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, i, n
from proveit.core_expr_types import v_1_to_n, v_i, w_1_to_n, w_i
from proveit.linear_algebra import OrthoNormBases, Unitary, VecSum
from proveit.logic import CartExp, Forall, InSet, Set
from proveit.numbers import Complex, Interval, NaturalPos, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets, w_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = OrthoNormBases(CartExp(Complex, n))
expr = Conditional(Forall(instance_param_or_params = [v_1_to_n, w_1_to_n], instance_expr = InSet(VecSum(index_or_indices = [i], summand = Qmult(Ket(v_i), Bra(w_i)), domain = Interval(one, n)), Unitary(n)), conditions = [InSet(Set(v_1_to_n_kets), sub_expr1), InSet(Set(w_1_to_n_kets), sub_expr1)]).with_wrapping(), InSet(n, NaturalPos))

# 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_{v_{1}, v_{2}, \ldots, v_{n}, w_{1}, w_{2}, \ldots, w_{n}~|~\left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right), \left\{\left\{\lvert w_{1} \rangle, \lvert w_{2} \rangle, \ldots, \lvert w_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right)}~\\
\left(\left(\sum_{i=1}^{n} \left(\lvert v_{i} \rangle \thinspace \langle w_{i} \rvert\right)\right) \in \textrm{U}\left(n\right)\right)\end{array} \textrm{ if } n \in \mathbb{N}^+\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 operand: 6
2	Operation	operator: 45 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	68, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	10, 11
9	Conditional	value: 12 condition: 13
10	ExprRange	lambda_map: 14 start_index: 67 end_index: 68
11	ExprRange	lambda_map: 15 start_index: 67 end_index: 68
12	Operation	operator: 45 operands: 16
13	Operation	operator: 17 operands: 18
14	Lambda	parameter: 81 body: 76
15	Lambda	parameter: 81 body: 77
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22
19	Operation	operator: 23 operand: 29
20	Operation	operator: 25 operand: 68
21	Operation	operator: 45 operands: 27
22	Operation	operator: 45 operands: 28
23	Literal
24	ExprTuple	29
25	Literal
26	ExprTuple	68
27	ExprTuple	30, 32
28	ExprTuple	31, 32
29	Lambda	parameter: 75 body: 33
30	Operation	operator: 48 operand: 40
31	Operation	operator: 48 operand: 41
32	Operation	operator: 36 operand: 42
33	Conditional	value: 38 condition: 39
34	ExprTuple	40
35	ExprTuple	41
36	Literal
37	ExprTuple	42
38	Operation	operator: 43 operands: 44
39	Operation	operator: 45 operands: 46
40	Operation	operator: 48 operands: 47
41	Operation	operator: 48 operands: 49
42	Operation	operator: 50 operands: 51
43	Literal
44	ExprTuple	52, 53
45	Literal
46	ExprTuple	75, 54
47	ExprTuple	55
48	Literal
49	ExprTuple	56
50	Literal
51	ExprTuple	57, 68
52	Operation	operator: 73 operand: 65
53	Operation	operator: 59 operand: 66
54	Operation	operator: 61 operands: 62
55	ExprRange	lambda_map: 63 start_index: 67 end_index: 68
56	ExprRange	lambda_map: 64 start_index: 67 end_index: 68
57	Literal
58	ExprTuple	65
59	Literal
60	ExprTuple	66
61	Literal
62	ExprTuple	67, 68
63	Lambda	parameter: 81 body: 69
64	Lambda	parameter: 81 body: 70
65	IndexedVar	variable: 78 index: 75
66	IndexedVar	variable: 79 index: 75
67	Literal
68	Variable
69	Operation	operator: 73 operand: 76
70	Operation	operator: 73 operand: 77
71	ExprTuple	75
72	ExprTuple	76
73	Literal
74	ExprTuple	77
75	Variable
76	IndexedVar	variable: 78 index: 81
77	IndexedVar	variable: 79 index: 81
78	Variable
79	Variable
80	ExprTuple	81
81	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 ¶