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Trace moments of induced random density matrices
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Numbers
$N$
$K$
$q$ 
$M_q(N,K)$
2
2
2:
4/5
2
2
3:
7/10
2
2
4:
22/35
2
2
5:
4/7
2
2
6:
11/21
2
2
7:
29/60
2
2
8:
74/165
2
3
2:
5/7
2
3
3:
4/7
2
3
4:
10/21
2
3
5:
17/42
2
3
6:
23/66
2
3
7:
10/33
2
3
8:
38/143
2
4
2:
2/3
2
4
3:
1/2
2
4
4:
13/33
2
4
5:
7/22
2
4
6:
112/429
2
4
7:
31/143
2
4
8:
2/11
2
5
2:
7/11
2
5
3:
5/11
2
5
4:
49/143
2
5
5:
38/143
2
5
6:
30/143
2
5
7:
24/143
2
5
8:
30/221
2
6
2:
8/13
2
6
3:
11/26
2
6
4:
4/13
2
6
5:
3/13
2
6
6:
3/17
2
6
7:
121/884
2
6
8:
451/4199
2
7
2:
3/5
2
7
3:
2/5
2
7
4:
24/85
2
7
5:
7/34
2
7
6:
99/646
2
7
7:
11/95
2
7
8:
143/1615
2
8
2:
10/17
2
8
3:
13/34
2
8
4:
5/19
2
8
5:
121/646
2
8
6:
44/323
2
8
7:
65/646
2
8
8:
559/7429
2
9
2:
11/19
2
9
3:
7/19
2
9
4:
33/133
2
9
5:
23/133
2
9
6:
377/3059
2
9
7:
39/437
2
9
8:
143/2185
2
10
2:
4/7
2
10
3:
5/14
2
10
4:
38/161
2
10
5:
26/161
2
10
6:
13/115
2
10
7:
37/460
2
10
8:
4/69
2
11
2:
13/23
2
11
3:
8/23
2
11
4:
26/115
2
11
5:
7/46
2
11
6:
217/2070
2
11
7:
76/1035
2
11
8:
68/1305
2
12
2:
14/25
2
12
3:
17/50
2
12
4:
49/225
2
12
5:
13/90
2
12
6:
128/1305
2
12
7:
442/6525
2
12
8:
3196/67425
2
13
2:
5/9
2
13
3:
1/3
2
13
4:
55/261
2
13
5:
4/29
2
13
6:
748/8091
2
13
7:
170/2697
2
13
8:
1292/29667
2
14
2:
16/29
2
14
3:
19/58
2
14
4:
184/899
2
14
5:
119/899
2
14
6:
867/9889
2
14
7:
2337/39556
2
14
8:
399/9889
2
15
2:
17/31
2
15
3:
10/31
2
15
4:
68/341
2
15
5:
87/682
2
15
6:
57/682
2
15
7:
19/341
2
15
8:
475/12617
2
16
2:
6/11
2
16
3:
7/22
2
16
4:
15/77
2
16
5:
19/154
2
16
6:
228/2849
2
16
7:
43/814
2
16
8:
17/481
2
17
2:
19/35
2
17
3:
11/35
2
17
4:
247/1295
2
17
5:
31/259
2
17
6:
1/13
2
17
7:
121/2405
2
17
8:
253/7585
2
18
2:
20/37
2
18
3:
23/74
2
18
4:
90/481
2
18
5:
56/481
2
18
6:
1463/19721
2
18
7:
3795/78884
2
18
8:
26818/848003
2
19
2:
7/13
2
19
3:
4/13
2
19
4:
98/533
2
19
5:
121/1066
2
19
6:
253/3526
2
19
7:
1058/22919
2
19
8:
690/22919
2
20
2:
22/41
2
20
3:
25/82
2
20
4:
319/1763
2
20
5:
391/3526
2
20
6:
368/5289
2
20
7:
235/5289
2
20
8:
7150/248583
2
21
2:
23/43
2
21
3:
13/43
2
21
4:
23/129
2
21
5:
14/129
2
21
6:
410/6063
2
21
7:
260/6063
2
21
8:
390/14147
2
22
2:
8/15
2
22
3:
3/10
2
22
4:
124/705
2
22
5:
5/47
2
22
6:
65/987
2
22
7:
39/940
2
22
8:
741/27965
2
23
2:
25/47
2
23
3:
14/47
2
23
4:
400/2303
2
23
5:
481/4606
2
23
6:
5031/78302
2
23
7:
225/5593
2
