This file is indexed.

/usr/share/perl5/Quantum/Entanglement.pm is in libquantum-entanglement-perl 0.32-3.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

   1
   2
   3
   4
   5
   6
   7
   8
   9
  10
  11
  12
  13
  14
  15
  16
  17
  18
  19
  20
  21
  22
  23
  24
  25
  26
  27
  28
  29
  30
  31
  32
  33
  34
  35
  36
  37
  38
  39
  40
  41
  42
  43
  44
  45
  46
  47
  48
  49
  50
  51
  52
  53
  54
  55
  56
  57
  58
  59
  60
  61
  62
  63
  64
  65
  66
  67
  68
  69
  70
  71
  72
  73
  74
  75
  76
  77
  78
  79
  80
  81
  82
  83
  84
  85
  86
  87
  88
  89
  90
  91
  92
  93
  94
  95
  96
  97
  98
  99
 100
 101
 102
 103
 104
 105
 106
 107
 108
 109
 110
 111
 112
 113
 114
 115
 116
 117
 118
 119
 120
 121
 122
 123
 124
 125
 126
 127
 128
 129
 130
 131
 132
 133
 134
 135
 136
 137
 138
 139
 140
 141
 142
 143
 144
 145
 146
 147
 148
 149
 150
 151
 152
 153
 154
 155
 156
 157
 158
 159
 160
 161
 162
 163
 164
 165
 166
 167
 168
 169
 170
 171
 172
 173
 174
 175
 176
 177
 178
 179
 180
 181
 182
 183
 184
 185
 186
 187
 188
 189
 190
 191
 192
 193
 194
 195
 196
 197
 198
 199
 200
 201
 202
 203
 204
 205
 206
 207
 208
 209
 210
 211
 212
 213
 214
 215
 216
 217
 218
 219
 220
 221
 222
 223
 224
 225
 226
 227
 228
 229
 230
 231
 232
 233
 234
 235
 236
 237
 238
 239
 240
 241
 242
 243
 244
 245
 246
 247
 248
 249
 250
 251
 252
 253
 254
 255
 256
 257
 258
 259
 260
 261
 262
 263
 264
 265
 266
 267
 268
 269
 270
 271
 272
 273
 274
 275
 276
 277
 278
 279
 280
 281
 282
 283
 284
 285
 286
 287
 288
 289
 290
 291
 292
 293
 294
 295
 296
 297
 298
 299
 300
 301
 302
 303
 304
 305
 306
 307
 308
 309
 310
 311
 312
 313
 314
 315
 316
 317
 318
 319
 320
 321
 322
 323
 324
 325
 326
 327
 328
 329
 330
 331
 332
 333
 334
 335
 336
 337
 338
 339
 340
 341
 342
 343
 344
 345
 346
 347
 348
 349
 350
 351
 352
 353
 354
 355
 356
 357
 358
 359
 360
 361
 362
 363
 364
 365
 366
 367
 368
 369
 370
 371
 372
 373
 374
 375
 376
 377
 378
 379
 380
 381
 382
 383
 384
 385
 386
 387
 388
 389
 390
 391
 392
 393
 394
 395
 396
 397
 398
 399
 400
 401
 402
 403
 404
 405
 406
 407
 408
 409
 410
 411
 412
 413
 414
 415
 416
 417
 418
 419
 420
 421
 422
 423
 424
 425
 426
 427
 428
 429
 430
 431
 432
 433
 434
 435
 436
 437
 438
 439
 440
 441
 442
 443
 444
 445
 446
 447
 448
 449
 450
 451
 452
 453
 454
 455
 456
 457
 458
 459
 460
 461
 462
 463
 464
 465
 466
 467
 468
 469
 470
 471
 472
 473
 474
 475
 476
 477
 478
 479
 480
 481
 482
 483
 484
 485
 486
 487
 488
 489
 490
 491
 492
 493
 494
 495
 496
 497
 498
 499
 500
 501
 502
 503
 504
 505
 506
 507
 508
 509
 510
 511
 512
 513
 514
 515
 516
 517
 518
 519
 520
 521
 522
 523
 524
 525
 526
 527
 528
 529
 530
 531
 532
 533
 534
 535
 536
 537
 538
 539
 540
 541
 542
 543
 544
 545
 546
 547
 548
 549
 550
 551
 552
 553
 554
 555
 556
 557
 558
 559
 560
 561
 562
 563
 564
 565
 566
 567
 568
 569
 570
 571
 572
 573
 574
 575
 576
 577
 578
 579
 580
 581
 582
 583
 584
 585
 586
 587
 588
 589
 590
 591
 592
 593
 594
 595
 596
 597
 598
 599
 600
 601
 602
 603
 604
 605
 606
 607
 608
 609
 610
 611
 612
 613
 614
 615
 616
 617
 618
 619
 620
 621
 622
 623
 624
 625
 626
 627
 628
 629
 630
 631
 632
 633
 634
 635
 636
 637
 638
 639
 640
 641
 642
 643
 644
 645
 646
 647
 648
 649
 650
 651
 652
 653
 654
 655
 656
 657
 658
 659
 660
 661
 662
 663
 664
 665
 666
 667
 668
 669
 670
 671
 672
 673
 674
 675
 676
 677
 678
 679
 680
 681
 682
 683
 684
 685
 686
 687
 688
 689
 690
 691
 692
 693
 694
 695
 696
 697
 698
 699
 700
 701
 702
 703
 704
 705
 706
 707
 708
 709
 710
 711
 712
 713
 714
 715
 716
 717
 718
 719
 720
 721
 722
 723
 724
 725
 726
 727
 728
 729
 730
 731
 732
 733
 734
 735
 736
 737
 738
 739
 740
 741
 742
 743
 744
 745
 746
 747
 748
 749
 750
 751
 752
 753
 754
 755
 756
 757
 758
 759
 760
 761
 762
 763
 764
 765
 766
 767
 768
 769
 770
 771
 772
 773
 774
 775
 776
 777
 778
 779
 780
 781
 782
 783
 784
 785
 786
 787
 788
 789
 790
 791
 792
 793
 794
 795
 796
 797
 798
 799
 800
 801
 802
 803
 804
 805
 806
 807
 808
 809
 810
 811
 812
 813
 814
 815
 816
 817
 818
 819
 820
 821
 822
 823
 824
 825
 826
 827
 828
 829
 830
 831
 832
 833
 834
 835
 836
 837
 838
 839
 840
 841
 842
 843
 844
 845
 846
 847
 848
 849
 850
 851
 852
 853
 854
 855
 856
 857
 858
 859
 860
 861
 862
 863
 864
 865
 866
 867
 868
 869
 870
 871
 872
 873
 874
 875
 876
 877
 878
 879
 880
 881
 882
 883
 884
 885
 886
 887
 888
 889
 890
 891
 892
 893
 894
 895
 896
 897
 898
 899
 900
 901
 902
 903
 904
 905
 906
 907
 908
 909
 910
 911
 912
 913
 914
 915
 916
 917
 918
 919
 920
 921
 922
 923
 924
 925
 926
 927
 928
 929
 930
 931
 932
 933
 934
 935
 936
 937
 938
 939
 940
 941
 942
 943
 944
 945
 946
 947
 948
 949
 950
 951
 952
 953
 954
 955
 956
 957
 958
 959
 960
 961
 962
 963
 964
 965
 966
 967
 968
 969
 970
 971
 972
 973
 974
 975
 976
 977
 978
 979
 980
 981
 982
 983
 984
 985
 986
 987
 988
 989
 990
 991
 992
 993
 994
 995
 996
 997
 998
 999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
package Quantum::Entanglement;
use strict;
use warnings;
use Carp;

