Expression of type Conditional

from the theory of proveit.physics.quantum.algebra

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, ExprRange, IndexedVar, Variable, i, lambda_, m, n
from proveit.core_expr_types import a_1_to_m, a_i, b_1_to_n, b_i, lambda_i
from proveit.linear_algebra import OrthoNormBases, ScalarMult, TensorProd, VecSum
from proveit.logic import And, CartExp, Equals, Exists, Forall, InSet, Set
from proveit.numbers import Complex, Interval, Min, NaturalPos, RealNonNeg, one
from proveit.physics.quantum import Ket, Qmult, var_ket_psi
from proveit.physics.quantum.algebra import a_1_to_m_kets, b_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Min(m, n)
sub_expr3 = CartExp(Complex, n)
sub_expr4 = CartExp(Complex, m)
expr = Conditional(Forall(instance_param_or_params = [var_ket_psi], instance_expr = Exists(instance_param_or_params = [a_1_to_m], instance_expr = Exists(instance_param_or_params = [b_1_to_n], instance_expr = Exists(instance_param_or_params = [ExprRange(sub_expr1, IndexedVar(lambda_, sub_expr1), one, sub_expr2)], instance_expr = Equals(var_ket_psi, VecSum(index_or_indices = [i], summand = ScalarMult(lambda_i, Qmult(Ket(a_i), Ket(b_i))), domain = Interval(one, sub_expr2))), domain = RealNonNeg).with_wrapping(), condition = InSet(Set(b_1_to_n_kets), OrthoNormBases(sub_expr3))).with_wrapping(), condition = InSet(Set(a_1_to_m_kets), OrthoNormBases(sub_expr4))).with_wrapping(), domain = TensorProd(sub_expr4, sub_expr3)).with_wrapping(), And(InSet(m, NaturalPos), InSet(n, NaturalPos)))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left\{\begin{array}{l}\forall_{\lvert \psi \rangle \in \mathbb{C}^{m} {\otimes} \mathbb{C}^{n}}~\\
\left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{m}~|~\left\{\left\{\lvert a_{1} \rangle, \lvert a_{2} \rangle, \ldots, \lvert a_{m} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{m}\right)}~\\
\left[\begin{array}{l}\exists_{b_{1}, b_{2}, \ldots, b_{n}~|~\left\{\left\{\lvert b_{1} \rangle, \lvert b_{2} \rangle, \ldots, \lvert b_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right)}~\\
\left[\begin{array}{l}\exists_{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{{\rm Min}\left(m, n\right)} \in \mathbb{R}^{\ge 0}}~\\
\left(\lvert \psi \rangle = \left(\sum_{i=1}^{{\rm Min}\left(m, n\right)} \left(\lambda_{i} \cdot \left(\lvert a_{i} \rangle \thinspace \lvert b_{i} \rangle\right)\right)\right)\right)\end{array}\right]\end{array}\right]\end{array}\right]\end{array} \textrm{ if } m \in \mathbb{N}^+ ,  n \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 6
2	Operation	operator: 67 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	7, 8
6	Lambda	parameter: 74 body: 10
7	Operation	operator: 97 operands: 11
8	Operation	operator: 97 operands: 12
9	ExprTuple	74
10	Conditional	value: 13 condition: 14
11	ExprTuple	120, 15
12	ExprTuple	121, 15
13	Operation	operator: 43 operand: 18
14	Operation	operator: 97 operands: 17
15	Literal
16	ExprTuple	18
17	ExprTuple	74, 19
18	Lambda	parameters: 20 body: 21
19	Operation	operator: 22 operands: 23
20	ExprTuple	24
21	Conditional	value: 25 condition: 26
22	Literal
23	ExprTuple	41, 62
24	ExprRange	lambda_map: 27 start_index: 111 end_index: 120
25	Operation	operator: 43 operand: 30
26	Operation	operator: 97 operands: 29
27	Lambda	parameter: 103 body: 79
28	ExprTuple	30
29	ExprTuple	31, 32
30	Lambda	parameters: 33 body: 34
31	Operation	operator: 69 operand: 40
32	Operation	operator: 55 operand: 41
33	ExprTuple	37
34	Conditional	value: 38 condition: 39
35	ExprTuple	40
36	ExprTuple	41
37	ExprRange	lambda_map: 42 start_index: 111 end_index: 121
38	Operation	operator: 43 operand: 48
39	Operation	operator: 97 operands: 45
40	Operation	operator: 69 operands: 46
41	Operation	operator: 71 operands: 47
42	Lambda	parameter: 103 body: 94
43	Literal
44	ExprTuple	48
45	ExprTuple	49, 50
46	ExprTuple	51
47	ExprTuple	78, 120
48	Lambda	parameters: 52 body: 53
49	Operation	operator: 69 operand: 61
50	Operation	operator: 55 operand: 62
51	ExprRange	lambda_map: 57 start_index: 111 end_index: 120
52	ExprTuple	58
53	Conditional	value: 59 condition: 60
54	ExprTuple	61
55	Literal
56	ExprTuple	62
57	Lambda	parameter: 103 body: 63
58	ExprRange	lambda_map: 64 start_index: 111 end_index: 112
59	Operation	operator: 65 operands: 66
60	Operation	operator: 67 operands: 68
61	Operation	operator: 69 operands: 70
62	Operation	operator: 71 operands: 72
63	Operation	operator: 114 operand: 79
64	Lambda	parameter: 103 body: 92
65	Literal
66	ExprTuple	74, 75
67	Literal
68	ExprTuple	76
69	Literal
70	ExprTuple	77
71	Literal
72	ExprTuple	78, 121
73	ExprTuple	79
74	Variable
75	Operation	operator: 80 operand: 84
76	ExprRange	lambda_map: 82 start_index: 111 end_index: 112
77	ExprRange	lambda_map: 83 start_index: 111 end_index: 121
78	Literal
79	IndexedVar	variable: 122 index: 103
80	Literal
81	ExprTuple	84
82	Lambda	parameter: 103 body: 85
83	Lambda	parameter: 103 body: 86
84	Lambda	parameter: 125 body: 87
85	Operation	operator: 97 operands: 88
86	Operation	operator: 114 operand: 94
87	Conditional	value: 90 condition: 91
88	ExprTuple	92, 93
89	ExprTuple	94
90	Operation	operator: 95 operands: 96
91	Operation	operator: 97 operands: 98
92	IndexedVar	variable: 104 index: 103
93	Literal
94	IndexedVar	variable: 123 index: 103
95	Literal
96	ExprTuple	100, 101
97	Literal
98	ExprTuple	125, 102
99	ExprTuple	103
100	IndexedVar	variable: 104 index: 125
101	Operation	operator: 105 operands: 106
102	Operation	operator: 107 operands: 108
103	Variable
104	Variable
105	Literal
106	ExprTuple	109, 110
107	Literal
108	ExprTuple	111, 112
109	Operation	operator: 114 operand: 118
110	Operation	operator: 114 operand: 119
111	Literal
112	Operation	operator: 116 operands: 117
113	ExprTuple	118
114	Literal
115	ExprTuple	119
116	Literal
117	ExprTuple	120, 121
118	IndexedVar	variable: 122 index: 125
119	IndexedVar	variable: 123 index: 125
120	Variable
121	Variable
122	Variable
123	Variable
124	ExprTuple	125
125	Variable