23
8:
7569/296429
2
24
2:
26/49
2
24
3:
29/98
2
24
4:
143/833
2
24
5:
171/1666
2
24
6:
396/6307
2
24
7:
29/742
2
24
8:
1711/69377
2
25
2:
9/17
2
25
3:
5/17
2
25
4:
9/53
2
25
5:
91/901
2
25
6:
609/9911
2
25
7:
377/9911
2
25
8:
4495/188309
2
26
2:
28/53
2
26
3:
31/106
2
26
4:
98/583
2
26
5:
58/583
2
26
6:
667/11077
2
26
7:
31/836
2
26
8:
15128/653543
2
27
2:
29/55
2
27
3:
16/55
2
27
4:
174/1045
2
27
5:
41/418
2
27
6:
1457/24662
2
27
7:
2232/61655
2
27
8:
7688/341905
2
28
2:
10/19
2
28
3:
11/38
2
28
4:
185/1121
2
28
5:
217/2242
2
28
6:
3968/68381
2
28
7:
2420/68381
2
28
8:
1496/68381
2
29
2:
31/59
2
29
3:
17/59
2
29
4:
589/3599
2
29
5:
344/3599
2
29
6:
616/10797
2
29
7:
374/10797
2
29
8:
2992/140361
2
30
2:
32/61
2
30
3:
35/122
2
30
4:
208/1281
2
30
5:
121/1281
2
30
6:
935/16653
2
30
7:
323/9516
2
30
8:
85/4087
2
31
2:
11/21
2
31
3:
2/7
2
31
4:
44/273
2
31
5:
17/182
2
31
6:
289/5226
2
31
7:
29/871
2
31
8:
407/20033
2
32
2:
34/65
2
32
3:
37/130
2
32
4:
697/4355
2
32
5:
161/1742
2
32
6:
84/1541
2
32
7:
6549/200330
2
32
8:
2109/106145
3
3
2:
3/5
3
3
3:
23/55
3
3
4:
17/55
3
3
5:
101/429
3
3
6:
183/1001
3
3
7:
103/715
3
3
8:
493/4290
3
4
2:
7/13
3
4
3:
31/91
3
4
4:
3/13
3
4
5:
59/364
3
4
6:
103/884
3
4
7:
113/1326
3
4
8:
531/8398
3
5
2:
1/2
3
5
3:
5/17
3
5
4:
19/102
3
5
5:
7/57
3
5
6:
161/1938
3
5
7:
37/646
3
5
8:
15/374
3
6
2:
9/19
3
6
3:
5/19
3
6
4:
3/19
3
6
5:
62/627
3
6
6:
306/4807
3
6
7:
201/4807
3
6
8:
61/2185
3
7
2:
5/11
3
7
3:
61/253
3
7
4:
35/253
3
7
5:
21/253
3
7
6:
843/16445
3
7
7:
29/897
3
7
8:
31/1495
3
8
2:
11/25
3
8
3:
73/325
3
8
4:
121/975
3
8
5:
14/195
3
8
6:
242/5655
3
8
7:
737/28275
3
8
8:
363/22475
3
9
2:
3/7
3
9
3:
43/203
3
9
4:
23/203
3
9
5:
1199/18879
3
9
6:
33/899
3
9
7:
39/1798
3
9
8:
598/45849
3
10
2:
13/31
3
10
3:
25/124
3
10
4:
13/124
3
10
5:
241/4216
3
10
6:
949/29512
3
10
7:
39/2108
3
10
8:
845/77996
3
11
2:
7/17
3
11
3:
23/119
3
11
4:
5/51
3
11
5:
689/13209
3
11
6:
1027/35853
3
11
7:
577/35853
3
11
8:
110/11951
3
12
2:
15/37
3
12
3:
131/703
3
12
4:
65/703
3
12
5:
203/4218
3
12
6:
1491/57646
3
12
7:
10/703
3
12
8:
29614/3718167
3
13
2:
2/5
3
13
3:
37/205
3
13
4:
18/205
3
13
5:
79/1763
3
13
6:
458/19393
3
13
7:
3706/290895
3
13
8:
15606/2230195
3
14
2:
17/43
3
14
3:
83/473
3
14
4:
119/1419
3
14
5:
1372/32637
3
14
6:
33388/1533939
3
14
7:
5899/511313
3
14
8:
969/155617
3
15
2:
9/23
3
15
3:
185/1081
3
15
4:
87/1081
3
15
5:
901/22701
3
15
6:
153/7567
3
15
7:
57/5405