BEGIN {
  use Exporter   ();
  use Math::Complex;
  my @M_Complex = qw(i Re Im rho theta arg cplx cplxe);
  our ($VERSION, @ISA, @EXPORT, @EXPORT_OK, %EXPORT_TAGS);
  $VERSION     = 0.32;
  @ISA         = qw(Exporter);
  @EXPORT      = qw(&entangle &p_op &p_func &q_logic
		    &save_state &restore_state);
  %EXPORT_TAGS = (DEFAULT => [@EXPORT],
		  complex => [@M_Complex],
		  QFT => [qw(&QFT)],);
  @EXPORT_OK   = (@M_Complex, '&QFT');
}
our (@EXPORT_OK, @EXPORT);

$Quantum::Entanglement::destroy = 1; # true=> p(0) states stomped on
$Quantum::Entanglement::conform = 0; # true=> strives for truth when observing

## Contents:
# Constructors
# Utility Routines
# Overload table
# Overload routines
# parallel operators and functions
# methods for saving and restoring state
# pod

# =begin pretty pictures
#
# Things look a bit like this...
#
# $variable = [ref to var which itself refs to an annon array (the universe),
#	       offset of values of variable within universe,
#	       ref to var which itself refs to an annon array (the offsets)];
#
# $offsets =  [refs to all the offsets in a given universe, ...]
# $universe=  [ [prob1,val1,prob2,val2],
#	        [prob1,val1,prob2,val2], etc. ]
#
# =cut

# creates a new set of universes
sub _new {
  my $universe = [];
  my $offsets = [];
  my $var = [\$universe,1,\$offsets];
  $offsets->[0] = \ $var->[1];
  while (@_) {
    push @$universe, [shift,shift];
  }
  bless $var, 'Quantum::Entanglement';
  return $var;
}

# add a variable without adding values (ie. a derived value)
# returns the new variable
sub _add {
  my $current = $_[0];
  my $universe = ${ $current->[0]};
  my $offset = scalar(@{$universe->[0]}) + 1;
  my $var= [\$universe,$offset,\ ${$current->[2]}];
  push @{${$current->[2]}} , \$var->[1];
  bless $var, 'Quantum::Entanglement';
  return $var;
}

# joins together two previously unconnected universes
# takes two variables as args, gets the universes from those.
# should be used to modify objects in place.
sub _join {
  my ($uni1,$uni2) = (${$_[0]->[0]},${$_[1]->[0]});
  return () if $uni1 == $uni2;
  my $universe = [];
  foreach my $s2 (@$uni2) {
    foreach my $s1 (@$uni1) {
      push @$universe, [@$s1,@$s2];
    }
  }
  my $offsets1 = ${$_[0]->[2]};
  my $offsets2 = ${$_[1]->[2]};
  my $extra = scalar(@{$uni1->[0]});
  push @$offsets1, map {$$_+=$extra; $_} @$offsets2;
  ${$_[1]->[2]} = $offsets1;
  ${$_[0]->[0]} = $universe;
  ${$_[1]->[0]} = $universe;
  return (1);
}

# exported constructor
sub entangle {
  return _new(@_);
}

## Utility routines

# a view of global state space, might still show historical states which
# are no longer accessable, does not count as observation
sub show_states {
  my $rt;
  my $var = shift;
  my $universe = ${$var->[0]};
  if ($_[0]) {
    foreach (@$universe) { my $t;
      $rt .= (++$t % 2) ? "$_|" : overload::StrVal($_).">\t" foreach @$_;
      $rt .= "\n";
    }
  }
  else {
    my $os = $var->[1];
    $rt .= $_->[$os-1]."|".overload::StrVal($_->[$os]).">\t"
      foreach @$universe;
    substr($rt,-1,1,"\n");
  }
  return $rt;
}

# egads! (and don't tell anyone about the grep, it's a secret)
sub DESTROY {
  my ($universe, $offsets) = (${$_[0]->[0]}, ${$_[0]->[2]});
  my $os = $_[0]->[1];
  splice(@$_,$os-1,2) foreach @$universe;
  @$offsets = grep {if ($$_ != $os) {$$_ -= 2 if $$_ > $os;1;} else {0;}}
               @$offsets;
  _rationalise_states([\$universe])
          if $Quantum::Entanglement::destroy;
}

# takes two non normalised probabilities and returns true with prob(1/1+2)
sub _sel_output {
  my ($c, $d) = @_;
  $c = abs($c)**2;
  $d = abs($d)**2;
  return rand(1) < ($c/($c+$d)) ? 1 : 0;
}

# Gets a ref to a hash of complex probs, produces ref to hash of sequential
# probs and ref to array of ordering.
sub _normalise {
  my $hr = $_[0];
  my $h2 = {};
  my $muts = [keys %{$hr}];
  my $sum = 0;
  foreach (values %{$hr}) {
    $sum += abs($_)**2;
  }
  if ($sum <= 0) {
    croak "$0: Cannot behave probabilistically with -ve probs";
  }
  else {
    my $cum;
    @{$h2}{ @{$muts} } = map {$cum +=abs($_)**2;
			      $cum / $sum       } @{$hr}{ @{$muts} };
    return ($h2, $muts);
  }
}

