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, k, l, m
from proveit.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import NaturalPos
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__u_Akl_v, circuit__u_Akl_v__psi_m
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [var_ket_u, var_ket_v], instance_expr = 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__u_Akl_v__psi_m)).with_wrapping_at(2)).with_wrapping(), domain = m_ket_domain), domain = l_ket_domain), And(InSet(k, NaturalPos), InSet(l, NaturalPos), InSet(m, NaturalPos)))

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 u \rangle, \lvert v \rangle \in \mathbb{C}^{2^{l}}}~\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{
\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} \\
\qin{\lvert \psi \rangle} & { /^{m} } \qw & { /^{m} } \qw & \gate{\cdots} & { /^{m} } \qw & \qout{\lvert \psi \rangle}
} \end{array}\right) \end{array} \end{array}\right)\end{array}\right]\right] \textrm{ if } k \in \mathbb{N}^+ ,  l \in \mathbb{N}^+ ,  m \in \mathbb{N}^+\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: 30 operand: 5
2	Operation	operator: 18 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7, 8
5	Lambda	parameters: 9 body: 10
6	Operation	operator: 32 operands: 11
7	Operation	operator: 32 operands: 12
8	Operation	operator: 32 operands: 13
9	ExprTuple	116, 119
10	Conditional	value: 14 condition: 15
11	ExprTuple	72, 16
12	ExprTuple	128, 16
13	ExprTuple	129, 16
14	Operation	operator: 30 operand: 20
15	Operation	operator: 18 operands: 19
16	Literal
17	ExprTuple	20
18	Literal
19	ExprTuple	21, 22
20	Lambda	parameter: 120 body: 24
21	Operation	operator: 32 operands: 25
22	Operation	operator: 32 operands: 26
23	ExprTuple	120
24	Conditional	value: 27 condition: 28
25	ExprTuple	116, 29
26	ExprTuple	119, 29
27	Operation	operator: 30 operand: 35
28	Operation	operator: 32 operands: 33
29	Operation	operator: 40 operands: 34
30	Literal
31	ExprTuple	35
32	Literal
33	ExprTuple	120, 36
34	ExprTuple	46, 37
35	Lambda	parameters: 38 body: 39
36	Operation	operator: 40 operands: 41
37	Operation	operator: 51 operands: 42
38	ExprTuple	43
39	Operation	operator: 44 operands: 45
40	Literal
41	ExprTuple	46, 47
42	ExprTuple	56, 128
43	ExprRange	lambda_map: 48 start_index: 127 end_index: 72
44	Literal
45	ExprTuple	49, 50
46	Literal
47	Operation	operator: 51 operands: 52
48	Lambda	parameter: 117 body: 84
49	Operation	operator: 54 operand: 57
50	Operation	operator: 54 operand: 58
51	Literal
52	ExprTuple	56, 129
53	ExprTuple	57
54	Literal
55	ExprTuple	58
56	Literal
57	Operation	operator: 60 operands: 59
58	Operation	operator: 60 operands: 61
59	ExprTuple	62, 63, 64
60	Literal
61	ExprTuple	65, 66, 67
62	ExprTuple	69
63	ExprRange	lambda_map: 68 start_index: 127 end_index: 72
64	ExprTuple	73
65	ExprTuple	69, 70
66	ExprRange	lambda_map: 71 start_index: 127 end_index: 72
67	ExprTuple	73, 74
68	Lambda	parameter: 117 body: 75
69	ExprRange	lambda_map: 76 start_index: 127 end_index: 128
70	ExprRange	lambda_map: 77 start_index: 127 end_index: 129
71	Lambda	parameter: 117 body: 79
72	Variable
73	ExprRange	lambda_map: 80 start_index: 127 end_index: 128
74	ExprRange	lambda_map: 81 start_index: 127 end_index: 129
75	ExprTuple	84
76	Lambda	parameter: 121 body: 82
77	Lambda	parameter: 121 body: 83
78	ExprTuple	117
79	ExprTuple	84, 85
80	Lambda	parameter: 121 body: 86
81	Lambda	parameter: 121 body: 87
82	Operation	operator: 93 operands: 88
83	Operation	operator: 93 operands: 89
84	ExprRange	lambda_map: 90 start_index: 127 end_index: 128
85	ExprRange	lambda_map: 91 start_index: 127 end_index: 129
86	Operation	operator: 93 operands: 92
87	Operation	operator: 93 operands: 94
88	NamedExprs	element: 95 targets: 101
89	NamedExprs	element: 96 targets: 103
90	Lambda	parameter: 121 body: 97
91	Lambda	parameter: 121 body: 99
92	NamedExprs	element: 100 targets: 101
93	Literal
94	NamedExprs	element: 102 targets: 103
95	Operation	operator: 105 operands: 104
96	Operation	operator: 105 operands: 113
97	IndexedVar	variable: 106 indices: 107
98	ExprTuple	121
99	Operation	operator: 108 operands: 109
100	Operation	operator: 112 operands: 110
101	Operation	operator: 114 operands: 111
102	Operation	operator: 112 operands: 113
103	Operation	operator: 114 operands: 115
104	NamedExprs	state: 116 part: 121
105	Literal
106	Variable
107	ExprTuple	117, 121
108	Literal
109	NamedExprs	operation: 118
110	NamedExprs	state: 119 part: 121
111	ExprTuple	127, 128
112	Literal
113	NamedExprs	state: 120 part: 121
114	Literal
115	ExprTuple	122, 123
116	Variable
117	Variable
118	Literal
119	Variable
120	Variable
121	Variable
122	Operation	operator: 125 operands: 124
123	Operation	operator: 125 operands: 126
124	ExprTuple	128, 127
125	Literal
126	ExprTuple	128, 129
127	Literal
128	Variable
129	Variable