3
15
8:
551/98371
3
16
2:
19/49
3
16
3:
41/245
3
16
4:
19/245
3
16
5:
24/637
3
16
6:
456/24115
3
16
7:
703/72345
3
16
8:
1349/265265
3
17
2:
5/13
3
17
3:
113/689
3
17
4:
155/2067
3
17
5:
817/22737
3
17
6:
2831/159159
3
17
7:
205/22737
3
17
8:
93/19981
3
18
2:
21/55
3
18
3:
62/385
3
18
4:
4/55
3
18
5:
461/13398
3
18
6:
633/37642
3
18
7:
72/8555
3
18
8:
6716/1565565
3
19
2:
11/29
3
19
3:
271/1711
3
19
4:
121/1711
3
19
5:
119/3599
3
19
6:
847/53041
3
19
7:
25553/3235501
3
19
8:
102971/25884008
3
20
2:
23/61
3
20
3:
295/1891
3
20
4:
391/5673
3
20
5:
2893/90768
3
20
6:
17963/1179984
3
20
7:
4393/589992
3
20
8:
48875/13176488
3
21
2:
3/8
3
21
3:
2/13
3
21
4:
7/104
3
21
5:
161/5226
3
21
6:
1725/118456
3
21
7:
835/118456
3
21
8:
95/27336
4
4
2:
8/17
4
4
3:
9/34
4
4
4:
52/323
4
4
5:
33/323
4
4
6:
452/6783
4
4
7:
1891/42636
4
4
8:
7358/245157
4
5
2:
3/7
4
5
3:
17/77
4
5
4:
219/1771
4
5
5:
257/3542
4
5
6:
111/2530
4
5
7:
446/16445
4
5
8:
842/49335
4
6
2:
2/5
4
6
3:
5/26
4
6
4:
59/585
4
6
5:
1/18
4
6
6:
536/16965
4
6
7:
8/435
4
6
8:
1912/175305
4
7
2:
11/29
4
7
3:
5/29
4
7
4:
77/899
4
7
5:
161/3596
4
7
6:
3/124
4
7
7:
4503/336226
4
7
8:
231/30566
4
8
2:
4/11
4
8
3:
59/374
4
8
4:
14/187
4
8
5:
7/187
4
8
6:
134/6919
4
8
7:
491/47804
4
8
8:
863/155363
4
9
2:
13/37
4
9
3:
103/703
4
9
4:
47/703
4
9
5:
294/9139
4
9
6:
462/28823
4
9
7:
3069/374699
4
9
8:
143/33497
4
10
2:
14/41
4
10
3:
79/574
4
10
4:
107/1763
4
10
5:
17/602
4
10
6:
24/1763
4
10
7:
273/40549
4
10
8:
6474/1905803
4
11
2:
1/3
4
11
3:
3/23
4
11
4:
181/3243
4
11
5:
109/4324
4
11
6:
7501/635628
4
11
7:
429/75670
4
11
8:
715/257278
4
12
2:
16/49
4
12
3:
61/490
4
12
4:
216/4165
4
12
5:
19/833
4
12
6:
328/31535
4
12
7:
5527/1135260
4
12
8:
44/18921
4
13
2:
17/53
4
13
3:
19/159
4
13
4:
85/1749
4
13
5:
73/3498
4
13
6:
1853/199386
4
13
7:
12284/2891097
4
13
8:
338164/170574723
4
14
2:
6/19
4
14
3:
127/1102
4
14
4:
1491/32509
4
14
5:
1253/65018
4
14
6:
16656/1983049
4
14
7:
231064/61474519
4
14
8:
105536/61474519
4
15
2:
19/61
4
15
3:
211/1891
4
15
4:
247/5673
4
15
5:
407/22692
4
15
6:
2261/294996
4
15
7:
1819/540826
4
15
8:
4199/2787334
4
16
2:
4/13
4
16
3:
31/286
4
16
4:
398/9581
4
16
5:
161/9581
4
16
6:
1554/220363
4
16
7:
2679/881452
4
16
8:
20919/15645773
5
5
2:
5/13
5
5
3:
7/39
5
5
4:
25/273
5
5
5:
389/7917
5
5
6:
647/23751
5
5
7:
1627/105183
5
5
8:
3767/420732
5
6
2:
11/31
5
6
3:
19/124
5
6
4:
9/124
5
6
5:
1681/46376
5
6
6:
79/4216
5
6
7:
