Theorem schmidt_decomposition of type Forall

from the theory of proveit.physics.quantum.algebra

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.physics.quantum.algebra import schmidt_decomposition

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

Passed sanity check: schmidt_decomposition matches stored_expr

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

\forall_{m, n \in \mathbb{N}^+}~\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}\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 6 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 120 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 9
5	Operation	operator: 70 operands: 8
6	Literal
7	ExprTuple	9
8	ExprTuple	10, 11
9	Lambda	parameter: 77 body: 13
10	Operation	operator: 100 operands: 14
11	Operation	operator: 100 operands: 15
12	ExprTuple	77
13	Conditional	value: 16 condition: 17
14	ExprTuple	123, 18
15	ExprTuple	124, 18
16	Operation	operator: 46 operand: 21
17	Operation	operator: 100 operands: 20
18	Literal
19	ExprTuple	21
20	ExprTuple	77, 22
21	Lambda	parameters: 23 body: 24
22	Operation	operator: 25 operands: 26
23	ExprTuple	27
24	Conditional	value: 28 condition: 29
25	Literal
26	ExprTuple	44, 65
27	ExprRange	lambda_map: 30 start_index: 114 end_index: 123
28	Operation	operator: 46 operand: 33
29	Operation	operator: 100 operands: 32
30	Lambda	parameter: 106 body: 82
31	ExprTuple	33
32	ExprTuple	34, 35
33	Lambda	parameters: 36 body: 37
34	Operation	operator: 72 operand: 43
35	Operation	operator: 58 operand: 44
36	ExprTuple	40
37	Conditional	value: 41 condition: 42
38	ExprTuple	43
39	ExprTuple	44
40	ExprRange	lambda_map: 45 start_index: 114 end_index: 124
41	Operation	operator: 46 operand: 51
42	Operation	operator: 100 operands: 48
43	Operation	operator: 72 operands: 49
44	Operation	operator: 74 operands: 50
45	Lambda	parameter: 106 body: 97
46	Literal
47	ExprTuple	51
48	ExprTuple	52, 53
49	ExprTuple	54
50	ExprTuple	81, 123
51	Lambda	parameters: 55 body: 56
52	Operation	operator: 72 operand: 64
53	Operation	operator: 58 operand: 65
54	ExprRange	lambda_map: 60 start_index: 114 end_index: 123
55	ExprTuple	61
56	Conditional	value: 62 condition: 63
57	ExprTuple	64
58	Literal
59	ExprTuple	65
60	Lambda	parameter: 106 body: 66
61	ExprRange	lambda_map: 67 start_index: 114 end_index: 115
62	Operation	operator: 68 operands: 69
63	Operation	operator: 70 operands: 71
64	Operation	operator: 72 operands: 73
65	Operation	operator: 74 operands: 75
66	Operation	operator: 117 operand: 82
67	Lambda	parameter: 106 body: 95
68	Literal
69	ExprTuple	77, 78
70	Literal
71	ExprTuple	79
72	Literal
73	ExprTuple	80
74	Literal
75	ExprTuple	81, 124
76	ExprTuple	82
77	Variable
78	Operation	operator: 83 operand: 87
79	ExprRange	lambda_map: 85 start_index: 114 end_index: 115
80	ExprRange	lambda_map: 86 start_index: 114 end_index: 124
81	Literal
82	IndexedVar	variable: 125 index: 106
83	Literal
84	ExprTuple	87
85	Lambda	parameter: 106 body: 88
86	Lambda	parameter: 106 body: 89
87	Lambda	parameter: 128 body: 90
88	Operation	operator: 100 operands: 91
89	Operation	operator: 117 operand: 97
90	Conditional	value: 93 condition: 94
91	ExprTuple	95, 96
92	ExprTuple	97
93	Operation	operator: 98 operands: 99
94	Operation	operator: 100 operands: 101
95	IndexedVar	variable: 107 index: 106
96	Literal
97	IndexedVar	variable: 126 index: 106
98	Literal
99	ExprTuple	103, 104
100	Literal
101	ExprTuple	128, 105
102	ExprTuple	106
103	IndexedVar	variable: 107 index: 128
104	Operation	operator: 108 operands: 109
105	Operation	operator: 110 operands: 111
106	Variable
107	Variable
108	Literal
109	ExprTuple	112, 113
110	Literal
111	ExprTuple	114, 115
112	Operation	operator: 117 operand: 121
113	Operation	operator: 117 operand: 122
114	Literal
115	Operation	operator: 119 operands: 120
116	ExprTuple	121
117	Literal
118	ExprTuple	122
119	Literal
120	ExprTuple	123, 124
121	IndexedVar	variable: 125 index: 128
122	IndexedVar	variable: 126 index: 128
123	Variable
124	Variable
125	Variable
126	Variable
127	ExprTuple	128
128	Variable