn + 6 m − n − (cid:1) +2 m !( n − m − X k =1 ( n − k )!( m − − k )! 1 k ( k + 1) − m − X k =1 ( n − − k )!( m − − k )! 1 k ( k + 1)( k + 2)( k + 3) ! . (49) 2. Calculating I B Inserting (33) into (32) and using the symmetry of thecorrelation kernel, the integral I B can be represented as I B = 2 m − X j =1 m − j − X k =0 k !( k + j )! B k + j,k ( n − m + k )!( n − m + k + j )! + m − X k =0 k ! B k,k ( n − m + k )! , (50)where we have further defined (cf. (44)) B s,t = B ( n − m,n − m ) s,t ( n − m + 1) . (51)The identity (44) gives B k,k = k X j =0 (cid:18) k − j (cid:19) ( n − m + 1 + j )! j ! (cid:0) ψ (2) + ψ ( n − m + 2 + j ) − ψ (2 − k + j ) (cid:1) = ( n − m + k )! k ! (cid:0) ( n − m + 1 + 2 k ) × ψ ( n − m + 1 + k ) + 2 k + 1 (cid:1) , (52)where j = k − , k provides the nonzero contribution tothe sum and we have used (27a) for the simplification.In the same manner, one obtains B k +1 ,k = ( n − m + k )! k ! (cid:0) ( n − m + 1 + k ) × ψ ( n − m + 1 + k ) + n − m + 3 k/ (cid:1) (53)and the cases j = 2 , . . . , m − B k + j,k = ( n − m + k )! k ! j (cid:18) n − m + 1 + kj − − kj + 1 (cid:19) . (54)Inserting (52), (53), and (54) into (50), we arrive at I B = m − X k =0 (( n − m + 1 + 2 k ) ψ ( n − m + 1 + k ) + 2 k + 1) + m − X k =0 k + 1) n − m + 1 + k (( n − m + 1 + k ) ψ ( n − m + 1 + k ) + n − m + 2 + 3 k/ + m − X j =2 m − − j X k =0 n − m + k )!( k + j )!( n − m + k + j )! k ! j (cid:18) n − m + 1 + kj − − kj + 1 (cid:19) . (55) III. SIMPLIFICATION OF SUMMATIONS The remaining task is to simplify the sums appear in I A and I B to polygamma functions. This is a straight-forward but tedious task, for which we need several finitesum identities as listed in the Appendix. Some remarkson these identities are also provided in the Appendix.Though I A in (49) and I B in (55) are valid for any pos-itive integers m and n with m ≤ n , as will be seen it isconvenient to assume n > m ≥ I A and I B in the case n > m ≥ 3. The remaining special cases willbe considered at the end of this section.For ease of presentation, we cite the identities used ineach step on top of the equality symbol. The argumentof each of the resulting polygamma functions is shiftedto one of the following n − m + 2, m , n , 1, with thehelp of (27). In addition, simplification by combininglike terms is also performed in each step without beingexplicitly mentioned. We start with I A in (49), whereby using partial fraction decomposition the first sum issimplified as m − X k =1 ( n − k )!( m − − k )! 1 k ( k + 1) = m − X k =1 ( n − k )!( m − − k )! (cid:18) k + 1( k + 1) + 2 k + 1 − k (cid:19) = ( m + n ) m − X k =2 ( n − k )!( m − − k )! 1 k + 2( n − m + 1) m − X k =1 ( n − k )!