# this builds up a multi-layered hash so as to find the unique sets of
# states, it then uses _unravel to get them back out of the hash
sub _rationalise_states {
  my $universe = ${$_[0]->[0]};
  my $len = scalar(@{$universe->[0]})/2;
  my @p_os = map {$_*2  } (0..$len-1);
  my @v_os = map {$_*2+1} (0..$len-1);
  my $foo = {};
  foreach my $state (@$universe) { # build an icky data structure
    my $tref = $foo;
    foreach (@v_os) {
      my $val = ref($state->[$_]) ? overload::StrVal($state->[$_])
	                          : $state->[$_];
      if ($_==2*$len-1) { # last level of the structure
	if (exists $tref->{$val}) {
	  my @temp = @{$state}[@p_os];
	  $_+=shift @temp foreach @{$tref->{$val}}[@p_os];
	}
	else {
	  $tref->{$val} = [@{$state}];
	}
      }
      else { # an intermediate level
	if (exists $tref->{$val}) {
	  $tref = $tref->{$val};
	}
	else {
	  $tref = $tref->{$val} = {};
	}
      }
    }
  }
  # do something with it...
  @$universe =();
  while (1) {
    my $aref = _unravel($foo);
    last unless $aref;
    push @$universe, $aref;
  }
  return $universe;
}

sub _unravel {
  my $tref = $_[0];
  return undef unless (scalar keys %$tref);
  my @hrs;
  my($last_ref, $val);
  do {
    $last_ref = $tref;
    ($val,$tref) = %$tref;
    unshift @hrs, $val, $last_ref;
  } until (ref($tref) eq 'ARRAY');
  delete ${$last_ref}{$val};
  splice @hrs, 0,2;
  while (@hrs) {
    my $val = shift @hrs;
    my $h = shift @hrs;
    delete ${$h}{$val} if scalar(keys %{${$h}{$val}}) < 1;
  }
  return $tref;
}


##
# Overloading.  Everything except for assignment operators
# are overloaded specifically.  Need to specifically overload a lot
# of stuff so that pruning of states can happen as soon as poss

use overload
  '+'  => sub { binop(@_, sub{$_[0] + $_[1]} ) },
  '*'  => sub { binop(@_, sub{$_[0] * $_[1]} ) },
  '-'  => sub { binop(@_, sub{$_[0] - $_[1]} ) },
  '/'  => sub { binop(@_, sub{$_[0] / $_[1]} ) },
  '**' => sub { binop(@_, sub{$_[0] **$_[1]} ) },
  '%'  => sub { binop(@_, sub{$_[0] % $_[1]} ) },
  'x'  => sub { binop(@_, sub{$_[0] x $_[1]} ) },
  '.'  => sub { binop(@_, sub{$_[0] . $_[1]} ) },
  '<<' => sub { binop(@_, sub{$_[0] <<$_[1]} ) },
  '>>' => sub { binop(@_, sub{$_[0] >>$_[1]} ) },
  '&'  => sub { binop(@_, sub{$_[0] & $_[1]} ) },
  '|'  => sub { binop(@_, sub{$_[0] | $_[1]} ) },
  '^'  => sub { binop(@_, sub{$_[0] ^ $_[1]} ) },
  '~'  => sub { unnop($_[0], sub { ~$_[0]} ) },
  'neg'=> sub { unnop($_[0], sub { -$_[0]} ) },
  '!'  => sub { unnop($_[0], sub { !$_[0]} ) },
  '++' => sub { mutop($_[0], sub {++$_[0]} ) },
  '--' => sub { mutop($_[0], sub {--$_[0]} ) },
  '<'  => sub { bioop(@_, sub{$_[0] <  $_[1]} ) },
  '>'  => sub { bioop(@_, sub{$_[0] >  $_[1]} ) },
  '<=' => sub { bioop(@_, sub{$_[0] <= $_[1]} ) },
  '>=' => sub { bioop(@_, sub{$_[0] >= $_[1]} ) },
  '==' => sub { bioop(@_, sub{$_[0] == $_[1]} ) },
  '!=' => sub { bioop(@_, sub{$_[0] != $_[1]} ) },
  'lt' => sub { bioop(@_, sub{$_[0] lt $_[1]} ) },
  'le' => sub { bioop(@_, sub{$_[0] le $_[1]} ) },
  'ge' => sub { bioop(@_, sub{$_[0] ge $_[1]} ) },
  'gt' => sub { bioop(@_, sub{$_[0] gt $_[1]} ) },
  'eq' => sub { bioop(@_, sub{$_[0] eq $_[1]} ) },
  'ne' => sub { bioop(@_, sub{$_[0] ne $_[1]} ) },
  '<=>'=> sub { binop(@_, sub{$_[0] <=>$_[1]} ) },
  'cmp'=> sub { binop(@_, sub{$_[0] cmp$_[1]} ) },
  'cos'=> sub { unnop($_[0], sub{ cos $_[0]} ) },
  'sin'=> sub { unnop($_[0], sub{ sin $_[0]} ) },
  'exp'=> sub { unnop($_[0], sub{ exp $_[0]} ) },
  'abs'=> sub { unnop($_[0], sub{ abs $_[0]} ) },
  'log'=> sub { unnop($_[0], sub{ log $_[0]} ) },
  'sqrt'=>sub { unnop($_[0], sub{ sqrt $_[0]}) },
  'atan2'=>sub{ binop(@_, sub{atan2($_[0], $_[1])} ) },
  '&{}'=> \&sub_ent,
  'bool'=> \&bool_ent, q{""}  => \&str_ent,  '0+' => \&num_ent,
  '='   => \&copy_ent,
  'fallback' => 1;

# copying (not observation, clones states, does not increase state space)
sub copy_ent {
  my $os = $_[0]->[1];
  my $val = $_[0]->_add;
  my $universe = ${$_[0]->[0]};
  push(@$_, $_->[$os-1], $_->[$os]) foreach @$universe;
  return $val;
}

# running entangled subroutines
sub sub_ent {
  my $obj = $_[0];
  my $os = $obj->[1];
  my $universe = ${$obj->[0]};
  return sub {
    my $var = $obj->_add;
    foreach my $state (@$universe) {
      push(@$state, $state->[$os-1],
	   scalar( $state->[$os]->(@_) ));
    }
    return $var;
  }
}

