-
Notifications
You must be signed in to change notification settings - Fork 1
/
Q4L15d.txt
106 lines (105 loc) · 4.33 KB
/
Q4L15d.txt
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
#
# File: content-mit-8371x-subtitles/Q4L15d.txt
#
# Captions for 8.421x module
#
# This file has 96 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
The clean, quantum, bit commitment protocol fails.
We will now see that all quantum bit commitment protocols fail.
The most general possible protocol
involves multiple rounds and not just a single round,
as was the case in the clean protocol.
Alice may perform a measurement, shown here as m j 1.
Bob may perform another measurement shown here
as m j 2.
And what Bob does may depend on classical communication
from Alice, shown here as j 1.
Then Alice may be asked to perform yet another step, shown
here as m j 3.
And that may depend on a message from Bob, shown here as j 2.
And so forth, with multiple rounds going
back and forth between Alice and Bob.
Shown here is the dividing line.
And let me outline in green what Bob's steps are.
The key to analyzing such a multi-round protocol
is to show how a single round, say
which converts between psi and phi,
may be reduced to something simpler.
We use here a Lemma, known as the Lo-Chau reduction.
Suppose the transformation between psi and phi
can be done using local operations
and classical communication.
That is, no quantum communication,
and no entanglement creating steps.
The Lemma then holds that this transformation between psi
and phi can also be performed by a simpler equivalent
circuit, where Alice performs a measurement,
or generalized quantum operation n sub j,
followed by Bob performing a unitary operation u sub j.
This unitary may depend on a message from Alice
and we will show how this transforms between psi and phi.
The key step in showing this Lemma
is to prove that any measurement by Bob
can be simulated by Alice working locally.
We prove this by starting with our friend
the Schmidt decomposition, here written with lambda sub
k, a sub k, and b sub k.
In this basis, let Bob's measurement quantum operators
be m sub j, with the matrix elements m sub j kl.
Note how this operator maps b sub l to b sub k.
Acting on psi, Bob's measurement produces
psi sub j, a superposition involving
the measurement matrix elements, as well as a sub l,
and b sub k.
a sub l appears because of the delta function
that arises from matching b sub l, and b sub k.
The jth measurement result occurs
with probability given by a sum over kl lambda sub l,
and the matrix element m j k l squared.
Suppose that instead of Bob measuring m, Alice measures n.
Let this be n sub j, which looks like m sub j,
but has matrix elements a sub k, a sub l,
instead of b sub k, b sub l.
This operator, measured by Alice locally,
would produce a state psi tilda sub
j, which is a sum over matrix elements and a sub k, b sub l.
The probability of the jth measurement result occurring
would be a sum over kl of matrix elements squared.
And it turns out to be exactly the same as p sub j.
These states, though, are slightly different.
Previously, we had a sub l, but now we have a sub k.
These are different, but it turns out
that psi tilde sub j has the same Schmidt
decomposition as psi sub j.
This is evident by inspection because the two states are
the same under a relabelling of a sub k, and a sub l,
and b sub k, and b sub l.
This remapping is an operation which may be performed locally
by unitaries.
Call these u sub i, and u sub j.
And these unitaries map between psi sub j, and psi tilde sub j.
Let us absorb u into the definition of n prime.
We have thus shown that a measurement of m sub
j by Bob, can be equivalently done by Alice performing n sub
j prime, and then sending Bob a message via which
he performs v sub j.
Repeating this reduction equivalence thus
leads to the general conclusion that a multiple round quantum
bit commitment protocol, say with Alice measuring, Bob
measuring, and back and forth, can always
be reduced to a single round where
Alice does one measurement and Bob does
another unitary transform.
This single round protocol is exactly the clean qubit quantum
bit commitment protocol.
And since we have shown that protocol to be insecure,
then a perfectly secure quantum bit commitment protocol
is thus impossible.
I leave you to dwell on why this is the case
and what ramifications it might have for other quantum
protocols or how this might call for different definitions
for bit commitment.