( m − − k )! 1 k + ( m + n )( n − m + 1)!( m − − (2 n − m )( n − m − ( A. ) = ( m + n ) m − X k =1 ( n − k )!( m − − k )! 1 k + 2( n − m + 1) n !( m − ψ ( n ) − ψ ( n − m + 2)) + (5 n − mn − m + 2)( n − m − ( A. ) = ( m + n ) n !( m − m X k =1 ψ ( n − m + k ) k + ( m + n ) n !( m − (cid:18) ψ ( n − m + 2) − ψ ( n ) − ψ ( n − m + 2) − ψ ( n ) + ψ ( n − m + 2)( ψ ( n ) − ψ ( m ) + ψ (1)) + (cid:18) m − m + n + 1 n − m + 1 + 1 n − m + 1 n − (cid:19) ( ψ ( n − m + 2) − ψ ( n )) + (2 n − m + 1)( ψ ( m ) − ψ (1))( n − m )( n − m + 1) − ψ ( n ) m + 5 n − mn − m + 2( m + n ) n − n − mn ( n − m )( n − m + 1) (cid:19) . (56)Similarly, the second sum in (49) is simplified as m − X k =1 ( n − − k )!( m − − k )! 1 k ( k + 1)( k + 2)( k + 3) = m − X k =1 ( n − − k )!( m − − k )!2 (cid:18) k − k + 1 + 1 k + 2 − k + 3) (cid:19) ( A. ) = ( n − m ! (cid:18) m ( m − m − − n ( n + 1)( n + 2)3 + mn ( n − m + 2) (cid:19) (cid:18) ψ ( n ) − ψ ( n − m + 2) − n − m + 2 (cid:19) +( n − m ! ( m − (cid:0) m + 6 n − mn − m + 12 n + 4 (cid:1) . (57)Inserting (56) and (57) into (49), I A is simplified to I A = 2 mn ( m + n ) m X k =1 ψ ( n − m + k ) k + mn ( m + n ) (cid:18) ψ ( n − m + 2) − ψ ( n − m + 2) + 2 ψ ( n − m + 2)( ψ ( n ) − ψ ( m ) + ψ (1)) + 2(2 n − m + 1) ( ψ ( m ) − ψ (1))( n − m )( n − m + 1) (cid:19) + a ψ ( n ) + a ψ ( n − m + 2) + a ( n − m )( n − m + 1) , (58) a = n (cid:0) m n + 9 m − m n − m n − m + 7 mn − mn − mn − m + n − n − n + 2 n (cid:1) , (59) a = 13 (cid:0) m + 7 m n + 2 m − m n − m n − m + 26 m n + 18 m n + 3 m n − m − mn +10 mn + 15 mn + 4 mn − n − n − n − n (cid:1) , (60) a = 118 (cid:0) − m − m n + 14 m + 139 m n + 169 m n + 41 m − m n − m n − m n − m − mn + 87 mn − mn − mn − m + 12 n + 42 n + 42 n + 12 n (cid:1) . (61)We now simplify I B in (55), where the first two sums are m − X k =0 (( n − m + 1 + 2 k ) ψ ( n − m + 1 + k ) + 2 k + 1) + m − X k =0 k + 1) n − m + 1 + k (( n − m + 1 + k ) × ψ ( n − m + 1 + k ) + n − m + 2 + 3 k/ = 6 m − X k =0 k ψ ( n − m + 1 + k ) + 2(3 n − m + 4) m − X k =0 kψ ( n − m + 1 + k ) + ( n − m + 1)( n − m + 3) × m − X k =0 ψ ( n − m + 1 + k ) + 14 m − X k =0 k ψ ( n − m + 1 + k ) + 2(4 n − m + 11) m − X k =0 kψ ( n − m + 1 + k ) +2(3 n − m + 5) m − X k =0 ψ ( n − m + 1 + k ) + m − X k =0 k + 1)( n − m + 2 + 3 k/ n − m + 1 + k + m − X k =0 (2 k + 1) +(( n + m − ψ ( n ) + 2 m − A. ) − ( A. ) = mn ( m + n − ψ ( n ) + 16 (cid:0) m + 15 m n + 3 mn − mn − m − n + n (cid:1) ψ ( n ) + 16 ( n − m − × ( n − m )( n − m + 1) ψ ( n − m + 2) + 136 (cid:0) m + 21 m n + 6 mn − mn − m − n + 6 n + 6 (cid:1) . (62)The remaining double sums in I B needs some preprocess-ing before the sum of the types in the appendix appear. Specifically, by shifting the inner sum k → k − j , changingthe summation order, and using partial fraction decom-position, we have m − X j =2 m − j − X k =0 n − m + k )!