Download Entanglement and Quantum Teleportation

Entanglement and
Quantum Teleportation
Stephen Bartlett
Centre for Advanced Computing – Algorithms and Cryptography
Australian Centre of Excellence in Quantum Computer Technology
Macquarie University, Sydney, Australia
Lecture 2 on Quantum Computing
NITP Summer School 2003
Adelaide, Australia
28-31 January 2003
Outline - Entanglement
What is entanglement?
Coupled quantum systems
Classical and quantum correlations
Using entanglement
Superdense coding
Quantum teleportation
Entanglement as a resource
Creating entanglement
Quantum teleportation in the lab!
Entanglement and Teleportation - NITP 2003
2
An abstract quantum system
We describe it using:
a Hilbert space
basis of states
, with dimension
State of the system could be a superposition
Projective measurements in this basis
give result with prob.
Example: a qubit (a two-level system)
basis
e.g., spin ½ particle, photon with pol.
Entanglement and Teleportation - NITP 2003
3
Composite quantum systems
A+B
A
B
How do we describe two quantum systems together?
A: Hilbert space
dimension
B: Hilbert space
dimension
A+B: Hilbert space
basis:
dimension:
Product,
not sum
Let’s look at the form of states for A+B
Entanglement and Teleportation - NITP 2003
4
Product and entangled states
Product state: has the form
System A is in the state
regardless of B
Measurements on A and B will be uncorrelated
Entanglement: take a superposition of product states,
e.g.,
Leads to correlated measurements between A and B
Entanglement and Teleportation - NITP 2003
5
Example: coupled qubits
Let A and B be two-level (qubit) systems
Product state: e.g.,
Basis for
:
Many qubits: e.g.,
Basis for
:
“Computational
basis”
Use binary notation to label product state basis
Entanglement and Teleportation - NITP 2003
6
Example: the Bell states
Entangled state: e.g., for two qubits
Bell states: a basis of entangled states for two qubits
Check: they are orthogonal and cannot be expressed
as product states for A and B
Entanglement and Teleportation - NITP 2003
7
Different bases
Consider the Bell state
What if we changed bases for each qubit?
Rewrite the Bell state:
Entangled in any basis
Entanglement and Teleportation - NITP 2003
8
Using entanglement
Take an entangled system (e.g., in a Bell state)
Give system A to Alice and B to Bob
Bell state
Bob
Alice
Alice and Bob can:
transform their systems (quantum evolution)
perform measurements on their systems
Entanglement and Teleportation - NITP 2003
9
Measurements by Alice or Bob
What happens if Alice (or Bob) performs
projective measurements on their system?
Random result in any basis
Basis:
Random
results
Basis:
1
0
1
+
2
0
2
–
3
1
3
+
4
0
4
+
5
1
5
–
6
1
6
+
7
0
7
–
8
1
8
–
9
0
9
+
Measurements
on an ensemble
of the same Bell
state
Entanglement and Teleportation - NITP 2003
Random
results
10
Measurements by Alice or Bob
What happens if Alice and Bob both perform
projective measurements and compare?
Correlated results if in the same basis
Basis:
Different bases
A B
Basis:
A B
A B
1
0
1
1
1
+
1
+ +
2
0
1
2
0
+
2
–
–
3
1
0
3
1
–
3
–
–
4
0
1
4
1
–
4
+ +
5
1
0
5
0
–
5
+ +
Entanglement and Teleportation - NITP 2003
11
Entangled states can generate
classical correlations
Classical correlations can be useful for secret
communication
Example: private key cryptography (one-time pad)
Alice has a message m (00100) to send Bob
Alice and Bob have a channel
A
00100
B
If Alice and Bob share a private key of random
numbers (11010):
00100
+11010
A
11110
No transmitted information!
11110
B
11110
-11010
00100
Entanglement and Teleportation - NITP 2003
12
Power of quantum correlations
Quantum correlations (from entangled states) can be
useful for communication
Quantum correlations can lead to classical correlations
(one-time pads) which are powerful
Without converting to classical correlations, the
entangled states have even more power:
Tests of local realism (Bell)
Superdense coding
Quantum teleportation
Entanglement swapping
Quantum cryptography
Quantum computing (?)
Other applications ???
(field is still growing)
Entanglement and Teleportation - NITP 2003
13
Classical communication on a
quantum channel
Alice and Bob share a quantum channel
Quantum channel sends qubits instead of bits
Alice wants to send classical messages to Bob
They agree on a basis, say |0 , |1
Alice wants to communicate a bit b, so sends a qubit |b
Bob measures in the agreed basis
gets the result b with certainty
Entanglement and Teleportation - NITP 2003
14
Classical communication on a
quantum channel
Can we do better?
Qubits seem to store two complex numbers:
Number of distinguishable states is limited to the
dimension of the Hilbert space (for a qubit, it's 2)
Only one bit of information can be measured from a qubit
One qubit must be sent for every bit, right? NO!
