Expression of type Lambda

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 Lambda
from proveit.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import Equals
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 = Lambda([A11_to_Akl], Equals(Prob(circuit__u_Akl_v), Prob(circuit__psi_m__u_Akl_v)).with_wrapping_at(2))

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(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}\right) \mapsto \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)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameters: 1 body: 2
1	ExprTuple	3
2	Operation	operator: 4 operands: 5
3	ExprRange	lambda_map: 6 start_index: 92 end_index: 29
4	Literal
5	ExprTuple	7, 8
6	Lambda	parameter: 83 body: 46
7	Operation	operator: 10 operand: 12
8	Operation	operator: 10 operand: 13
9	ExprTuple	12
10	Literal
11	ExprTuple	13
12	Operation	operator: 15 operands: 14
13	Operation	operator: 15 operands: 16
14	ExprTuple	17, 18, 19
15	Literal
16	ExprTuple	20, 21, 22
17	ExprTuple	23
18	ExprRange	lambda_map: 24 start_index: 92 end_index: 29
19	ExprTuple	25
20	ExprTuple	26, 27
21	ExprRange	lambda_map: 28 start_index: 92 end_index: 29
22	ExprTuple	30, 31
23	ExprRange	lambda_map: 32 start_index: 92 end_index: 94
24	Lambda	parameter: 83 body: 33
25	ExprRange	lambda_map: 34 start_index: 92 end_index: 94
26	ExprRange	lambda_map: 35 start_index: 92 end_index: 93
27	ExprRange	lambda_map: 36 start_index: 92 end_index: 94
28	Lambda	parameter: 83 body: 38
29	Variable
30	ExprRange	lambda_map: 39 start_index: 92 end_index: 93
31	ExprRange	lambda_map: 40 start_index: 92 end_index: 94
32	Lambda	parameter: 86 body: 41
33	ExprTuple	46
34	Lambda	parameter: 86 body: 42
35	Lambda	parameter: 86 body: 43
36	Lambda	parameter: 86 body: 44
37	ExprTuple	83
38	ExprTuple	45, 46
39	Lambda	parameter: 86 body: 47
40	Lambda	parameter: 86 body: 48
41	Operation	operator: 56 operands: 49
42	Operation	operator: 56 operands: 50
43	Operation	operator: 56 operands: 51
44	Operation	operator: 56 operands: 52
45	ExprRange	lambda_map: 53 start_index: 92 end_index: 93
46	ExprRange	lambda_map: 54 start_index: 92 end_index: 94
47	Operation	operator: 56 operands: 55
48	Operation	operator: 56 operands: 57
49	NamedExprs	element: 60 targets: 58
50	NamedExprs	element: 66 targets: 58
51	NamedExprs	element: 59 targets: 65
52	NamedExprs	element: 60 targets: 67
53	Lambda	parameter: 86 body: 61
54	Lambda	parameter: 86 body: 63
55	NamedExprs	element: 64 targets: 65
56	Literal
57	NamedExprs	element: 66 targets: 67
58	Operation	operator: 79 operands: 68
59	Operation	operator: 69 operands: 75
60	Operation	operator: 69 operands: 70
61	Operation	operator: 71 operands: 72
62	ExprTuple	86
63	IndexedVar	variable: 73 indices: 74
64	Operation	operator: 77 operands: 75
65	Operation	operator: 79 operands: 76
66	Operation	operator: 77 operands: 78
67	Operation	operator: 79 operands: 80
68	ExprTuple	92, 94
69	Literal
70	NamedExprs	state: 81 part: 86
71	Literal
72	NamedExprs	operation: 82
73	Variable
74	ExprTuple	83, 86
75	NamedExprs	state: 84 part: 86
76	ExprTuple	92, 93
77	Literal
78	NamedExprs	state: 85 part: 86
79	Literal
80	ExprTuple	87, 88
81	Variable
82	Literal
83	Variable
84	Variable
85	Variable
86	Variable
87	Operation	operator: 90 operands: 89
88	Operation	operator: 90 operands: 91
89	ExprTuple	93, 92
90	Literal
91	ExprTuple	93, 94
92	Literal
93	Variable
94	Variable