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, U, i, j, n
from proveit.core_expr_types import 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 = 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))))

# 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\{\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()

# 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: 46 operands: 5
3	Literal
4	ExprTuple	6, 7
5	ExprTuple	8, 9
6	Operation	operator: 10 operand: 16
7	Operation	operator: 46 operands: 12
8	Operation	operator: 23 operand: 18
9	Operation	operator: 14 operand: 19
10	Literal
11	ExprTuple	16
12	ExprTuple	64, 17
13	ExprTuple	18
14	Literal
15	ExprTuple	19
16	Lambda	parameters: 44 body: 20
17	Operation	operator: 21 operand: 67
18	Operation	operator: 23 operands: 24
19	Operation	operator: 25 operands: 26
20	Conditional	value: 27 condition: 28
21	Literal
22	ExprTuple	67
23	Literal
24	ExprTuple	29
25	Literal
26	ExprTuple	30, 67
27	Operation	operator: 31 operands: 32
28	Operation	operator: 33 operands: 34
29	ExprRange	lambda_map: 35 start_index: 66 end_index: 67
30	Literal
31	Literal
32	ExprTuple	36, 37
33	Literal
34	ExprTuple	38, 39
35	Lambda	parameter: 68 body: 40
36	Operation	operator: 41 operands: 42
37	Operation	operator: 43 operands: 44
38	Operation	operator: 46 operands: 45
39	Operation	operator: 46 operands: 47
40	Operation	operator: 58 operand: 53
41	Literal
42	ExprTuple	49, 50, 64, 51
43	Literal
44	ExprTuple	72, 73
45	ExprTuple	72, 52
46	Literal
47	ExprTuple	73, 52
48	ExprTuple	53
49	Operation	operator: 54 operand: 63
50	Operation	operator: 56 operand: 64
51	Operation	operator: 58 operand: 65
52	Operation	operator: 60 operands: 61
53	IndexedVar	variable: 70 index: 68
54	Literal
55	ExprTuple	63
56	Literal
57	ExprTuple	64
58	Literal
59	ExprTuple	65
60	Literal
61	ExprTuple	66, 67
62	ExprTuple	68
63	IndexedVar	variable: 70 index: 72
64	Variable
65	IndexedVar	variable: 70 index: 73
66	Literal
67	Variable
68	Variable
69	ExprTuple	72
70	Variable
71	ExprTuple	73
72	Variable
73	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 ¶