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_1_to_m, 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 a_1_to_m_kets, 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 = [a_1_to_m], instance_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(), condition = InSet(Set(a_1_to_m_kets), OrthoNormBases(CartExp(Complex, m)))).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_{a_{1}, a_{2}, \ldots, a_{m}~|~\left\{\left\{\lvert a_{1} \rangle, \lvert a_{2} \rangle, \ldots, \lvert a_{m} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{m}\right)}~\\
\left[\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}\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: 24 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: 92 end_index: 101
6	Operation	operator: 24 operand: 11
7	Operation	operator: 78 operands: 10
8	Lambda	parameter: 84 body: 60
9	ExprTuple	11
10	ExprTuple	12, 13
11	Lambda	parameters: 14 body: 15
12	Operation	operator: 50 operand: 21
13	Operation	operator: 36 operand: 22
14	ExprTuple	18
15	Conditional	value: 19 condition: 20
16	ExprTuple	21
17	ExprTuple	22
18	ExprRange	lambda_map: 23 start_index: 92 end_index: 102
19	Operation	operator: 24 operand: 29
20	Operation	operator: 78 operands: 26
21	Operation	operator: 50 operands: 27
22	Operation	operator: 52 operands: 28
23	Lambda	parameter: 84 body: 75
24	Literal
25	ExprTuple	29
26	ExprTuple	30, 31
27	ExprTuple	32
28	ExprTuple	59, 101
29	Lambda	parameters: 33 body: 34
30	Operation	operator: 50 operand: 42
31	Operation	operator: 36 operand: 43
32	ExprRange	lambda_map: 38 start_index: 92 end_index: 101
33	ExprTuple	39
34	Conditional	value: 40 condition: 41
35	ExprTuple	42
36	Literal
37	ExprTuple	43
38	Lambda	parameter: 84 body: 44
39	ExprRange	lambda_map: 45 start_index: 92 end_index: 93
40	Operation	operator: 46 operands: 47
41	Operation	operator: 48 operands: 49
42	Operation	operator: 50 operands: 51
43	Operation	operator: 52 operands: 53
44	Operation	operator: 95 operand: 60
45	Lambda	parameter: 84 body: 73
46	Literal
47	ExprTuple	55, 56
48	Literal
49	ExprTuple	57
50	Literal
51	ExprTuple	58
52	Literal
53	ExprTuple	59, 102
54	ExprTuple	60
55	Variable
56	Operation	operator: 61 operand: 65
57	ExprRange	lambda_map: 63 start_index: 92 end_index: 93
58	ExprRange	lambda_map: 64 start_index: 92 end_index: 102
59	Literal
60	IndexedVar	variable: 103 index: 84
61	Literal
62	ExprTuple	65
63	Lambda	parameter: 84 body: 66
64	Lambda	parameter: 84 body: 67
65	Lambda	parameter: 106 body: 68
66	Operation	operator: 78 operands: 69
67	Operation	operator: 95 operand: 75
68	Conditional	value: 71 condition: 72
69	ExprTuple	73, 74
70	ExprTuple	75
71	Operation	operator: 76 operands: 77
72	Operation	operator: 78 operands: 79
73	IndexedVar	variable: 85 index: 84
74	Literal
75	IndexedVar	variable: 104 index: 84
76	Literal
77	ExprTuple	81, 82
78	Literal
79	ExprTuple	106, 83
80	ExprTuple	84
81	IndexedVar	variable: 85 index: 106
82	Operation	operator: 86 operands: 87
83	Operation	operator: 88 operands: 89
84	Variable
85	Variable
86	Literal
87	ExprTuple	90, 91
88	Literal
89	ExprTuple	92, 93
90	Operation	operator: 95 operand: 99
91	Operation	operator: 95 operand: 100
92	Literal
93	Operation	operator: 97 operands: 98
94	ExprTuple	99
95	Literal
96	ExprTuple	100
97	Literal
98	ExprTuple	101, 102
99	IndexedVar	variable: 103 index: 106
100	IndexedVar	variable: 104 index: 106
101	Variable
102	Variable
103	Variable
104	Variable
105	ExprTuple	106
106	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 ¶