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 Conditional, Lambda, m, n
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Abs, Exp, two
from proveit.physics.quantum import NumBra, Qmult, m_bit_interval, m_ket_domain, normalized_var_ket_psi, var_ket_psi
from proveit.physics.quantum.circuits import register_meas
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [var_ket_psi], instance_expr = Equals(Prob(register_meas), Exp(Abs(Qmult(NumBra(n, m), var_ket_psi)), two)), domain = m_ket_domain, condition = normalized_var_ket_psi), InSet(n, m_bit_interval)))

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())

n \mapsto \left\{\forall_{\lvert \psi \rangle \in \mathbb{C}^{2^{m}}~|~\left \|\lvert \psi \rangle\right \| = 1}~\left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert \psi \rangle} & \meter & \multiqout{3}{\lvert n \rangle_{m}} \\
\ghostqin{\lvert \psi \rangle} & \meter & \ghostqout{\lvert n \rangle_{m}} \\
\ghostqin{\lvert \psi \rangle} & \measure{\begin{array}{c}:\\ \left(m - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert n \rangle_{m}} \\
\ghostqin{\lvert \psi \rangle} & \meter & \ghostqout{\lvert n \rangle_{m}}
} \end{array}\right) = \left|{_{m}}\langle n \rvert \thinspace \lvert \psi \rangle\right|^{2}\right) \textrm{ if } n \in \{0~\ldotp \ldotp~2^{m} - 1\}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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