Expression of type Conditional ¶

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 Conditional, Function, f, 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 Dim, HilbertSpaces, Hspace, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import And, Equals, Forall, Functions, InClass, 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 = [i]
sub_expr2 = Interval(one, n)
sub_expr3 = Qmult(Ket(v_i), Bra(v_i))
expr = Conditional(Forall(instance_param_or_params = [v_1_to_n], instance_expr = Forall(instance_param_or_params = [a_1_to_n], instance_expr = Forall(instance_param_or_params = [f], instance_expr = Equals(Function(f, [VecSum(index_or_indices = sub_expr1, summand = ScalarMult(a_i, sub_expr3), domain = sub_expr2)]), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Function(f, [a_i]), sub_expr3), domain = sub_expr2)), domain = Functions(Complex, Complex)), domain = Complex), condition = InSet(Set(v_1_to_n_kets), OrthoNormBases(Hspace))).with_wrapping(), And(InClass(Hspace, HilbertSpaces), Equals(Dim(Hspace), 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())

\left\{\begin{array}{l}\forall_{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[\forall_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}}~\left[\forall_{f \in \left[\mathbb{C} \rightarrow \mathbb{C}\right]}~\left(f\left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right) = \left(\sum_{i=1}^{n} \left(f\left(a_{i}\right) \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\right]\right]\end{array} \textrm{ if } \mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  \textrm{Dim}\left(\mathcal{H}\right) = n\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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

Expression of type Conditional¶

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

Expression of type Conditional ¶

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