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, j, n
from proveit.core_expr_types import v_1_to_n, v_i, v_j
from proveit.linear_algebra import Adj, OrthoNormBases, Unitary
from proveit.logic import CartExp, Equals, Forall, Implies, InSet, Set
from proveit.numbers import Complex, Interval, KroneckerDelta, 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
expr = Lambda([v_1_to_n], Conditional(Implies(Forall(instance_param_or_params = [i, j], instance_expr = Equals(Qmult(Bra(v_i), Adj(U), U, Ket(v_j)), KroneckerDelta(i, j)), domain = Interval(one, n)), InSet(U, Unitary(n))), InSet(Set(v_1_to_n_kets), OrthoNormBases(CartExp(Complex, n)))))

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