( k + j )!( n − m + k + j )! k ! j (cid:18) n − m + 1 + kj − − kj + 1 (cid:19) = m − X k =2 k X j =2 n − m + k − j )! k !( n − m + k )!( k − j )! j (cid:18) n − m + 1 + k − jj − − k − jj + 1 (cid:19) = m − X k =2 k X j =2 n − m + k − j )! k !( n − m + k )!( k − j )! (cid:18) ( n − m + 1 + k − j ) (cid:18) j − + 1 j + 2 j − j − (cid:19) + ( k − j ) (cid:18) j + 1) + 1 j +2 j + 1 − j (cid:19) − ( n − m + 1 + k − j )( k − j ) (cid:18) j − − j + 1 − j (cid:19)(cid:19) = I + I , (63)where I and I collect terms involving 1 /j and 1 /j , respectively, as (the terms involving 1 /j cancel) I = m − X k =2 k !( n − m + k )! k +1 X j =3 ( n − m + 2 + k − j )!( k − j )! j − k − X j =1 ( n − m + k − j )!( k − − j )! j + (2 k + 1) k +1 X j =3 ( n − m + 1 + k − j )!( k − j )! j − k X j =2 ( n − m + k − j )!( k − − j )! j ! + 2( n − m + 1 / k ) k X j =2 ( n − m + 1 + k − j )!( k − j )! j − k − X j =1 ( n − m + k − j )!( k − − j )! j !! , (64) I = m − X k =2 k !( n − m + k )! k X j =2 ( n − m + 1 + k − j )!( k − − j )! j + k k X j =2 ( n − m + k − j )!( k − − j )! j + ( k + 1) k +1 X j =3 ( n − m + 1 + k − j )!( k − j )! j +( n − m + k ) k − X j =1 ( n − m + k − j )!( k − − j )! j + ( n − m + 1 + k ) k X j =2 ( n − m + 1 + k − j )!( k − j )! j ! . (65)The sums in I are further simplified as I = m − X k =2 n − m + 4)( n − m + 1 + 2 k ) k !( n − m + k )! k X j =2 ( n − m + k − j )!( k − j )! j − m − X k =2 n − m + 2) k !( n − m + k )! × k X j =2 ( n − m + k − j )!( k − j )! − m − X k =2 k ( k − n − m − k ) n − m + k ( A. ) = m − X k =2 n − m + 4)( n − m + 1 + 2 k ) k !( n − m + k )! k X j =2 ( n − m + k − j )!( k − j )! j + ( n − m ) (cid:0) m − mn − m + 4 n + n − (cid:1) ( ψ ( n ) − ψ ( n − m + 2)) + m − n − m + 1) (cid:0) m − m n − m + 28 mn + 19 mn − m − n − n + 13 n + 7 (cid:1) ( A. ) = 2(3 n − m + 4) m − X k =2 ( n − m + 1 + 2 k ) ψ ( n − m + 1 + k ) − ( mn + 2 m − n − ψ ( n − m + 2) ! − (2 n − m − n − m )( n − m + 1)( ψ ( n ) − ψ ( n − m + 2)) + 6 m − mn − m + 4 n + 6 n + 72( m − − A. ) − ( A. ) = (cid:0) m − m n − m + 12 mn + 10 mn − m − n − n + n (cid:1) ( ψ ( n ) − ψ ( n − m + 2)) +12 ( m − (cid:0) m − mn − m + 4 n − (cid:1) . (66)The sums in I are further simplified as I = m − X k =2 (cid:0) m − mn − m + n + 3 n + 2 − m − n − k + 6 k (cid:1) k !( n − m + k )! k X j =2 ( n − m + k − j )!( k − j )! j − m − X k =2 n − m + 1 + 2 k ) k !( n − m + k )! k X j =2 ( n − m + k − j )!( k − j )! j + m − X k =2 k !( n − m + k )! k X j =2 ( n − m + k − j )!