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 OrthoNormBases, ScalarMult, VecSum
from proveit.logic import And, CartExp, 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(CartExp(Complex, 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}\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(\mathbb{C}^{n}\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: 65 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: 79 end_index: 80
16	ExprRange	lambda_map: 22 start_index: 79 end_index: 80
17	Operation	operator: 24 operands: 23
18	Operation	operator: 24 operands: 25
19	ExprRange	lambda_map: 26 start_index: 79 end_index: 80
20	Operation	operator: 27 operands: 28
21	Lambda	parameter: 61 body: 52
22	Lambda	parameter: 61 body: 53
23	ExprTuple	29, 30
24	Literal
25	ExprTuple	31, 32
26	Lambda	parameter: 61 body: 33
27	Literal
28	ExprTuple	54, 80
29	Operation	operator: 35 operands: 34
30	Operation	operator: 35 operands: 36
31	ExprRange	lambda_map: 37 start_index: 79 end_index: 80
32	ExprRange	lambda_map: 38 start_index: 79 end_index: 80
33	IndexedVar	variable: 85 index: 61
34	ExprTuple	39, 40
35	Literal
36	ExprTuple	41, 42
37	Lambda	parameter: 61 body: 43
38	Lambda	parameter: 61 body: 44
39	Variable
40	Operation	operator: 46 operand: 50
41	Variable
42	Operation	operator: 46 operand: 51
43	Operation	operator: 65 operands: 48
44	Operation	operator: 65 operands: 49
45	ExprTuple	50
46	Literal
47	ExprTuple	51
48	ExprTuple	52, 54
49	ExprTuple	53, 54
50	Lambda	parameter: 87 body: 55
51	Lambda	parameter: 87 body: 56
52	IndexedVar	variable: 71 index: 61
53	IndexedVar	variable: 72 index: 61
54	Literal
55	Conditional	value: 58 condition: 60
56	Conditional	value: 59 condition: 60
57	ExprTuple	61
58	Operation	operator: 63 operands: 62
59	Operation	operator: 63 operands: 64
60	Operation	operator: 65 operands: 66
61	Variable
62	ExprTuple	67, 69
63	Literal
64	ExprTuple	68, 69
65	Literal
66	ExprTuple	87, 70
67	IndexedVar	variable: 71 index: 87
68	IndexedVar	variable: 72 index: 87
69	Operation	operator: 73 operands: 74
70	Operation	operator: 75 operands: 76
71	Variable
72	Variable
73	Literal
74	ExprTuple	77, 78
75	Literal
76	ExprTuple	79, 80
77	Operation	operator: 81 operand: 84
78	Operation	operator: 82 operand: 84
79	Literal
80	Variable
81	Literal
82	Literal
83	ExprTuple	84
84	IndexedVar	variable: 85 index: 87
85	Variable
86	ExprTuple	87
87	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 ¶