Consequences and Limits of Nonlocal Strategies

1
Superdense coding
2
How much classical information in n qubits?

• Observe that 2n?1 complex numbers apparently
  needed to describe an arbitrary n-qubit pure
  quantum state
• because ?000?000? ?001?001? ?010?010? ?
  ?111?111?
• 2n is exponential so does this mean that an
  exponential amount of classical information is
  somehow stored in n qubits?
• Not in an operational sense ...
• For example, Holevos Theorem (from 1973)
  implies
• one cannot convey more than n classical bits of
  information in n qubits

3
Holevos Theorem
Easy case
Hard case (the general case)
measurement
b1b2 ... bn certainly cannot convey more than n
bits!
The difficult proof is beyond the scope of this
course
We can use only n classical bits
4
Superdense coding (prelude)
Can we convey two classical bits by sending just
one qubit?
Suppose that Alice wants to convey two classical
bits to Bob sending just one qubit
ab
Alice
Bob
ab
By Holevos Theorem, this is impossible
5
Superdense coding
In superdense coding, Bob is allowed to send a
qubit to Alice first
ab
Alice
Bob
ab
How can this help?
The idea is to use entanglement!
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How superdense coding works?

1. Bob creates the state ?00? ?11? and sends the
   first qubit to Alice

2. Alice wants to send two bits a and b
No change
ab state
00 ?00? ?11?
01 ?00? - ?11?
10 ?01? ?10?
11 ?01? - ?10?
So let us analyze what Alice sends back to Bob?
Alice applies X to first qubit
Bell basis

1. Bob measures the two qubits in the Bell basis

To analyze this communication scheme we need to
use our known methods for quantum circuit analysis
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1. Bob creates the state ?00? ?11? and sends the
   first qubit to Alice

?00? ?11?
X
Bob does measurement In Bell basis
Bob
Alice
Similarly other cases can be calculated
0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
0 1 1 0
1 0 0 1
x

ab state
00 ?00? ?11?
01 ?00? - ?11?
10 ?01? ?10?
11 ?01? - ?10?
0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
1 0 0 1
01 10

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Measurement in the Bell basis
Specifically, Bob applies
input output
?00? ?11? ?00?
?01? ?10? ?01?
?00? - ?11? ?10?
?01? - ?10? ?11?
to his two qubits ...
and then measures them, yielding ab
Homework or exam
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Alice
0 0
X
Z
In this scheme the measurement is only here
Bob Creates EPR entanglement
Bob measures in Bell Base
to his two qubits ...
Results of Bell Base measurement
and then measures them, yielding ab
input output
?00? ?11? ?00?
?01? ?10? ?01?
?00? - ?11? ?10?
?01? - ?10? ?11?
This concludes superdense coding
Observe that Bob knows what Alice has done. This
is used in Quantum Games. Bob knows more than
Alice if he sends her quantum entangled info.
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Incomplete measurements

• von Neumann Measurement associated with a
  partition of the space into mutually orthogonal
  subspaces
• When the measurement is performed, the state
  collapses to each subspace with probability the
  square of the length of its projection on that
  subspace

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Incomplete measurements
Measurements up until now are with respect to
orthogonal one-dimensional subspaces
The orthogonal subspaces can have other
dimensions
(qutrit)
Such a measurement on ?0 ?0? ?1 ?1? ?2 ?2?
(renormalized)
results in ?0?0? ?1?1? with prob ??0?2
??1?2
?2? with prob ??2?2
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Teleportation
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Measuring the first qubit of a two-qubit system
?00?00? ?01?01? ?10?10? ?11?11?

• Measuring this first qubit is defined as the
  incomplete measurement with respect to the two
  dimensional subspaces
• span of ?00? ?01? (all states with first qubit
  0), and
• span of ?10? ?11? (all states with first qubit
  1)

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Easy exercise show that measuring the first
qubit and then measuring the second qubit gives
the same result as measuring both qubits at once
Hint continue calculations from last slide
Homework or exam
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Teleportation (prelude)
Suppose Alice wishes to convey a qubit to Bob by
sending just classical bits
If Alice knows ? and ?, she can send
approximations of them ?but this still requires
infinitely many bits for perfect precision
Moreover, if Alice does not know ? or ?, she can
at best acquire one bit about them by a
measurement
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Teleportation scenario
In teleportation, Alice and Bob also start with a
Bell state
(1/?2)(?00? ?11?)
??0? ??1?
and Alice can send two classical bits to Bob
Note that the initial state of the three qubit
system is (1/?2)(??0? ??1?)(?00? ?11?)
(1/?2)(??000? ??011? ??100? ??111?)
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teleportation circuit
Measurement by Alice in Bell Basis
??0? ??1?
Bob
H
Alice
a
?00? ?11?
b
??0? ??1?
Bob
X
Z
These two qubits are entangled so Alice changing
her private qubit changes the qubit in Bobs
possession
Results of Bell Base measurement
input output
?00? ?11? ?00?
?01? ?10? ?01?
?00? - ?11? ?10?
?01? - ?10? ?11?
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You can multiply the lower formula to get the
upper formula
16 terms after multiplication. Some cancel
This is state of the qubit in Bob possession
which is changed by entanglement with changes in
Alice circuit
These qubits are measured by Alice
Protocol Alice measures her two qubits in the
Bell basis and sends the result to Bob (who then
corrects his state)
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How teleportation works
This is state of the qubit in Bob possession
which is changed by entanglement with changes in
Alice circuit
These are classical bits that Alice sends to Bob
Protocol Alice measures her two qubits in the
Bell basis and sends the result to Bob (who then
corrects his state)
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What Alice does specifically
Alice applies
to her two qubits, yielding
Before measurement
After measurement
This is what Alice sends to Bob
Then Alice sends her two classical bits to Bob,
who then adjusts his qubit to be ??0? ??1?
whatever case occurs
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Bobs adjustment procedure
Bob receives two classical bits a, b from Alice,
and
if b 1 he applies X to qubit if a 1 he
applies Z to qubit
yielding
Note that Bob acquires the correct state in each
case
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Summary of teleportation circuit
This circuit works correctly regardless the
randomness of measurements
measurement
??0? ??1?
H
Alice
a
?00? ?11?
b
Bob
??0? ??1?
X
Z
Homework or exam
Quantum circuit exercise try to work through the
details of the analysis of this teleportation
protocol
23
No-cloning theorem
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Classical information can be copied
a
a
a
0
What about quantum information?
?
25
Candidate
works fine for ??? ?0? and ??? ?1?
... but it fails for ??? (1/?2)(?0? ?1?) ...
... where it yields output (1/?2)(?00? ?11?)
instead of ?????? (1/4)(?00? ?01? ?10?
?11?)
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No-cloning theorem
Theorem there is no valid quantum operation that
maps an arbitrary state ??? to ??????
Proof
Let ??? and ??'? be two input states, yielding
outputs ???????g? and ??'???'??g'? respectively
Since U preserves inner products ????'?
????'?????'??g?g'? so ????'?(1- ????'??g?g'?) 0
so ?????'?? 0 or 1
Homework or exam
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Introduction to Quantum Information
ProcessingCS 467 / CS 667Phys 667 / Phys
767CO 481 / CO 681
Source of slides
Lecture 2 (2005)

• Richard Cleve
• DC 3524
• cleve_at_cs.uwaterloo.ca

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• Used in 2007, easy to explain, students can do
  detailed calculations for circuits to analyze
  them. Good for similar problems.