Expression of type Forall ¶

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 M, i, n
from proveit.core_expr_types import a_1_to_n, a_i, v_1_to_n, v_i
from proveit.linear_algebra import Adj, Hspace, LinMap, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import Equals, Exists, Forall, Iff, InSet, Set
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = Adj(M)
expr = Forall(instance_param_or_params = [M], instance_expr = Iff(Equals(Qmult(sub_expr1, M), Qmult(M, sub_expr1)), Exists(instance_param_or_params = [v_1_to_n], instance_expr = Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(M, VecSum(index_or_indices = [i], summand = ScalarMult(a_i, Qmult(Ket(v_i), Bra(v_i))), domain = Interval(one, n))), domain = Complex).with_wrapping(), condition = InSet(Set(v_1_to_n_kets), OrthoNormBases(Hspace))).with_wrapping()).with_wrapping_at(2), domain = LinMap(Hspace, Hspace))

# 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())

\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)

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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 54 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 75 operands: 9
7	Literal
8	ExprTuple	10, 11
9	ExprTuple	54, 12
10	Operation	operator: 48 operands: 13
11	Operation	operator: 31 operand: 19
12	Operation	operator: 15 operands: 16
13	ExprTuple	17, 18
14	ExprTuple	19
15	Literal
16	ExprTuple	46, 46
17	Operation	operator: 83 operands: 20
18	Operation	operator: 83 operands: 21
19	Lambda	parameters: 22 body: 23
20	ExprTuple	24, 54
21	ExprTuple	54, 24
22	ExprTuple	25
23	Conditional	value: 26 condition: 27
24	Operation	operator: 28 operand: 54
25	ExprRange	lambda_map: 30 start_index: 89 end_index: 90
26	Operation	operator: 31 operand: 34
27	Operation	operator: 75 operands: 33
28	Literal
29	ExprTuple	54
30	Lambda	parameter: 81 body: 72
31	Literal
32	ExprTuple	34
33	ExprTuple	35, 36
34	Lambda	parameters: 37 body: 38
35	Operation	operator: 52 operand: 45
36	Operation	operator: 40 operand: 46
37	ExprTuple	42
38	Conditional	value: 43 condition: 44
39	ExprTuple	45
40	Literal
41	ExprTuple	46
42	ExprRange	lambda_map: 47 start_index: 89 end_index: 90
43	Operation	operator: 48 operands: 49
44	Operation	operator: 50 operands: 51
45	Operation	operator: 52 operands: 53
46	Variable
47	Lambda	parameter: 81 body: 70
48	Literal
49	ExprTuple	54, 55
50	Literal
51	ExprTuple	56
52	Literal
53	ExprTuple	57
54	Variable
55	Operation	operator: 58 operand: 62
56	ExprRange	lambda_map: 60 start_index: 89 end_index: 90
57	ExprRange	lambda_map: 61 start_index: 89 end_index: 90
58	Literal
59	ExprTuple	62
60	Lambda	parameter: 81 body: 63
61	Lambda	parameter: 81 body: 64
62	Lambda	parameter: 97 body: 65
63	Operation	operator: 75 operands: 66
64	Operation	operator: 91 operand: 72
65	Conditional	value: 68 condition: 69
66	ExprTuple	70, 71
67	ExprTuple	72
68	Operation	operator: 73 operands: 74
69	Operation	operator: 75 operands: 76
70	IndexedVar	variable: 82 index: 81
71	Literal
72	IndexedVar	variable: 95 index: 81
73	Literal
74	ExprTuple	78, 79
75	Literal
76	ExprTuple	97, 80
77	ExprTuple	81
78	IndexedVar	variable: 82 index: 97
79	Operation	operator: 83 operands: 84
80	Operation	operator: 85 operands: 86
81	Variable
82	Variable
83	Literal
84	ExprTuple	87, 88
85	Literal
86	ExprTuple	89, 90
87	Operation	operator: 91 operand: 94
88	Operation	operator: 92 operand: 94
89	Literal
90	Variable
91	Literal
92	Literal
93	ExprTuple	94
94	IndexedVar	variable: 95 index: 97
95	Variable
96	ExprTuple	97
97	Variable

Expression of type Forall¶

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

Expression of type Forall ¶

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