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 Conditional, 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 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 = Conditional(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(), InSet(Set(a_1_to_m_kets), OrthoNormBases(CartExp(Complex, m))))

# 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_{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} \textrm{ if } \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)\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: 18 operand: 5
2	Operation	operator: 72 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7
5	Lambda	parameters: 8 body: 9
6	Operation	operator: 44 operand: 15
7	Operation	operator: 30 operand: 16
8	ExprTuple	12
9	Conditional	value: 13 condition: 14
10	ExprTuple	15
11	ExprTuple	16
12	ExprRange	lambda_map: 17 start_index: 86 end_index: 96
13	Operation	operator: 18 operand: 23
14	Operation	operator: 72 operands: 20
15	Operation	operator: 44 operands: 21
16	Operation	operator: 46 operands: 22
17	Lambda	parameter: 78 body: 69
18	Literal
19	ExprTuple	23
20	ExprTuple	24, 25
21	ExprTuple	26
22	ExprTuple	53, 95
23	Lambda	parameters: 27 body: 28
24	Operation	operator: 44 operand: 36
25	Operation	operator: 30 operand: 37
26	ExprRange	lambda_map: 32 start_index: 86 end_index: 95
27	ExprTuple	33
28	Conditional	value: 34 condition: 35
29	ExprTuple	36
30	Literal
31	ExprTuple	37
32	Lambda	parameter: 78 body: 38
33	ExprRange	lambda_map: 39 start_index: 86 end_index: 87
34	Operation	operator: 40 operands: 41
35	Operation	operator: 42 operands: 43
36	Operation	operator: 44 operands: 45
37	Operation	operator: 46 operands: 47
38	Operation	operator: 89 operand: 54
39	Lambda	parameter: 78 body: 67
40	Literal
41	ExprTuple	49, 50
42	Literal
43	ExprTuple	51
44	Literal
45	ExprTuple	52
46	Literal
47	ExprTuple	53, 96
48	ExprTuple	54
49	Variable
50	Operation	operator: 55 operand: 59
51	ExprRange	lambda_map: 57 start_index: 86 end_index: 87
52	ExprRange	lambda_map: 58 start_index: 86 end_index: 96
53	Literal
54	IndexedVar	variable: 97 index: 78
55	Literal
56	ExprTuple	59
57	Lambda	parameter: 78 body: 60
58	Lambda	parameter: 78 body: 61
59	Lambda	parameter: 100 body: 62
60	Operation	operator: 72 operands: 63
61	Operation	operator: 89 operand: 69
62	Conditional	value: 65 condition: 66
63	ExprTuple	67, 68
64	ExprTuple	69
65	Operation	operator: 70 operands: 71
66	Operation	operator: 72 operands: 73
67	IndexedVar	variable: 79 index: 78
68	Literal
69	IndexedVar	variable: 98 index: 78
70	Literal
71	ExprTuple	75, 76
72	Literal
73	ExprTuple	100, 77
74	ExprTuple	78
75	IndexedVar	variable: 79 index: 100
76	Operation	operator: 80 operands: 81
77	Operation	operator: 82 operands: 83
78	Variable
79	Variable
80	Literal
81	ExprTuple	84, 85
82	Literal
83	ExprTuple	86, 87
84	Operation	operator: 89 operand: 93
85	Operation	operator: 89 operand: 94
86	Literal
87	Operation	operator: 91 operands: 92
88	ExprTuple	93
89	Literal
90	ExprTuple	94
91	Literal
92	ExprTuple	95, 96
93	IndexedVar	variable: 97 index: 100
94	IndexedVar	variable: 98 index: 100
95	Variable
96	Variable
97	Variable
98	Variable
99	ExprTuple	100
100	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 ¶