Expression of type Exists ¶

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, 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 = Exists(instance_param_or_params = [v_1_to_n], instance_expr = 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(), condition = InSet(Set(v_1_to_n), OrthoNormBases(Hspace))).with_wrapping()

# 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())

\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}~|~\left\{v_{1}, v_{2}, \ldots, v_{n}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\right)}~\\
\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}\right]\end{array}

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

Expression of type Exists¶

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

Expression of type Exists ¶

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