Expression of type Lambda

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, Lambda, U, 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, VecSum
from proveit.logic import And, CartExp, Equals, 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, w_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = OrthoNormBases(CartExp(Complex, n))
expr = Lambda([v_1_to_n, w_1_to_n], Conditional(Equals(U, VecSum(index_or_indices = [i], summand = Qmult(Ket(v_i), Bra(w_i)), domain = Interval(one, n))), And(InSet(Set(v_1_to_n_kets), sub_expr1), InSet(Set(w_1_to_n_kets), sub_expr1))))

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