Theorem spectral_decomposition_as_outer_prods of type Forall ¶

from the theory of proveit.physics.quantum.algebra ¶

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 spectral_decomposition_as_outer_prods

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

Passed sanity check: spectral_decomposition_as_outer_prods matches stored_expr

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

\forall_{n \in \mathbb{N}^+}~\left[\begin{array}{l}\forall_{\mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}~|~\textrm{Dim}\left(\mathcal{H}\right) = n}~\\
\left[\forall_{M \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)}~\left(\begin{array}{c} \begin{array}{l} \left(\left(M^{\dagger} \thinspace M\right) = \left(M \thinspace M^{\dagger}\right)\right) \Leftrightarrow  \\ \left[\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}~|~\left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\right)}~\\
\left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}}~\\
\left(M = \left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\end{array}\right]\end{array}\right] \end{array} \end{array}\right)\right]\end{array}\right]

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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',)

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

Theorem spectral_decomposition_as_outer_prods of type Forall¶

from the theory of proveit.physics.quantum.algebra¶

Theorem spectral_decomposition_as_outer_prods of type Forall ¶

from the theory of proveit.physics.quantum.algebra ¶