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 import U_1_to_i, V_1_to_j
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import 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 = Conditional(Forall(instance_param_or_params = [U_1_to_i, V_1_to_j], instance_expr = Implies(Equals(Prob(circuit_psi_m_Vj), one), Equals(Prob(circuit_Ui_Vj), Prob(circuit_Ui_psi_m)).with_wrapping_at(2)).with_wrapping_at(2)).with_wrapping(), InSet(var_ket_psi, m_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\{\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{
\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) = 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{
& \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) \end{array} \end{array}\right) \end{array} \end{array}\right)\end{array} \textrm{ if } \lvert \psi \rangle \in \mathbb{C}^{2^{m}}\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: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	69, 8
7	Lambda	parameters: 9 body: 10
8	Operation	operator: 11 operands: 12
9	ExprTuple	41, 40
10	Operation	operator: 13 operands: 14
11	Literal
12	ExprTuple	15, 16
13	Literal
14	ExprTuple	17, 18
15	Literal
16	Operation	operator: 19 operands: 20
17	Operation	operator: 22 operands: 21
18	Operation	operator: 22 operands: 23
19	Literal
20	ExprTuple	24, 72
21	ExprTuple	25, 71
22	Literal
23	ExprTuple	26, 27
24	Literal
25	Operation	operator: 30 operand: 32
26	Operation	operator: 30 operand: 33
27	Operation	operator: 30 operand: 34
28	ExprTuple	32
29	ExprTuple	33
30	Literal
31	ExprTuple	34
32	Operation	operator: 37 operands: 35
33	Operation	operator: 37 operands: 36
34	Operation	operator: 37 operands: 38
35	ExprTuple	39, 40
36	ExprTuple	41, 40
37	Literal
38	ExprTuple	41, 42
39	ExprTuple	43
40	ExprRange	lambda_map: 44 start_index: 71 end_index: 45
41	ExprRange	lambda_map: 46 start_index: 71 end_index: 47
42	ExprTuple	48
43	ExprRange	lambda_map: 49 start_index: 71 end_index: 72
44	Lambda	parameter: 70 body: 50
45	Variable
46	Lambda	parameter: 70 body: 51
47	Variable
48	ExprRange	lambda_map: 52 start_index: 71 end_index: 72
49	Lambda	parameter: 70 body: 53
50	IndexedVar	variable: 54 index: 70
51	IndexedVar	variable: 55 index: 70
52	Lambda	parameter: 70 body: 57
53	Operation	operator: 59 operands: 58
54	Variable
55	Variable
56	ExprTuple	70
57	Operation	operator: 59 operands: 60
58	NamedExprs	element: 61 targets: 63
59	Literal
60	NamedExprs	element: 62 targets: 63
61	Operation	operator: 64 operands: 66
62	Operation	operator: 65 operands: 66
63	Operation	operator: 67 operands: 68
64	Literal
65	Literal
66	NamedExprs	state: 69 part: 70
67	Literal
68	ExprTuple	71, 72
69	Variable
70	Variable
71	Literal
72	Variable