# stringification (observation)
sub str_ent {
  my $c = $_[0];
  my $os = $c->[1];
  my $universe = ${$c->[0]};
  my %str_vals;
  # work out which state we want to retain
  foreach my $state (@$universe) {
    $str_vals{$state->[$os]} = $state->[$os-1] + ($str_vals{$state->[$os]}||0);
  }

  my ($hr, $ar) = _normalise(\%str_vals);
  my $rand = rand(1);
  my $rt;
 LOOP: foreach (@$ar) {
    if ( $rand < ${$hr}{$_}) {
      $rt = $_;
      last LOOP;
    }
  }
  # retain only that state
  my @retains;
  for (0..(@$universe-1)) {
    my $state = $universe->[$_];
    my $foo = $state->[$os];
    push(@retains, $_) if ("$foo" eq $rt);
  }
  if ($Quantum::Entanglement::destroy) {
    @$universe = @$universe[@retains];
    return $rt;
  }

  # set all non retained states to zero probability, leave others alone
  my $next_retain = shift @retains;
 PURGE: foreach my $snum ( 0..(@$universe-1) ) {
    if ($snum == $next_retain) {
      $next_retain = shift(@retains) || -1;
      next PURGE;
    }
    my $state = ${$universe}[$snum];
    $$state[$_] = 0 foreach grep {!($_ % 2)} (0..(@$state-1))
  }
  return $rt;
}

# numification (have to coerce things into numbers then strings for
# probability hash purposes, ick) (observation)
sub num_ent {
  my $c = $_[0];
  my $os = $c->[1];
  my $universe = ${$c->[0]};
  my %str_vals;
  # work out which state we want to retain
  foreach my $state (@$universe) {
    $str_vals{+$state->[$os]} =
               $state->[$os-1] + ($str_vals{+$state->[$os]}||0);
  }
  my ($hr, $ar) = _normalise(\%str_vals);
  my $rand = rand(1);
  my $rt;
 LOOP: foreach (@$ar) {
    if ( $rand < ${$hr}{$_}) {
      $rt = +$_;
      last LOOP;
    }
  }
  # retain only that state
  my @retains;
  for (0..(@$universe-1)) {
    my $state = $universe->[$_];
    my $foo = +$state->[$os];
    push(@retains, $_) if ($foo == $rt);
  }

  if ($Quantum::Entanglement::destroy) {
    @$universe = @$universe[@retains];
    return $rt;
  }

  # set probabilty to zero for each state we know can't be so
  my $next_retain = shift @retains;
 PURGE: foreach my $snum ( 0..(@$universe-1) ) {
    if ($snum == $next_retain) {
      $next_retain = shift(@retains) || -1;
      next PURGE;
    }
    my $state = ${$universe}[$snum];
    $$state[$_] = 0 foreach grep {!($_ % 2)} ( 0..(@$state-1) )
  }
  return $rt;
}

# boolean context (observation)
sub bool_ent {
  my $c = $_[0];
  my $os = $c->[1];
  my $universe = ${$c->[0]};
  my ($rt,$ft,$p_true, $p_false) = (0,0,0,0);
  my (@true, @false);

  foreach (0..(@$universe-1)) {
    my $state = $universe->[$_];
    my $c2 = $state->[$os];
    if ($c2) {
      $rt++;
      push @true, $_;
      $p_true += $state->[$os-1];
    }
    else {
      $ft++;
      push @false, $_;
      $p_false += $state->[$os-1];
    }
  }

  return 0 unless $rt;   # no states are true, so must be false
  return $rt unless $ft; # no states are false, so must be true
  # if it can be true, decide if it will end up being true or not
  my @retains;
  if ( _sel_output( $p_true,$p_false)
       or $Quantum::Entanglement::conform) {
    @retains = @true;
    $rt = $rt;
  }
  else {
    @retains = @false;
    $rt = 0;
  }

  if ($Quantum::Entanglement::destroy) {
    @$universe = @$universe[@retains];
    return $rt;
  }

  my $next_retain = shift @retains;
 PURGE: foreach my $snum ( 0..(@$universe-1) ) {
    if ($snum == $next_retain) {
      $next_retain = shift(@retains) || -1;
      next PURGE;
    }
    my $state = ${$universe}[$snum];
    $$state[$_] = 0 foreach grep {!($_ % 2)} (0..(@$state-1))
  }
  return $rt;
}

### any BInary, Non-observational OPeration
sub binop {
  my ($c,$d,$r,$code) = @_;
  my $var;
  my $universe;
  if ( ref($d)
       && UNIVERSAL::isa($d, 'Quantum::Entanglement')) {
    _join($c,$d);
    my $od = $d->[1]; my $oc = $c->[1];
    $var = _add($c);
    $universe = ${$c->[0]};
    foreach my $state (@$universe) {
      push @$state, ($state->[$oc-1] * $state->[$od-1],
                     &$code($state->[$oc],$state->[$od]) );
    }
  }
  else {        # adding something to one state
    my $oc = $c->[1];
    $var = _add($c);
    $universe = ${$c->[0]};
    if ($r) {
      push(@$_, ($_->[$oc-1], &$code($d,$_->[$oc]))) foreach @$universe;
    }
    else {
      push(@$_, ($_->[$oc-1], &$code($_->[$oc],$d))) foreach @$universe;
    }
  }
  return $var;
}

# any BInary Observational OPeration
sub bioop {
  my ($c, $d, $reverse, $code) = @_;
  my $rt = 0;
  my $ft = 0;
  my (@true, @false);
  my ($p_true, $p_false) = (0,0);
  my $universe;
  if (ref($d) && UNIVERSAL::isa($d, 'Quantum::Entanglement')) {
    $c->_join($d);
    $universe = ${$c->[0]};
    foreach (0..(@$universe-1)) {
      my $state = $universe->[$_];
      my $oc = $c->[1]; my $od = $d->[1];
      my $d2 = $state->[$od];
      my $c2 = $state->[$oc];
      if (&$code($c2, $d2)) {
        $rt++;
        push @true, $_;
        $p_true += $state->[$oc-1]* $state->[$od-1];
      }
      else {
        $ft++;
        push @false, $_;
        $p_false += $state->[$oc-1]* $state->[$od-1];
      }
    }
  }
  else {
    $universe = ${$c->[0]};
    foreach (0..(@$universe-1)) {
      my $state = $universe->[$_];
      my $d2 = $d;
      my $os = $c->[1];
      my $c2 = $state->[$os];
      ($c2, $d2) = ($d2, $c2) if $reverse;
      if (&$code($c2,$d2)) {
        $rt++;
        push @true, $_;
        $p_true += $state->[$os-1];
      }
      else {
        $ft++;
        push @false, $_;
        $p_false += $state->[$os-1];
      }
    }
  }

  return 0 unless $rt; # no states are true, so must be false
  return $rt unless $ft; # no states are false, so must be true
  my @retains;
  # if it can be true, decide if it will end up being true or not
  if ( _sel_output( $p_true,$p_false)
       or $Quantum::Entanglement::conform) {
    @retains = @true;
    $rt = $rt;
  }
  else {
    @retains = @false;
    $rt = 0;
  }

  if ($Quantum::Entanglement::destroy) {
    @$universe = @$universe[@retains];
    return $rt;
  }

  my $next_retain = shift @retains;
 PURGE: foreach my $snum ( 0..(@$universe-1) ) {
    if ($snum == $next_retain) {
      $next_retain = shift(@retains) || -1;
      next PURGE;
    }
    my $state = ${$universe}[$snum];
    $$state[$_] = 0 foreach grep {!($_ % 2)} (0..(@$state-1))
  }
  return $rt;

