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 U, i, n
from proveit.core_expr_types import v_1_to_n, v_i, w_1_to_n, w_i
from proveit.linear_algebra import OrthoNormBases, Unitary, VecSum
from proveit.logic import CartExp, Equals, Exists, Forall, 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, w_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = OrthoNormBases(CartExp(Complex, n))
expr = Forall(instance_param_or_params = [U], instance_expr = Exists(instance_param_or_params = [v_1_to_n, w_1_to_n], instance_expr = Equals(U, VecSum(index_or_indices = [i], summand = Qmult(Ket(v_i), Bra(w_i)), domain = Interval(one, n))), conditions = [InSet(Set(v_1_to_n_kets), sub_expr1), InSet(Set(w_1_to_n_kets), sub_expr1)]).with_wrapping(), domain = Unitary(n))

# 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_{U \in \textrm{U}\left(n\right)}~\left[\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}, w_{1}, w_{2}, \ldots, w_{n}~|~\left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right), \left\{\left\{\lvert w_{1} \rangle, \lvert w_{2} \rangle, \ldots, \lvert w_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right)}~\\
\left(U = \left(\sum_{i=1}^{n} \left(\lvert v_{i} \rangle \thinspace \langle w_{i} \rvert\right)\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 27 body: 5
4	ExprTuple	27
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operand: 11
7	Operation	operator: 51 operands: 10
8	Literal
9	ExprTuple	11
10	ExprTuple	27, 12
11	Lambda	parameters: 13 body: 14
12	Operation	operator: 15 operand: 74
13	ExprTuple	17, 18
14	Conditional	value: 19 condition: 20
15	Literal
16	ExprTuple	74
17	ExprRange	lambda_map: 21 start_index: 73 end_index: 74
18	ExprRange	lambda_map: 22 start_index: 73 end_index: 74
19	Operation	operator: 23 operands: 24
20	Operation	operator: 25 operands: 26
21	Lambda	parameter: 87 body: 82
22	Lambda	parameter: 87 body: 83
23	Literal
24	ExprTuple	27, 28
25	Literal
26	ExprTuple	29, 30
27	Variable
28	Operation	operator: 31 operand: 35
29	Operation	operator: 51 operands: 33
30	Operation	operator: 51 operands: 34
31	Literal
32	ExprTuple	35
33	ExprTuple	36, 38
34	ExprTuple	37, 38
35	Lambda	parameter: 81 body: 39
36	Operation	operator: 54 operand: 46
37	Operation	operator: 54 operand: 47
38	Operation	operator: 42 operand: 48
39	Conditional	value: 44 condition: 45
40	ExprTuple	46
41	ExprTuple	47
42	Literal
43	ExprTuple	48
44	Operation	operator: 49 operands: 50
45	Operation	operator: 51 operands: 52
46	Operation	operator: 54 operands: 53
47	Operation	operator: 54 operands: 55
48	Operation	operator: 56 operands: 57
49	Literal
50	ExprTuple	58, 59
51	Literal
52	ExprTuple	81, 60
53	ExprTuple	61
54	Literal
55	ExprTuple	62
56	Literal
57	ExprTuple	63, 74
58	Operation	operator: 79 operand: 71
59	Operation	operator: 65 operand: 72
60	Operation	operator: 67 operands: 68
61	ExprRange	lambda_map: 69 start_index: 73 end_index: 74
62	ExprRange	lambda_map: 70 start_index: 73 end_index: 74
63	Literal
64	ExprTuple	71
65	Literal
66	ExprTuple	72
67	Literal
68	ExprTuple	73, 74
69	Lambda	parameter: 87 body: 75
70	Lambda	parameter: 87 body: 76
71	IndexedVar	variable: 84 index: 81
72	IndexedVar	variable: 85 index: 81
73	Literal
74	Variable
75	Operation	operator: 79 operand: 82
76	Operation	operator: 79 operand: 83
77	ExprTuple	81
78	ExprTuple	82
79	Literal
80	ExprTuple	83
81	Variable
82	IndexedVar	variable: 84 index: 87
83	IndexedVar	variable: 85 index: 87
84	Variable
85	Variable
86	ExprTuple	87
87	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 ¶