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 ExprRange, IndexedVar, Variable, i, lambda_, m, n
from proveit.core_expr_types import a_i, b_1_to_n, b_i, lambda_i
from proveit.linear_algebra import OrthoNormBases, ScalarMult, VecSum
from proveit.logic import CartExp, Equals, Exists, InSet, Set
from proveit.numbers import Complex, Interval, Min, RealNonNeg, one
from proveit.physics.quantum import Ket, Qmult, var_ket_psi
from proveit.physics.quantum.algebra import b_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Min(m, n)
expr = Exists(instance_param_or_params = [b_1_to_n], instance_expr = Exists(instance_param_or_params = [ExprRange(sub_expr1, IndexedVar(lambda_, sub_expr1), one, sub_expr2)], instance_expr = Equals(var_ket_psi, VecSum(index_or_indices = [i], summand = ScalarMult(lambda_i, Qmult(Ket(a_i), Ket(b_i))), domain = Interval(one, sub_expr2))), domain = RealNonNeg).with_wrapping(), condition = InSet(Set(b_1_to_n_kets), OrthoNormBases(CartExp(Complex, n)))).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_{b_{1}, b_{2}, \ldots, b_{n}~|~\left\{\left\{\lvert b_{1} \rangle, \lvert b_{2} \rangle, \ldots, \lvert b_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right)}~\\
\left[\begin{array}{l}\exists_{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{{\rm Min}\left(m, n\right)} \in \mathbb{R}^{\ge 0}}~\\
\left(\lvert \psi \rangle = \left(\sum_{i=1}^{{\rm Min}\left(m, n\right)} \left(\lambda_{i} \cdot \left(\lvert a_{i} \rangle \thinspace \lvert b_{i} \rangle\right)\right)\right)\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: 9 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	5
4	Conditional	value: 6 condition: 7
5	ExprRange	lambda_map: 8 start_index: 70 end_index: 80
6	Operation	operator: 9 operand: 12
7	Operation	operator: 56 operands: 11
8	Lambda	parameter: 62 body: 53
9	Literal
10	ExprTuple	12
11	ExprTuple	13, 14
12	Lambda	parameters: 15 body: 16
13	Operation	operator: 30 operand: 23
14	Operation	operator: 18 operand: 24
15	ExprTuple	20
16	Conditional	value: 21 condition: 22
17	ExprTuple	23
18	Literal
19	ExprTuple	24
20	ExprRange	lambda_map: 25 start_index: 70 end_index: 71
21	Operation	operator: 26 operands: 27
22	Operation	operator: 28 operands: 29
23	Operation	operator: 30 operands: 31
24	Operation	operator: 32 operands: 33
25	Lambda	parameter: 62 body: 51
26	Literal
27	ExprTuple	34, 35
28	Literal
29	ExprTuple	36
30	Literal
31	ExprTuple	37
32	Literal
33	ExprTuple	38, 80
34	Variable
35	Operation	operator: 39 operand: 43
36	ExprRange	lambda_map: 41 start_index: 70 end_index: 71
37	ExprRange	lambda_map: 42 start_index: 70 end_index: 80
38	Literal
39	Literal
40	ExprTuple	43
41	Lambda	parameter: 62 body: 44
42	Lambda	parameter: 62 body: 45
43	Lambda	parameter: 84 body: 46
44	Operation	operator: 56 operands: 47
45	Operation	operator: 73 operand: 53
46	Conditional	value: 49 condition: 50
47	ExprTuple	51, 52
48	ExprTuple	53
49	Operation	operator: 54 operands: 55
50	Operation	operator: 56 operands: 57
51	IndexedVar	variable: 63 index: 62
52	Literal
53	IndexedVar	variable: 82 index: 62
54	Literal
55	ExprTuple	59, 60
56	Literal
57	ExprTuple	84, 61
58	ExprTuple	62
59	IndexedVar	variable: 63 index: 84
60	Operation	operator: 64 operands: 65
61	Operation	operator: 66 operands: 67
62	Variable
63	Variable
64	Literal
65	ExprTuple	68, 69
66	Literal
67	ExprTuple	70, 71
68	Operation	operator: 73 operand: 77
69	Operation	operator: 73 operand: 78
70	Literal
71	Operation	operator: 75 operands: 76
72	ExprTuple	77
73	Literal
74	ExprTuple	78
75	Literal
76	ExprTuple	79, 80
77	IndexedVar	variable: 81 index: 84
78	IndexedVar	variable: 82 index: 84
79	Variable
80	Variable
81	Variable
82	Variable
83	ExprTuple	84
84	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 ¶