}

# any MUTating OPerator
sub mutop {
  my $c = $_[0];
  my $code = $_[1];
  my $os = $c->[1];
  my $universe = ${$c->[0]};
  foreach my $state (@$universe) {
    $state->[$os] = &$code($state->[$os]);
  }
  return $c;
}

sub unnop {
  my $c = $_[0];
  my $code = $_[1];
  my $os = $c->[1];
  my $val = $c->_add; my $universe = ${$c->[0]};
  foreach my $state (@$universe) {
    push(@$state, $state->[$os-1], &$code($state->[$os]) );
  }
  return $val;
}

##
# performing a conditional in paralell on the states (ie. without looking)
# returns a new variable

sub p_op {
  my ($arg1, $op, $arg2, $true_cf, $false_cf) = @_;
  $true_cf  = ref($true_cf)  ? $true_cf  : sub {1};
  $false_cf = ref($false_cf) ? $false_cf : sub {0};
  my $r = 0;
  unless (ref($arg1) && UNIVERSAL::isa($arg1, 'Quantum::Entanglement')) {
    $r = 1;
    ($arg1, $arg2) = ($arg2, $arg1);
  }
  my $tcref;
  eval "
     \$tcref = sub {
       local \*QE::arg1 = \\\$_[0];
       local \*QE::arg2 = \\\$_[1];
       if (\$_[0] $op \$_[1]) {
         return \&\$true_cf;
       }
       else {
         return \&\$false_cf;
       }
     }
  "; croak "$0: something wrong in p_op $@" if $@;

  return binop($arg1, $arg2, $r, $tcref);
}

# allows for other functions to be performed accross states, can take
# as many entangled variables as you like...
# can take code ref, or "symbolic" function name (eg. p_func('substr', ..))
sub p_func {
  my $func = shift;
  my $package = (caller)[0];
  # build up the function call by shifting off
  # entangled variables until something isn't entangled
  my $foo = ref($func) ? "&\$func(" : "$func(";
  my @temp = @_;
  my $first = $temp[0];
  do {
    my $c = shift @temp;
    _join($first,$c);
  } while (ref($temp[0]) && UNIVERSAL::isa($temp[0],'Quantum::Entanglement'));
  my @p_codes = ();
  do {
    my $c = shift;
    $foo .= '$state->[' . $c->[1] . '],';
    push @p_codes, $c->[1]-1;
  } while ( ref($_[0]) && UNIVERSAL::isa($_[0], 'Quantum::Entanglement'));
  $foo .= scalar(@_)? '@args);' : ');';
  my @args = @_;
  # loop over states, evaluating function in caller's package
  my $var = $first->_add;
  my $p_code = join('*', map {"\$state->[$_]"} @p_codes);
  my $universe = ${$first->[0]};
  foreach my $state (@$universe) {
    my $new_prob = eval $p_code;
    push(@$state, $new_prob, eval "package $package; $foo");
    croak "Internal error: $@" if $@;
  }
  return $var;
}

# This allows the introduction of new states into the system, based
# on the current values and probability amplitudes of current states
# must be given a code ref, followed by a list of entangled vars whose
# states will be passed to the function.
sub q_logic {
  my $func = shift;
  my (@offsets);
  my $first = $_[0];
  _join($first,$_) foreach @_;
  @offsets = map {$_->[1]-1, $_->[1]} @_;
  my $var = $first->_add;
  my $universe = ${$first->[0]};
  my @resultant_space;
  foreach my $state (@$universe) {
    my @new_states = &$func(@{$state}[@offsets]);
    do {
      push @resultant_space, [@$state, splice(@new_states,0,2)];
    } while (@new_states);
  }
  @{$universe} = @resultant_space;
  return $var;
}

# takes ft of amplitudes of a var, creates new state with the
# transformed amplitudes and the values from the first state.
sub QFT {
  my $c = $_[0];
  my $var = $c->_add;
  my $os = $c->[1];
  my $universe = ${$c->[0]};
  my @inputs = map {$_->[$os-1]} @$universe; # get current probs
  my $num = scalar @inputs;
  foreach my $r (0..($num-1)) {
    my $prob = 0;
    foreach my $x (0..($num-1)) {
      $prob += cplxe(1,(-2*pi*$r*$x / $num)) * $inputs[$x];
    }
    push @{$universe->[$r]}, $prob, $universe->[$r]->[$os];
  }
  return $var;
}

sub save_state{
  my @os;
  my $stash = [];

  foreach (@_) {
    carp "Can only save state of Quantum::Entanglement variables"
      unless (ref($_) && UNIVERSAL::isa($_, 'Quantum::Entanglement'));
  }

  my $first = $_[0];
  _join($first, $_) foreach @_;
  push(@os, $_->[1]) foreach @_;
  my $universe = ${$_[0]->[0]};
  foreach my $state (@$universe) {
    push @$stash, [ @{$state}[map {$_-1,$_} @os] ];
  }
  return bless $stash, 'Quantum::Entanglement::State';
}

# completely clobbers current state with whatever was saved previously
sub restore_state {
  my $stash = shift;

  my $num_saved = scalar(@{$stash->[0]}) /2;
  carp "You don't have any states saved!" unless $num_saved;
  my @newvars;
  $newvars[0] = _new();
  ${$newvars[0]->[0]}->[0] = ['fake','fake']; # no hackery here, no.
  if ($num_saved > 1) {
    for (2..$num_saved) {
      push(@newvars, $newvars[0]->_add());
      push @{${$newvars[0]->[0]}->[0]}, qw(fake fake); # or here, never
    }
  }
  my $universe = ${$newvars[0]->[0]};
  shift @$universe;
  foreach (@$stash) {
    push @$universe, [@$_];
  }
  return wantarray ? @newvars : $newvars[0];
}

