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, i, j, m
from proveit.core_expr_types import U_1_to_i, V_1_to_j
from proveit.logic import And, Equals, Forall, Implies, InSet
from proveit.numbers import NaturalPos, one
from proveit.physics.quantum import m_ket_domain, var_ket_psi
from proveit.physics.quantum.circuits import circuit_Ui_Vj, circuit_Ui_psi_m, circuit_psi_m_Vj
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Lambda([i, j, m], Conditional(Forall(instance_param_or_params = [var_ket_psi], instance_expr = Forall(instance_param_or_params = [U_1_to_i, V_1_to_j], instance_expr = Implies(Equals(Prob(circuit_Ui_psi_m), one), Equals(Prob(circuit_Ui_Vj), Prob(circuit_psi_m_Vj)).with_wrapping_at(2)).with_wrapping_at(2)).with_wrapping(), domain = m_ket_domain).with_wrapping(), And(InSet(i, NaturalPos), InSet(j, 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(i, j, m\right) \mapsto \left\{\begin{array}{l}\forall_{\lvert \psi \rangle \in \mathbb{C}^{2^{m}}}~\\
\left[\begin{array}{l}\forall_{U_{1}, U_{2}, \ldots, U_{i}, V_{1}, V_{2}, \ldots, V_{j}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\U_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{i} \\ \downarrow\end{array}} & \qout{\lvert \psi \rangle}
} \end{array}\right) = 1\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\U_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{i} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{j} \\ \downarrow\end{array}} & \qw
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert \psi \rangle} & \gate{\begin{array}{c} \uparrow \\V_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{j} \\ \downarrow\end{array}} & { /^{m} } \qw
} \end{array}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\end{array}\right]\end{array} \textrm{ if } i \in \mathbb{N}^+ ,  j \in \mathbb{N}^+ ,  m \in \mathbb{N}^+\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	62, 65, 89
2	Conditional	value: 3 condition: 4
3	Operation	operator: 20 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	9, 10, 11
8	Lambda	parameter: 86 body: 13
9	Operation	operator: 22 operands: 14
10	Operation	operator: 22 operands: 15
11	Operation	operator: 22 operands: 16
12	ExprTuple	86
13	Conditional	value: 17 condition: 18
14	ExprTuple	62, 19
15	ExprTuple	65, 19
16	ExprTuple	89, 19
17	Operation	operator: 20 operand: 24
18	Operation	operator: 22 operands: 23
19	Literal
20	Literal
21	ExprTuple	24
22	Literal
23	ExprTuple	86, 25
24	Lambda	parameters: 26 body: 27
25	Operation	operator: 28 operands: 29
26	ExprTuple	57, 59
27	Operation	operator: 30 operands: 31
28	Literal
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34, 35
32	Literal
33	Operation	operator: 36 operands: 37
34	Operation	operator: 39 operands: 38
35	Operation	operator: 39 operands: 40
36	Literal
37	ExprTuple	41, 89
38	ExprTuple	42, 88
39	Literal
40	ExprTuple	43, 44
41	Literal
42	Operation	operator: 47 operand: 49
43	Operation	operator: 47 operand: 50
44	Operation	operator: 47 operand: 51
45	ExprTuple	49
46	ExprTuple	50
47	Literal
48	ExprTuple	51
49	Operation	operator: 54 operands: 52
50	Operation	operator: 54 operands: 53
51	Operation	operator: 54 operands: 55
52	ExprTuple	57, 56
53	ExprTuple	57, 59
54	Literal
55	ExprTuple	58, 59
56	ExprTuple	60
57	ExprRange	lambda_map: 61 start_index: 88 end_index: 62
58	ExprTuple	63
59	ExprRange	lambda_map: 64 start_index: 88 end_index: 65
60	ExprRange	lambda_map: 66 start_index: 88 end_index: 89
61	Lambda	parameter: 87 body: 67
62	Variable
63	ExprRange	lambda_map: 68 start_index: 88 end_index: 89
64	Lambda	parameter: 87 body: 69
65	Variable
66	Lambda	parameter: 87 body: 70
67	IndexedVar	variable: 71 index: 87
68	Lambda	parameter: 87 body: 72
69	IndexedVar	variable: 73 index: 87
70	Operation	operator: 76 operands: 75
71	Variable
72	Operation	operator: 76 operands: 77
73	Variable
74	ExprTuple	87
75	NamedExprs	element: 78 targets: 80
76	Literal
77	NamedExprs	element: 79 targets: 80
78	Operation	operator: 81 operands: 83
79	Operation	operator: 82 operands: 83
80	Operation	operator: 84 operands: 85
81	Literal
82	Literal
83	NamedExprs	state: 86 part: 87
84	Literal
85	ExprTuple	88, 89
86	Variable
87	Variable
88	Literal
89	Variable