Quantum (non-classical) correlations can be used to
send two bits with every qubit
superdense coding
Entanglement and Teleportation - NITP 2003
15
Superdense coding
Let Alice and Bob share a Bell state
Bell state
Alice wants to send two bits b1 and b2 to Bob
Alice performs a unitary operation on her qubit
Check:
X and Z
are
unitary
operators
If b1=1, then “flip” the qubit:
If b2=1, then change the
relative phase by π:
Entanglement and Teleportation - NITP 2003
16
Superdense coding
Result of Alice’s operations:
bits
apply
bits
result
00
I
00
|Ψ +
01
X
01
|Ψ –
10
Z
10
|Φ +
11
XZ
11
|Φ –
Resulting effect on
total state of both
parties
Alice then sends her qubit to Bob
Bob performs a measurement in the Bell basis:
with both qubits, Bob can perform a 4-outcome
measurement and obtain two bits of information
Entanglement and Teleportation - NITP 2003
17
Results from superdense coding
Superdense coding transfers two bits of info per qubit
b1,b2
b1,b2
The qubit transferred from Alice to Bob is half of one of
the four Bell states:
Contains no information on its own
All the information is in the quantum correlations
This coding has the properties of the classical one-time
pad, plus the remarkable advantage of sending two
classical bits with every qubit!
Entanglement and Teleportation - NITP 2003
18
Interpreting superdense coding
Alice has managed to communicate two bits of
information to Bob by sending only one qubit, provided
they shared a Bell state to start
To create and share a Bell state, they must have (at
some point) transmitted a qubit, although this
transmission could be in either direction
The important point: the act of sharing the quantum
correlation (Bell state) could be long prior to the protocol,
and does not involve the transmission of information
All the information about the two bits is transmitted with a
single qubit... yet somehow this qubit doesn't contain any
information either!
Quantum correlations (entanglement) are a resource
Entanglement and Teleportation - NITP 2003
19
Sending quantum information
Say Alice wants to send Bob a qubit
(i.e., quantum information rather than classical)
001001011101101
Quantum channels are hard to make and maintain!
Can Alice send the qubit over a classical channel
(i.e., the telephone)?
Option 1: -measure the qubit
-send the measurement results to Bob
Entanglement and Teleportation - NITP 2003
20
Sending quantum information
If Alice has complete information about the qubit:
Alice tells Bob all of this information
Bob performs a preparation to create this state
If Alice has NO information about the qubit:
for instance, the qubit is prepared by a third party
Could perform a measurement, e.g., in basis
If the qubit were in
, no information is
gained and the qubit is “destroyed” in the process
Without knowledge of the preparation procedure of
a qubit, no measurement can determine its state
Entanglement and Teleportation - NITP 2003
21
Quantum teleportation
Again, entanglement provides a solution!
Let Alice and Bob share a Bell state
1
3
2
Bell state
Alice takes the qubit to send (1) and the qubit from the
Bell state (2) and measures them in the Bell basis
One of four possible outcomes
two bits of information
Send these bits to Bob, who operates on his qubit (3)
Entanglement and Teleportation - NITP 2003
22
Quantum teleportation
Result of Alice’s measurements:
result
bits
|Ψ +
00
|Ψ –
01
|Φ +
10
|Φ –
11
bits
apply
Send bits to Bob,
who must apply
00
I
01
X
b1,b2
10
Z
11
XZ
Looks like the opposite of superdense coding!
Result: any measurement predictions involving the
original qubit (1) now apply to Bob’s qubit (3)
The qubit has been quantum teleported to Bob
Entanglement and Teleportation - NITP 2003
23
Interpreting quantum teleportation
The quantum system has not been teleported, only the
state of the system
The two bits contain no information about the qubit
If qubit (1) was entangled with another system before
quantum teleportation, qubit (3) is entangled after
After teleportation, qubit (1) contains no information
Entanglement and Teleportation - NITP 2003
24
Quantum teleportation: reality
Quantum teleportation has been performed in the lab!
1997: Innsbrook, Austria
Qubit: polarization state of a
single photon
Bell state: generated through
parametric down conversion
1998: Caltech, USA
“Qubit”: coherent state of
electromagnetic field mode
“Bell state”: generated through
two-mode squeezing
Entanglement and Teleportation - NITP 2003
25
Quantum teleportation in Oz
2002: Ping-Koy Lam’s group at ANU
Similar to
Caltech exp.
“Hi-Fi” QT
Demonstrates:
Entanglement
was used
Alice gains no
info about the
system
Entanglement and Teleportation - NITP 2003
26
Summary
Entanglement is a resource
Quantum correlations (from entangled states) can be
useful for communication
Tests of local realism (Bell)
Superdense coding
Bell state
Quantum teleportation
Entanglement swapping
Quantum cryptography
Quantum computing (?)
Other applications ??? (field is still growing)
Next lecture: quantum algorithms...
Entanglement and Teleportation - NITP 2003
27