691/69564
5
6
8:
419/77996
5
7
2:
1/3
5
7
3:
5/37
5
7
4:
127/2109
5
7
5:
7/247
5
7
6:
379/27417
5
7
7:
2593/374699
5
7
8:
1325/374699
5
8
2:
13/41
5
8
3:
5/41
5
8
4:
91/1763
5
8
5:
448/19393
5
8
6:
208/19393
5
8
7:
2281/446039
5
8
8:
4745/1905803
5
9
2:
7/23
5
9
3:
121/1081
5
9
4:
49/1081
5
9
5:
21/1081
5
9
6:
327/37835
5
9
7:
363/91885
5
9
8:
15422/8361535
5
10
2:
5/17
5
10
3:
23/221
5
10
4:
475/11713
5
10
5:
196/11713
5
10
6:
84/11713
5
10
7:
37/11713
5
10
8:
73/51357
5
11
2:
2/7
5
11
3:
13/133
5
11
4:
142/3857
5
11
5:
3343/227563
5
11
6:
198/32509
5
11
7:
5148/1983049
5
11
8:
69498/61474519
5
12
2:
17/61
5
12
3:
175/1891
5
12
4:
1343/39711
5
12
5:
8321/635376
5
12
6:
3349/635376
5
12
7:
33/15128
5
12
8:
935/1013576
6
6
2:
12/37
6
6
3:
181/1406
6
6
4:
514/9139
6
6
5:
712/27417
6
6
6:
4659/374699
6
6
7:
9161/1498796
6
6
8:
148286/48336171
6
7
2:
13/43
6
7
3:
53/473
6
7
4:
65/1419
6
7
5:
647/32637
6
7
6:
13663/1533939
6
7
7:
6308/1533939
6
7
8:
6942/3579191
6
8
2:
2/7
6
8
3:
1/10
6
8
4:
23/595
6
8
5:
7/442
6
8
6:
2764/409955
6
8
7:
61/20670
6
8
8:
5953/4509505
6
9
2:
3/11
6
9
3:
1/11
6
9
4:
7/209
6
9
5:
238/18183
6
9
6:
1902/357599
6
9
7:
795/357599
6
9
8:
5651/5949147
6
10
2:
16/61
6
10
3:
317/3782
6
10
4:
56/1891
6
10
5:
21/1891
6
10
6:
213/49166
6
10
7:
171/98332
6
10
8:
2351/3294122
7
7
2:
7/25
7
7
3:
41/425
7
7
4:
203/5525
7
7
5:
869/58565
7
7
6:
3283/527085
7
7
7:
77947/28989675
7
7
8:
34393/28989675
7
8
2:
5/19
7
8
3:
47/551
7
8
4:
995/32509
7
8
5:
379/32509
7
8
6:
9161/1983049
7
8
7:
115795/61474519
7
8
8:
48277/61474519
7
9
2:
1/4
7
9
3:
1/13
7
9
4:
15/572
7
9
5:
7/737
7
9
6:
2333/651508
7
9
7:
801/576334
7
9
8:
8267/14984684
8
8
2:
16/65
8
8
3:
107/1430
8
8
4:
1208/47905
8
8
5:
1471/162877
8
8
6:
12616/3746171
8
8
7:
96871/74923420
8
8
8:
675244/1329890705
Definition
The trace moment $M_q(N,K)=\mathbb E\operatorname{Tr}(\rho^q)$ is taken over $\rho=\operatorname{Tr}_{\mathbb C^K}(|\psi\rangle\langle\psi|)$, where $\psi$ is Haar-uniform on the unit sphere of $\mathbb C^N\otimes\mathbb C^K$ [4].
Parameters
$N$
—   subsystem dimension ($N\geq2$)
$K$
—   environment dimension ($K\geq N$)
$q$
—   trace-power order ($q\geq2$)
Formulas
(1)
$\displaystyle M_q=\frac{(2q-1)(N+K)}{(NK+q-1)(q+1)}M_{q-1}+\frac{(q-2)((q-1)^2-(K-N)^2)}{(NK+q-1)(NK+q-2)(q+1)}M_{q-2}$ for integers $q\geq2$, with $M_0=N$ and $M_1=1$ [2], equations (37)–(38). The arguments $(N,K)$ are fixed in this recurrence.
