Expression of type Exists ¶

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 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 CartExp, Equals, Exists, 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 = Exists(instance_param_or_params = [v_1_to_n, w_1_to_n], instance_expr = Equals(U, VecSum(index_or_indices = [i], summand = Qmult(Ket(v_i), Bra(w_i)), domain = Interval(one, n))), conditions = [InSet(Set(v_1_to_n_kets), sub_expr1), InSet(Set(w_1_to_n_kets), sub_expr1)]).with_wrapping()

# 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())

\begin{array}{l}\exists_{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(U = \left(\sum_{i=1}^{n} \left(\lvert v_{i} \rangle \thinspace \langle w_{i} \rvert\right)\right)\right)\end{array}

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

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

Expression of type Exists¶

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

Expression of type Exists ¶

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