( k − j )! + m − X k =2 k ( k − n − m − k )2( n − m + k ) ( A. ) − ( A. ) = m − X k =2 (cid:0) m − mn − m + n + 3 n + 2 − m − n − k + 6 k (cid:1) k !( n − m + k )! k X j =2 ( n − m + k − j )!( k − j )! j − (cid:0) m − m n − m + 3 mn + 48 mn − m − n − n + 9 n (cid:1) ( ψ ( n ) − ψ ( n − m + 2)) + m − n − m + 1) (cid:0) m − m n − m + 7 mn + 112 mn + 19 m − n − n + 2 n + 9 (cid:1) ( A. ) = m − X k =2 (cid:0) ( n − m + 2)( n − m + 1) + 6( n − m + 1) k + 6 k (cid:1) k X j =1 ψ ( n − m + j ) j − ψ ( n − m + 1 + k ) − ψ ( n − m + 1 + k ) + ψ ( n − m + 1) + ψ ( n − m + 1) + 2 ψ ( n − m )( ψ ( n − m + 1 + k ) − ψ ( k + 1) − ψ ( n − m + 1) + ψ (1)) ! − (cid:0) m − m n − m + 15 mn + 24 mn − m − n − n + n (cid:1) ( ψ ( n ) − ψ ( n − m + 2)) + ( m − (cid:0) m − m n − m + 31 mn + 48 mn + 27 m − n − n − n − (cid:1) n − m + 1) ( A. ) − ( A. ) = 2 mn ( m + n ) m X k =1 ψ ( n − m + k ) k + mn ( m + n ) (cid:18) − ψ ( n ) + ψ ( n − m + 2) − ψ ( n ) − ψ ( n − m + 2) +2 ψ ( n − m + 2)( ψ ( n ) − ψ ( m ) + ψ (1)) + 2(2 n − m + 1) ( ψ ( m ) − ψ (1))( n − m )( n − m + 1) (cid:19) + b ψ ( n ) + b ψ ( n − m + 2) + b ( n − m )( n − m + 1) , (67)where we also changed the summation order between j and k to arrive at the last equality, and b , b , b are b = 12 (cid:0) − m + 29 m n + 5 m − m n − m n + m + 62 m n + 86 m n + 17 m n − m − mn − mn − mn + 2 mn + 5 n + 5 n − n − n (cid:1) , (68) b = 12 (cid:0) m − m n − m + 62 m n + 36 m n − m − m n − m n − m n + m + 29 mn +60 mn + 25 mn − mn − n − n − n + n (cid:1) , (69) b = 14 (cid:0) − m + 83 m n + 95 m − m n − m n − m + 45 m n + 243 m n + 187 m n + 29 m − mn − mn − mn − mn + 2 m + 20 n + 26 n + 4 n − n (cid:1) . (70)With I and I being simplified as in (66) and (67), re- spectively, we now insert (62) and (63) into (55) to obtain I B = 2 mn ( m + n ) m X k =1 ψ ( n − m + k ) k + mn ( m + n ) (cid:18) − ψ ( n ) + ψ ( n − m + 2) − ψ ( n ) m + n − ψ ( n − m + 2) +2 ψ ( n − m + 2)( ψ ( n ) − ψ ( m ) + ψ (1)) + 2(2 n − m + 1) ( ψ ( m ) − ψ (1))( n − m )( n − m + 1) (cid:19) + b ψ ( n ) + b ψ ( n − m + 2) + b ( n − m )( n − m + 1) , (71)0 b = 13 (cid:0) − m + 9 m n + 9 m n − m n + 3 m n + 8 m n + 3 m + 7 mn − mn − mn − mn + n − n − n + 2 n (cid:1) , (72) b = 13 (cid:0) m + 7 m n + 2 m − m n − m n − m + 26 m n + 18 m n + 3 m n − m − mn +10 mn + 15 mn + 4 mn − n − n − n − n (cid:1) , (73) b = 118 (cid:0) − m − m n + 5 m + 139 m n + 187 m n + 41 m − m n − m n − m n + 7 m − mn + 87 mn − mn − mn − m + 12 n + 42 n + 42 n + 12 n (cid:1) . (74)We observe that I A in (58) and I B in (71) share many common terms, where by inserting (58) and (71) into (30)the remaining terms of the induced variance E g (cid:2) T (cid:3) are E g (cid:2) T (cid:3) = mn ( m + n ) ψ ( n ) + mn ( mn + 1) ψ ( n ) + m ( m + 1) nψ ( n ) + 14 m ( m + 1) +( a − b ) ψ ( n ) + ( a − b ) ψ ( n − m + 2) + a − b ( n − m )( n − m + 1)= mn ( m + n ) ψ ( n ) + mn ( mn + 1) ψ ( n ) + m (cid:0) m n + mn + m + 2 n + 1 (cid:1) ψ ( n ) + 14 m ( m + 1) (cid:0) m + m + 2 (cid:1) , where we have used the results a − b = m ( n − m )( n − m + 1)(2 n + m + 1) ,a − b = 0 ,a − b = 12 m ( m + 1)( n − m )( n − m + 1) , obtained by comparing (59)–(61) to (72)–(74). This com-pletes the proof of the induced conjecture (26) in the case n > m ≥ m ≤ n , the remaining cases to be shown are m = 1, m = 2, and m = n , where I A in (49) and I B in (55) can be directly computed. We list the simplifiedexpressions for I A , I B , and the induced variance E g (cid:2) T (cid:3) in Table I as shown on top of the next page. Each of thespecial cases is proven by comparing the expression of E g (cid:2) T (cid:3) in Table I with that of the corresponding inducedconjecture (26). We complete the proof of the VPO’sconjecture (7). ACKNOWLEDGMENTS The author wishes to thank Michael Milgram, GregorySchehr, and Yu Xiang for the inspiring discussion. Appendix: Finite Sum Identities Useful in Section III n X k =1 ψ ( k + l ) = ( n + l ) ψ ( n + l ) − lψ ( l ) − n. (A.1) n X k =1 kψ ( k + l ) = 12 (cid:0) n + n − l + l (cid:1) ψ ( n + l ) + 12 l ( l − ψ ( l ) + 14 n ( − n + 2 l − . (A.2) n X k =1 k ψ ( k + l ) = 16 (cid:0) n + 3 n + n + 2 l − l + l (cid:1) ψ ( n + l ) − l (cid:0) l − l + 1 (cid:1) ψ ( l ) +136 n (cid:0) − n + 6 nl − n − l + 12 l + 1 (cid:1) . (A.3)1 TABLE I. Special Cases m = 1 I A n ( n + 1) ψ ( n ) + n ( n + 1) ψ ( n ) + (4 n + 2) ψ ( n ) + 2 I B ( nψ ( n ) + 1) E g (cid:2) T (cid:3) n ( n + 1) ψ ( n ) + n ( n + 1) ψ ( n ) + (4 n + 2) ψ ( n ) + 2 m = 2 I A (cid:0) n ( n + 2) ψ ( n ) + n ( n + 2) ψ ( n ) + (7 n + 4) ψ ( n ) + n + 5 (cid:1) I B n ( n + 1) ψ ( n ) + 2(5 n + 1) ψ ( n ) + 2 n + 7 E g (cid:2) T (cid:3) (cid:0) n ( n + 2) ψ ( n ) + n (2 n + 1) ψ ( n ) + (8 n + 3) ψ ( n ) + 6 (cid:1) m = n I A (cid:0) − n ψ ( n ) + 36 n ψ (1) + 18 n ψ ( n ) + 6 n (cid:0) n + 3 n + 1 (cid:1) ψ ( n ) − n + 33 n + 22 n + 6 (cid:1) I B (cid:0) − n ψ ( n ) + 72 n ψ (1) + 18(2 n − n ψ ( n ) + 6 n (cid:0) n − n − (cid:1) ψ ( n ) − n + 57 n + 35 n + 12 (cid:1) E g (cid:2) T (cid:3) (cid:0) n ψ ( n ) + 4 n (cid:0) n + 1 (cid:1) ψ ( n ) + 4 n (cid:0) n + n + 3 n + 1 (cid:1) ψ ( n ) + n ( n + 1) (cid:0) n + n + 2 (cid:1)(cid:1) n X k =1 ψ ( k + l ) = ( n + l ) ψ ( n + l ) − (2 n + 2 l − ψ ( n + l ) − lψ ( l ) + (2 l − ψ ( l ) + 2 n. (A.4) n X k =1 kψ ( k + l ) = 12 (cid:0) n + n − l + l (cid:1) ψ ( n + l ) + 12 (cid:0) − n + 2 nl − n + 3 l − l + 1 (cid:1) ψ ( n + l ) +12 l ( l − ψ ( l ) − (cid:0) l − l + 1 (cid:1) ψ ( l ) + 14 n ( n − l + 3) . (A.5) n X k =1 k ψ ( k + l ) = 16 (cid:0) n + 3 n + n + 2 l − l + l (cid:1) ψ ( n + l ) − (cid:0) n − n l + 3 n + 12 nl −− nl − n + 22 l − l + 17 l − (cid:1) ψ ( n + l ) − l (cid:0) l − l + 1 (cid:1) ψ ( l ) +118 (cid:0) l − l + 17 l − (cid:1) ψ ( l ) + 1108 n (cid:0) n − nl + 15 n + 132 l − l + 25 (cid:1) . (A.6) n X k =1 ψ ( k + l ) = ( n + l ) ψ ( n + l ) − lψ ( l ) + ψ ( n + l ) − ψ ( l ) . (A.7) n X k =1 kψ ( k + l ) = 12 (cid:0) n + n − l + l (cid:1) ψ ( n + l ) + 12 l ( l − ψ ( l ) − 12 (2 l − ψ ( n + l ) +12 (2 l − ψ ( l ) + 12 n. (A.8) n X k =1 k ψ ( k + l ) = 16 (cid:0) n + 3 n + n + 2 l − l + l (cid:1) ψ ( n + l ) − l (cid:0) l − l + 1 (cid:1) ψ ( l ) +16 (cid:0) l − l + 1 (cid:1) ψ ( n + l ) − (cid:0) l − l + 1 (cid:1) ψ ( l ) + 16 n ( n − l + 2) . (A.9)2 m X k =1 ( n − k )!( m − k )! = ( n − m − nn − m + 1 . (A.10) m X k =1 ( n − k )!( m − k )! 1 k = n ! m ! ( ψ ( n + 1) − ψ ( n − m + 1)) . (A.11) m X k =1 ( n − k )!( m − k )! 1 k = n ! m ! m X k =1 ψ ( n − m + k ) k + n !2 m ! (cid:0) ψ ( n − m + 1) − ψ ( n + 1) + ψ ( n − m + 1) − ψ ( n + 1) (cid:1) + n ! m ! ψ ( n − m ) ( ψ ( n + 1) − ψ ( m + 1) − ψ ( n − m + 1) + ψ (1)) . (A.12) Some Remarks on the Identities in the Appendix The formulas of finite sums of polygamma functions ofthe types (A.1)–(A.9) are straightforward to show. Theproofs essentially involve changing the order of the sumsand making use of the lower order sums already obtainedin a recursive manner. In particular, the formulas (A.1)–(A.4) are available in [15, ch. 5.1]. The formulas (A.5)–(A.9) can be read off from the expressions in [16, p. 861]by keeping in mind the difference between polygammafunctions (6), (8) and harmonic numbers.The last three formulas (A.10)–(A.12) play a crucialrole in the simplification in Sec. III as they connect someof the sums in (49) and (55) to polygamma functions.The first of them (A.10) is known as Chu-Vandermondeidentity [3, p. 99]. The next formula (A.11) can be estab-lished as follows. First, the identity (27a) implies that m X k =1 ( n − k )!( m − k )! 1 k = m X k =1 ( n − k )!( m − k )! ( ψ ( k + 1) − ψ ( k )) . (A.13)By using the definition of digamma function (6), chang-ing the order of sums, and evoking Chu-Vandermondeidentity (A.10), the first term in (A.13) is represented as m X k =1 ( n − k )!( m − k )! ψ ( k + 1) = n + 1 n − m + 1 m X k =1 ( n − k )!( m − k )! 1 k − n !( n − m + 1)( m − (cid:18) γ + 1 n − m + 1 (cid:19) . (A.14)Similarly, we have m X k =1 ( n − k )!( m − k )! ψ ( k ) = nn − m + 1 m − X k =1 ( n − − k )!( m − − k )! 