Expression of type Iff ¶

from the theory of proveit.physics.quantum.algebra ¶

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 M, i, n
from proveit.core_expr_types import a_1_to_n, a_i, v_1_to_n, v_i
from proveit.linear_algebra import Adj, Hspace, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import Equals, Exists, Iff, InSet, Set
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = Adj(M)
expr = Iff(Equals(Qmult(sub_expr1, M), Qmult(M, sub_expr1)), Exists(instance_param_or_params = [v_1_to_n], instance_expr = Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(M, VecSum(index_or_indices = [i], summand = ScalarMult(a_i, Qmult(Ket(v_i), Bra(v_i))), domain = Interval(one, n))), domain = Complex).with_wrapping(), condition = InSet(Set(v_1_to_n_kets), OrthoNormBases(Hspace))).with_wrapping()).with_wrapping_at(2)

# 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())

\begin{array}{c} \begin{array}{l} \left(\left(M^{\dagger} \thinspace M\right) = \left(M \thinspace M^{\dagger}\right)\right) \Leftrightarrow  \\ \left[\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}~|~\left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\right)}~\\
\left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}}~\\
\left(M = \left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\end{array}\right]\end{array}\right] \end{array} \end{array}

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

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

Expression of type Iff¶

from the theory of proveit.physics.quantum.algebra¶

Expression of type Iff ¶

from the theory of proveit.physics.quantum.algebra ¶