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 A, B, Conditional, i, n
from proveit.core_expr_types import a_1_to_n, a_i, b_1_to_n, b_i, v_1_to_n, v_i
from proveit.linear_algebra import Hspace, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import And, Equals, Exists, InSet, Set
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult

# 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(Exists(instance_param_or_params = [a_1_to_n, b_1_to_n], instance_expr = And(Equals(A, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(a_i, sub_expr3), domain = sub_expr2)), Equals(B, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(b_i, sub_expr3), domain = sub_expr2))).with_wrapping_at(2), domain = Complex).with_wrapping(), InSet(Set(v_1_to_n), OrthoNormBases(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())

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