# this is needed for simplicity of exporting save_states
package Quantum::Entanglement::State;
@Quantum::Entanglement::State::ISA = qw(Quantum::Entanglement);
sub DESTROY {}

1;

__END__;

=head1 NAME

Quantum::Entanglement - QM entanglement of variables in perl

=head1 SYNOPSIS

 use Quantum::Entanglement qw(:DEFAULT :complex :QFT);

 my $c = entangle(1,0,i,1);    # $c = |0> + i|1>
 my $d = entangle(1,0,1,1);    # $d = |0> + |1>

 $e = $c * $d; # $e now |0*0> + i|0*1> + |1*0> + i|1*1>, connected to $c, $d

 if ($e == 1) { # observe, probabilistically chose an outcome
   # if we are here, ($c,$d) = i|(1,1)>
   print "* \$e == 1\n";
 }
 else { # one of the not 1 versions of $e chosen
   # if we are here, ($c,$d) = |(0,0)> + i|(1,0)> + |(0,1)>
   print "* \$e != 1\n";
 }

=head1 BACKGROUND

 "Quantum Mechanics - the dreams that stuff is made of."

Quantum mechanics is one of the stranger things to have emerged from science
over the last hundred years.  It has led the way to new understanding
of a diverse range of fundamental physical phenomena and, should recent
developments prove fruitful, could also lead to an entirely new mode
of computation where previously intractable problems find themselves open
to easy solution.

While the detailed results of quantum theory are hard to prove, and
even harder to understand, there are a handful of concepts from the
theory which are more easily understood.  Hopefully this module will
shed some light on a few of these and their consequences.

One of the more popular interpretations of quantum mechanics holds that
instead of particles always being in a single, well defined, state
they instead exist as an almost ghostly overlay of many different
states (or values) at the same time.  Of course, it is our experience
that when we look at something, we only ever find it in one single state.
This is explained by the many states of the particle collapsing to a
single state and highlights the importance of observation.

In quantum mechanics, the
state of a system can be described by a set of numbers which have
a probability amplitude associated with them.
This probability amplitude is similar to the normal idea of probability
except for two differences.  It can be a complex number, which leads
to interference between states, and the probability with which we might
observe a system in a particular state is given by the modulus squared
of this amplitude.

Consider the simple system, often called a I<qubit>, which can take
the value of 0 or 1.  If we prepare it in the following superposition
of states (a fancy way of saying that we want it to have many possible
values at once):

  particle = 1 * (being equal to 1) + (1-i) * (being equal to 0)

we can then measure (observe) the value of the particle.  If we do
this, we find that it will be equal to 1 with a probability of

  1**2 / (1**2 + (1-i)(1+i) )

and equal to zero with a probability of

 (1+i)(1-i) / (1**2 + (1-i)(1+i) )

the factors on the bottom of each equation being necessary so that the chance
of the particle ending up in any state at all is equal to one.

Observing a particle in this way is said to collapse the wave-function,
or superposition of values, into a single value, which it will retain
from then onwards.  A simpler way of writing the equation above is
to say that

 particle = 1 |1> + (1-i) |0>

where the probability amplitude for a state is given as a 'multiplier'
of the value of the state, which appears inside the C<< | > >> pattern (this
is called a I<ket>, as sometimes the I<bra> or C<< <  | >>, pattern appears
to the left of the probability amplitudes in these equations).

Much of the power of quantum computation comes from collapsing states
and modifying the probability with which a state might collapse to a
particular value as this can be done to each possible state at the same
time, allowing for fantastic degrees of parallelism.

Things also get interesting when you have multiple particles together
in the same system.  It turns out that if two particles which exist
in many states at once interact, then after doing so, they will be
linked to one another so that when you measure the value of one
you also affect the possible values that the other can take.  This
is called entanglement and is important in many quantum algorithms.

=head1 DESCRIPTION

Essentially, this allows you to put variables into a superposition
of states, have them interact with each other (so that all states
interact) and then observe them (testing to see if they satisfy
some comparison operator, printing them) which will collapse
the entire system so that it is consistent with your knowledge.

As in quantum physics, the outcome of an observation will be the result
of selecting one of the states of the system at random.  This might
affect variables other than the ones observed, as they are able to
remember their history.

For instance, you can say:

 $foo = entangle(1,0,1,1); # foo = |0> + |1>
 $bar = entangle(1,0,1,1); # bar = |0> + |1>

if at this point we look at the values of $foo or $bar, we will
see them collapse to zero half of the time and one the other half of
the time.  We will also find that us looking at $foo will have no
effect on the possible values, or chance of getting any one of those
values, of $bar.

If we restrain ourselves a little and leave $foo and $bar unobserved
we can instead play some games with them.  We can use our entangled
variables just as we would any other variable in perl, for instance,

 $c = $foo * $bar;

will cause $c to exist in a superposition of all the possible outcomes
of multiplying each state of $foo with each state in $bar.  If we
now measure the value of $c, we will find that one quarter of the time
it will be equal to one, and three quarters of the time it will be equal
to zero.

Lets say we do this, and $c turns out to be equal to zero this time, what
does that leave $foo and $bar as?  Clearly we cannot have both $foo and
$bar both equal to one, as then $c would have been equal to one, but all
the other possible values of $foo and $bar can still occur.  We say
that the state of $foo is now entangled with the state of $bar so that

 ($foo, $bar ) = |0,0> + |0,1> + |1,0>.

If we now measure $foo, one third of the time it will be equal to one and
two thirds of the time, it will come out as zero.  If we do this and get
one, this means that should we observe $bar it will be equal to zero so
that our earlier measurement of $c still makes sense.

=head1 Use of this module

To use this module in your programs, simply add a

 use Quantum::Entanglement;

line to the top of your code,  if you want to use complex probability
amplitudes, you should instead say:

 use Quantum::Entanglement qw(:complex :DEFAULT);

which will import the C<Math::Complex i Re Im rho theta arg cplx cplxe>
functions / constants into your package.

You can also import a Quantum Fourier transform, which acts on the
probability amplitudes of a state (see below) by adding a C<:QFT>
tag.

This module adds an C<entangle> function to perl, this puts a
variable into multiple states simultaneously.  You can then
cause this variable to interact with other entangled, or normal,
values the result of which will also be in many states at once.

The different states which a variable can take each have an associated
complex probability amplitude, this can lead to interesting behaviour,
for instance, a root-not logic gate (see q_logic, below).