1 k − ( n − n − m + 1)( m − (cid:18) γn + m − n − m + 1 (cid:19) . (A.15)Inserting (A.14) and (A.15) into (A.13), we obtain a re- currence relation of the sum (A.11) as s ( m, n ) = ( n − m ! + nm s ( m − , n − , (A.16)where we denote s ( m, n ) = m X k =1 ( n − k )!( m − k )! 1 k . (A.17)Finally, by iterating m − s ( m, n ) = n ! m ! (cid:18) n + 1 n − · · · + 1 n − m + 1 (cid:19) + n ( n − · · · ( n − m + 1) m ( m − · · · s (0 , n − m )= n ! m ! ( ψ ( n + 1) − ψ ( n − m + 1)) , (A.18)where we have used the fact that s (0 , n − m ) = 0. Notethat the formula (A.11) can be also obtained via its con-nection to a hypergeometric function of unit argumentas [3, p. 111] m X k =1 ( n − k )!( m − k )! 1 k = ( n − m − F (1 , , − m ; 2 , − n ; 1)= n ! m ! ( ψ ( n + 1) − ψ ( n − m + 1)) . To prove the last formula (A.12), we first observefrom (27b) that m X k =1 ( n − k )!( m − k )! 1 k = m X k =1 ( n − k )!( m − k )! ( ψ ( k ) − ψ ( k + 1)) . (A.19)Following the same idea that has led to (A.16), we alsoobtain a recurrence relation in this case as t ( m, n ) = ( n − n − m ) m ! m ( ψ ( n ) − ψ ( n − m )) + nm t ( m − , n − , (A.20)3where we denote t ( m, n ) = m X k =1 ( n − k )!( m − k )! 1 k . (A.21)Iterating m − m X k =1 ( n − k )!( m − k )! 1 k = n ! m ! m X k =1 ψ ( n − m + k ) k − n ! m ! m X k =1 ψ ( n − m + k ) n − m + k + n ! m ! ψ ( n − m )( ψ ( n + 1) − ψ ( m + 1) − ψ ( n − m + 1) + ψ (1)) , (A.22)where by using the identity [17, eq. (23)] m X k =1 ψ ( n − m + k ) n − m + k = 12 (cid:0) ψ ( n + 1) − ψ ( n − m + 1) + ψ ( n + 1) − ψ ( n − m + 1) (cid:1) , (A.23)we obtain the claimed formula (A.12). Though the ex-pression (A.12) still contains a sum of digamma functions that may not be further simplified, it is sufficient for thesimplification purpose. As shown in Sec. III, the termsinvolving this remaining sum cancel each other. Finally,we note that as a result of the relation to the hypergeo-metric function m X k =1 ( n − k )!( m − k )! 1 k = ( n − m − F (1 , , , − m ; 2 , , − n ; 1)the formula (A.12) implies a byproduct that generalizesa result of Luke [3, p. 111] as F (1 , , , − m ; 2 , , − n ; 1) = nm m X k =1 ψ ( n − m + k ) k + n m (cid:0) ψ ( n − m + 1) − ψ ( n + 1) + ψ ( n − m + 1) − ψ ( n + 1) (cid:1) + nm ψ ( n − m )( ψ ( n + 1) − ψ ( m + 1) − ψ ( n − m + 1) + ψ (1)) , (A.24)which may be of independent interest. [1] D. N. Page, Phys. Rev. Lett. , 1291 (1993).[2] S. N. Majumdar, in The Oxford Handbook of RandomMatrix Theory , edited by G. Akemann, J. Baik, andP. Di Francesco, Chap. 37.[3] Y. L. Luke, The Special Functions and Their Approxima-tions , Vol. 1 (Academic Press, New York, 1969).[4] S. K. Foong and S. Kanno, Phys. Rev. Lett. , 1148(1994).[5] J. S´anchez-Ruiz, Phys. Rev. E , 5653 (1995).[6] S. Sen, Phys. Rev. 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