(2)
$\displaystyle M_q(N,K)=\frac{1}{(NK)_q}\sum_{\sigma\in S_q}N^{c(\gamma\sigma)}K^{c(\sigma)}$, where $\gamma=(1\ 2\ \cdots\ q)$, $c(\sigma)$ counts permutation cycles, and $(NK)_q=\prod_{j=0}^{q-1}(NK+j)$. Composition applies $\sigma$ first. This follows from the complex Wick rule [3], Theorem 3.9, and radial normalization [2], Proposition 5.
(3)
$\displaystyle M_2(N,K)=\frac{N+K}{NK+1}$ [1], equation (3.20) in the PDF, numbered (44) in the arXiv HTML.
(4)
$\displaystyle M_q(2,2)=6\left(\frac4{q+3}-\frac4{q+2}+\frac1{q+1}\right)$. The unordered eigenvalue $x$ has density $3(2x-1)^2$ on $[0,1]$, and the other eigenvalue is $1-x$ [1], equation (3.6) in the PDF, numbered (30) in the arXiv HTML.
(5)
$\displaystyle N^{1-q}<M_q(N,K)<1,\qquad M_{q+1}(N,K)<M_q(N,K)$ for $2\leq N\leq K$ and integer $q\geq2$.
Comments
(6)
Equivalently, $\rho=GG^*/\operatorname{Tr}(GG^*)$, where $G_{ia}=(X_{ia}+\mathrm iY_{ia})/\sqrt2$ and all entries of the real $N\times K$ matrices $X$ and $Y$ are independent standard normal variables [5]. The symbol $G^*$ denotes the conjugate transpose. The trace is the ordinary matrix trace, without division by $N$; $M_2$ is mean purity.
(7)
The two reduced states have identical nonzero eigenvalues, so their positive trace moments agree upon interchanging $N$ and $K$. The convention $N\leq K$ records each dimension pair once [1].
(8)
At $K=N$, the ensemble is the normalized complex full-rank Hilbert–Schmidt volume measure [1], whose total volumes are in Hilbert–Schmidt volumes of quantum states of fixed rank.
(9)
Page average entropies concern the same induced ensemble. The quantity $-\log M_2$ is not the mean Rényi entropy of order two, because taking an expectation does not commute with taking a logarithm.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ

def trace_moment(N, K, q):
    if not (2 <= N <= K and q >= 2):
        raise ValueError("Require 2 <= N <= K and q >= 2")
    previous, current = QQ(N), QQ(1)
    for j in range(2, q+1):
        a = QQ((2*j-1)*(N+K)) / QQ((N*K+j-1)*(j+1))
        b = QQ((j-2)*((j-1)**2-(K-N)**2)) / QQ((N*K+j-1)*(N*K+j-2)*(j+1))
        previous, current = current, a*current+b*previous
    return current

trace_moment(2, 2, 2)  # 4/5
References
[1]
K. Życzkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, Journal of Physics A 34 (2001), 7111–7125. (arXiv) (doi)
[2]
I. Nechita, Asymptotics of random density matrices, Annales Henri Poincaré 8 (2007), 1521–1538. (arXiv) (doi)
[3]
B. Collins and I. Nechita, Random matrix techniques in quantum information theory, Journal of Mathematical Physics 57 (2016), 015215. (arXiv) (doi)
Links
Similar tables
Page average entropies of quantum subsystems —   Mean von Neumann entropies of the same complex induced ensemble.
Hilbert–Schmidt volumes of quantum states of fixed rank —   Normalizing complex full-rank volume gives the induced measure at $K=N$.
Laguerre–Selberg integrals for classical Wishart ensembles —   At $\beta=2$, $n=N$ and $r=K-N$, these normalize the Wishart eigenvalue density before the trace is set to one.
Unitary Weingarten functions —   Coefficients for Haar-unitary integration, which also generates the uniform pure states underlying these induced ensembles.
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds every pair $2\leq N\leq K$ with $NK\leq64$ and every integer order $2\leq q\leq8$, covering balanced and unequal bipartitions within a total Hilbert-space dimension budget; the noncanonical order bound gives seven consecutive spectral moments while permitting exhaustive Wick-permutation verification. Orders zero and one are the trivial controls $N$ and $1$)
How they were obtained:

Every entry was computed over the rational field with the three-term recurrence and compared with a separate exhaustive Wick-permutation sum. All 217 qubit entries were also checked by exact beta integration of the eigenvalue density. Purity, third and fourth moments, dimension interchange, the recurrence and strict density-matrix bounds were checked across their full stored ranges.

more

The attached checks.py repeats these tests on generated or stored entries. The finite sum in Nechita equation (36), which fails the qubit purity control, was not used.



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