Expression of type Conditional

from the theory of proveit.physics.quantum.circuits

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
from proveit.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import And, Equals, Forall, InSet
from proveit.physics.quantum import l_ket_domain, m_ket_domain, var_ket_psi, var_ket_u, var_ket_v
from proveit.physics.quantum.circuits import circuit__psi_m__u_Akl_v, circuit__u_Akl_v
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [var_ket_psi], instance_expr = Forall(instance_param_or_params = [A11_to_Akl], instance_expr = Equals(Prob(circuit__u_Akl_v), Prob(circuit__psi_m__u_Akl_v)).with_wrapping_at(2)).with_wrapping(), domain = m_ket_domain), And(InSet(var_ket_u, l_ket_domain), InSet(var_ket_v, l_ket_domain)))

expr:

# 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\{\forall_{\lvert \psi \rangle \in \mathbb{C}^{2^{m}}}~\left[\begin{array}{l}\forall_{A_{1, 1}, A_{1, 2}, \ldots, A_{1, l}, A_{2, 1}, A_{2, 2}, \ldots, A_{2, l}, \ldots\ldots, A_{k, 1}, A_{k, 2}, \ldots, A_{k, l}}~\\
\left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle}
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert \psi \rangle} & { /^{m} } \qw & { /^{m} } \qw & \gate{\cdots} & { /^{m} } \qw & \qout{\lvert \psi \rangle} \\
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle}
} \end{array}\right) \end{array} \end{array}\right)\end{array}\right] \textrm{ if } \lvert u \rangle \in \mathbb{C}^{2^{l}} ,  \lvert v \rangle \in \mathbb{C}^{2^{l}}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 16 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	7, 8
6	Lambda	parameter: 115 body: 10
7	Operation	operator: 18 operands: 11
8	Operation	operator: 18 operands: 12
9	ExprTuple	115
10	Conditional	value: 13 condition: 14
11	ExprTuple	112, 15
12	ExprTuple	116, 15
13	Operation	operator: 16 operand: 21
14	Operation	operator: 18 operands: 19
15	Operation	operator: 26 operands: 20
16	Literal
17	ExprTuple	21
18	Literal
19	ExprTuple	115, 22
20	ExprTuple	32, 23
21	Lambda	parameters: 24 body: 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 37 operands: 28
24	ExprTuple	29
25	Operation	operator: 30 operands: 31
26	Literal
27	ExprTuple	32, 33
28	ExprTuple	42, 125
29	ExprRange	lambda_map: 34 start_index: 123 end_index: 60
30	Literal
31	ExprTuple	35, 36
32	Literal
33	Operation	operator: 37 operands: 38
34	Lambda	parameter: 114 body: 77
35	Operation	operator: 40 operand: 43
36	Operation	operator: 40 operand: 44
37	Literal
38	ExprTuple	42, 124
39	ExprTuple	43
40	Literal
41	ExprTuple	44
42	Literal
43	Operation	operator: 46 operands: 45
44	Operation	operator: 46 operands: 47
45	ExprTuple	48, 49, 50
46	Literal
47	ExprTuple	51, 52, 53
48	ExprTuple	54
49	ExprRange	lambda_map: 55 start_index: 123 end_index: 60
50	ExprTuple	56
51	ExprTuple	57, 58
52	ExprRange	lambda_map: 59 start_index: 123 end_index: 60
53	ExprTuple	61, 62
54	ExprRange	lambda_map: 63 start_index: 123 end_index: 125
55	Lambda	parameter: 114 body: 64
56	ExprRange	lambda_map: 65 start_index: 123 end_index: 125
57	ExprRange	lambda_map: 66 start_index: 123 end_index: 124
58	ExprRange	lambda_map: 67 start_index: 123 end_index: 125
59	Lambda	parameter: 114 body: 69
60	Variable
61	ExprRange	lambda_map: 70 start_index: 123 end_index: 124
62	ExprRange	lambda_map: 71 start_index: 123 end_index: 125
63	Lambda	parameter: 117 body: 72
64	ExprTuple	77
65	Lambda	parameter: 117 body: 73
66	Lambda	parameter: 117 body: 74
67	Lambda	parameter: 117 body: 75
68	ExprTuple	114
69	ExprTuple	76, 77
70	Lambda	parameter: 117 body: 78
71	Lambda	parameter: 117 body: 79
72	Operation	operator: 87 operands: 80
73	Operation	operator: 87 operands: 81
74	Operation	operator: 87 operands: 82
75	Operation	operator: 87 operands: 83
76	ExprRange	lambda_map: 84 start_index: 123 end_index: 124
77	ExprRange	lambda_map: 85 start_index: 123 end_index: 125
78	Operation	operator: 87 operands: 86
79	Operation	operator: 87 operands: 88
80	NamedExprs	element: 91 targets: 89
81	NamedExprs	element: 97 targets: 89
82	NamedExprs	element: 90 targets: 96
83	NamedExprs	element: 91 targets: 98
84	Lambda	parameter: 117 body: 92
85	Lambda	parameter: 117 body: 94
86	NamedExprs	element: 95 targets: 96
87	Literal
88	NamedExprs	element: 97 targets: 98
89	Operation	operator: 110 operands: 99
90	Operation	operator: 100 operands: 106
91	Operation	operator: 100 operands: 101
92	Operation	operator: 102 operands: 103
93	ExprTuple	117
94	IndexedVar	variable: 104 indices: 105
95	Operation	operator: 108 operands: 106
96	Operation	operator: 110 operands: 107
97	Operation	operator: 108 operands: 109
98	Operation	operator: 110 operands: 111
99	ExprTuple	123, 125
100	Literal
101	NamedExprs	state: 112 part: 117
102	Literal
103	NamedExprs	operation: 113
104	Variable
105	ExprTuple	114, 117
106	NamedExprs	state: 115 part: 117
107	ExprTuple	123, 124
108	Literal
109	NamedExprs	state: 116 part: 117
110	Literal
111	ExprTuple	118, 119
112	Variable
113	Literal
114	Variable
115	Variable
116	Variable
117	Variable
118	Operation	operator: 121 operands: 120
119	Operation	operator: 121 operands: 122
120	ExprTuple	124, 123
121	Literal
122	ExprTuple	124, 125
123	Literal
124	Variable
125	Variable