=head2 entangle

This sets up a new entangled variable:

 $foo = entangle(prob1, val1, prob2, val2, ...);

The probability values are strictly speaking probability amplitudes,
and can be complex numbers (corresponding to a phase or wave-ish
nature (this is stretching things slightly...)).  To use straight
numbers, just use them, to use complex values, supply a Math::Complex
number.

Thus

 $foo = entangle(1,  0, 1+4*i, 1);

corresponds to:

 foo = 1|0> + (1 + 4i)|1>

The probabilities do not need to be normalized, this is done
by the module whenever required (ie. when observing variables).

=head2 Non-observational operations

We can now use our entangled variable just as we would any normal
variable in perl.  Much of the time we will be making it do things
where we do not find anything out about the value of our variable,
if this is the case, then the variable does not collapse, although
any result of its interactions will be entangled with itself.

=head2 Observational Operators

Whenever you perform an operation on an entangled variable which
should increase your level of knowledge about the value of the variable
you will cause it to collapse into a single state or set of states.
All logical comparison (C<==>, C<gt> ....) operators, as well as
string and num -ifying and boolean observation will cause collapse.

When an entangled variable is observed in this way, sets of states which
would satisfy the operator are produced (ie. for $a < 2, all states <2 and
all >= 2).  One of these sets of states is then selected randomly, using
the probability amplitudes associated with the states.  The result of
operating on this state is then returned.  Any other states are then
destroyed.

For instance, if

 $foo = entangle(1,2,1,3,1,5,1,7);
        # |2> +|3> + |5> +|7>
then saying

 print '$foo is greater than four' if ($foo > 4);

will cause $foo to be either C<< |2> + |3> >> B<or> C<< |5> +7> >>.

Of course, if you had said instead:

  $foo = entangle(-1,2,1,3,1,5,1,7);
           # -1|2> + |3> + |5> +|7>

then if C<$foo> was measured here, it would come out as any one of 2,3,5,7
with equal likelyhood (remember, amplitude squared).  But saying

 print '$foo is greater than four' if ($foo > 4);

will cause foo to be C<< |2> or 3> >> with a probability of C<(-1 + 1) == 0> or
C<< |5 or 7> >> with probability of C<(1 + 1)/2 == 1>.  Thus C<< $foo > 4 >>
will B<always> be true.

It is possible to perform operations like these on an entangled
variable without causing collapse by using C<p_op> (below).

When performing an observation, the module can do two things to
the states which can no longer be valid (those to which it did not collapse,
|2 or 3> in the example above).  It can either internally
set the probability of them collapsing to be zero or it can delete
them entirely.  This could have consequences if you are writing parallel
functions that rely on there being a certain number of states in
a variable, even after collapse.

The default is for collapsed states to be destroyed, to alter this
behaviour, set the C<$Quantum::Entanglement::destroy> variable to
a false value.  In general though, you can leave this alone.

=head2 Dammit Jim, I can't change the laws of physics

Although not the default, it is possible to cause observation (for
boolean context or with comparison operators only) to act in a more
purposeful manner.  If the variable:

 $Quantum::Entanglement::conform

has a true value, then the overloaded operations provided by this
module will try their very best to return "truth" instead of
selecting randomly from both "true" and "false" outcomes.

For example:

 $foo = entangle(1,0,1,1,1,3); # foo = |0> + |1> + |3>
 $Quantum::Entanglement::conform = 1;
 print "\$foo > 0\n" if $foo > 0;
                               # foo now = |1> + |3>
 print "\$foo == 3\n" if $foo == 3;
                               # foo now = |3>

will always output:

 $foo > 0
 $foo == 3

Of course, setting this variable somewhat defeats the point of
the module, but it could lead to some interesting pre-calculating
algorithms which are fed with entangled input, which is then
later defined (by testing ==, say )with the answer of the calculation
appearing, as if by magic, in some other variable.  See also the
section L<save_state>.

=head2 p_op

This lets you perform conditional operations on variables in a
superposition of states B<without actually looking at them>.
This returns a new superposed variable, with states given by
the outcome of the p_op.  You cannot, of course, gain any information
about the variables involved in the p_op by doing this.

 $rt = p_op(var1, op, var2, code if true, code if false).

C<op> should be a string representing the operation to be performed
(eg. C<"==">).  The two code arguments should be references to subs
the return values of which will be used as the value of the
corresponding state should the expression be true or false.

If no code is provided, the return value of the operator itself is
evaluated in boolean context, if true, 1 or if false, 0 is
used as the corresponding state of the returned variable.  Only one
of var1 and var2 need to be entangled states.  The values of the states
being tested are placed into the $QE::arg1 and $QE::arg2 variables
should the subroutines want to play with them (these are localized
aliases to the actual values, so modify at your peril (or pleasure)).

The semantics are best shown by example:

 $gas = entangle(1, 'bottled', 1, 'released');
   # gas now in states |bottled> + |released>

 $cat_health = p_op($gas, 'eq', 'released',
                         sub {'Dead'},
                         sub {'Alive'});
   # cat,gas now in states |Alive, bottled> + |Dead, released>

This is similar to parallel execution of the following psuedo code:

 if (gas is in bottle) { # not probabilistic, as we don't look
   cat is still alive
 }
 else {
   cat is dead
 }

The cat can now be observed (with a conditional test say) and doing so will
collapse both the cat and the gas:

 if ($cat_health eq 'Dead') {# again, outcome is probabilistic
   # thus gas = |released>
 }
 else {
   # thus gas = |bottled>
 }

This also lets you use some other 'binary' operators on a superposition
of states by immediatly observing the return value of the parallel op.

 $string = entangle(1,'aa', 1, 'bb', 1, 'ab', 1, 'ba');
 $regex = qr/(.)\1/;

 if (q_op($string, '=~', $regex)) { # again, probabilistic
   # if here, string = |aa> + |bb>
 }
 else {
   # if here, string = |ab> + |ba>
 }

=head2 p_func

This lets you perform core functions and subs through the states
of a superposition without observing and produce a new variable
corresponding to a superposition of the results of the function.

 p_func("func" ,entangled var,[more vars,] [optional args])

Any number of entangled variables can be passed to the function,
optional args begin with the first non-entangled var.

The optional args will be passed to the subroutine or function unmodified.

eg. C<p_func('substr', $foo, 1,1)> will perform C<substr($state, 1,1)>
on each state in $foo.  Saying C<p_func('substr', $foo,$bar,1)> will
evaluate C<substr($s_foo, $s_bar,1)> for each state in $foo and $bar.

