Expression of type Conditional ¶

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 Conditional, 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, LinMap, 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 = Conditional(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), InSet(M, LinMap(Hspace, Hspace)))

# 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}{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} \textrm{ if } M \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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

Expression of type Conditional¶

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

Expression of type Conditional ¶

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