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, NaturalPos, 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 = [n], instance_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)), domain = NaturalPos)

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