You can also specify a subroutine, either in the same package that C<p_func>
is called from, or with a fully qualified name.

 sub foo {my $state = $_[0]; return ${$_[1]}[$state]}
 @foo = qw(one two three);
 $foo = entangle(1,1,1,2,1,3); # |1> + |2> + |3>
 $bar = p_func('foo', $foo, \@foo);

 # bar now |one> + |two> + |three>

You can also pass a code reference as first arg (cleaner)...

 $bar = p_func(\&foo, $foo, \@foo);

=head2 q_logic

This allows you to create new states, increasing the amount of
global state as you do so.  This lets you apply weird quantum
logic gates to your variables, amongst other things.

 q_logic(code ref, entangled var [,more vars] );

The code ref is passed a list of probabilities and values corresponding
to the state currently being examined. (prob, val, [prob, val..])
code ref must return a list of the following format:

 (prob, val, prob, val ...) # as entangle basically

For instance, this is a root-not gate:

 sub root_not {
   my ($prob, $val) = @_;
   return( $prob * (i* (1/sqrt(2))), $val,
	   $prob * (1/sqrt(2)), !$val ? 1 : 0);
 }

 $foo = entangle(1,0);
 $foo = q_logic(\&root_not, $foo);

 # if $foo is observed here, it will collapse to both 0 and 1, at random

 $foo = q_logic(\&root_not, $foo);

 print "\$foo is 1\n" if $foo; # always works, $foo is now 1.

This corresponds to the following:

 foo = |0>

 root_not( foo )

 foo is now in state: sqrt(2)i |0> + sqrt(2) |1>

 root_not (foo)

 foo in state: (0.5 - 0.5) |0> + (0.5i + 0.5i) |1>

 which if observed gives

 foo = 0|0> + i|1> which must collapse to 1.

Neat, huh?

=head2 save_state

Having set up a load of entangled variables, you might wish to
store their superposed state for later restoration.  This is acheived
using the C<save_state> function:

 $state = save_state( [list of entangled variables] );

To restore the states of the entangled variables, simply call
the C<restore_state> method on the C<$state>:

  ($foo, $bar) = $state->restore_state;

The variables return by C<restore_state> will no longer be entangled to
anything they were previously connected to.  If multiple variables have
their state saved at once, then any connections between them will remain.

See the demo calc_cache for an example of use.

=head2 QFT

This provides a quantum fourier transform which acts on the probability
amplitudes of a state, creating a new state with the same values as the
initial state but with new probability amplitudes.  FTs like this are
used in many quantum algorithms where it is important to find the
periodicity of some function (for instance, Shor).

This will only work if you have carefully populated your states, essentially
if all seperately C<entangle>d variables do not interact.  This
sort of breaks encapsulation, so might change in the future!

See C<~/demo/shor.pl> for an example of the use of this function.

=head2 Quantum::Entanglement::show_states

This allows you to find out the states that your variables are in, it
does not count as observation.

If called as a method it will
only return the states available to that variable, thus:

 $foo = entangle(1,0,1,1);
 print $foo->show_states;

outputs:

 1|0>   1|1>

If a variable is entangled with other superposed values, then calling
C<save_state> with an additional true argument will display the states
of all the variables which have interacted together.

 print $foo->show_states(1);

If two variables have not yet interacted, then they will not appear in
the state space of the other.

The ordering of the output of this function may change in later versions
of this module.

=head2 Entangling subroutines

It is possible to entangle a set of subroutine references and later
call them in parallel with the same set of arguments.  The subroutines
will always be called in scalar context.  The return values of the
subroutines will be present in the entangled variable returned.

eg.

 $subs = entangle(1 => sub {return $_[0}, 1=>sub {return $_[1]});
 $return = $subs->(qw(chalk cheese));
  # $return now |chalk> + |cheese>

=head1 EXPORT

This module exports quite a bit, C<entangle>, C<save_state>,
C<p_op>, C<p_func> and C<q_logic>.  If used with qw(:complex) it will
also export the following functions / constants from the Math::Complex
module: C<i Re Im rho theta arg cplx cplxe>.

=head1 AUTHOR

Alex Gough (F<alex@earth.li>).  Any comments, suggestions or bug
reports are warmly welcomed.

=head1 SEE ALSO

perl(1).  L<Quantum::Superpositions>. L<Math::Complex>.
L<http://www.qubit.org/resource/deutsch85.ps>
 - 1985 Paper by David Deutsch.
L<http://xxx.lanl.gov/abs/math.HO/9911150>
 - Machines, Logic and Quantum Physics,
      David Deutsch, Artur Ekert, Rossella Lupacchini.

Various examples are provided in the C<~/demo/> directory of the
distribution.  An article on the module is available at
L<http://the.earth.li/~alex/quant_ent.html>.

=head1 BUGS

This is slow(ish) but fun, so hey!

=head2 Shortcomings

This module does fall short of physical reality in a few important
areas, some of which are listed below:

=over 4

=item No eigenfunction like behaviour

All operators share the same set of eigenfunctions, in real QM this
is sometimes not the case, so observing one thing would cause some
other thing (even if already observed) to fall into a superposition
of states again.

=item Certain observables cannot simultaneously have precisely defined values.

This follows from the point above.  The famous uncertainty
principle follows from the fact that position and momentum have different
sets of eigenfunctions.  In this module, it is always possible to collapse
the system so that a value is known for every entangled variable.

=item Perl is not a quantum computing device

Perl, alas, is currently only implemented on classical computers, this
has the disadvantage that any quantum algorithm will not run in constant
time but will quite likely run in exponential time.  This might be
remedied in future releases of perl.  Just not anytime soon.

=item Quantum information cannot be copied with perfect fidelity

It is impossible to perfectly clone a real entangled state without
'damaging' in some way either the original or the copy.  In this
module, it is possible for this to happen as we have special
access to the states of our variables.

=item Cannot generate perfectly random numbers

It is well known that classical computers cannot produce a perfectly
random sequence of numbers, as this module runs on one of these, it
also suffers the same fate.  It is possible to give a classical computer
access to a perfect random number generator though (essentially by
linking it to a suitable physical system) in which case this is no
longer a problem.

=back

=head1 COPYRIGHT

This code is copyright (c) Alex Gough, 2001,2002.  All Rights
Reserved.  This module is free software.  It may be used,
redistributed and/or modified under the same terms as Perl itself.

=cut