Entanglement-enhanced classical communication without a shared frame of reference 1st CPTPN conference, Lethbridge, 25-26 August 2010

1. Entanglement-enhanced classical communication
without a shared frame of reference
M. Skotiniotis1 A. Roy1,2 G. Gour1,3 B. C. Sanders1
1 Institute for Quantum Information Science
University of Calgary
2 School
of Mathematical Sciences
Queen Mary, University of London
3 Department of Mathematics and Statistics
University of Calgary
CPTPN, Lethbridge
August 25-26, 2010

2. O UTLINE
M OTIVATION
BACKGROUND
R ESULTS
S UMMARY

3. R EFERENCE F RAMES AND I NFORMATION
Alice and Bob have a conversation but they do not speak a
common language
Alice sends binary signals to Bob who doesn’t know what
pulse is 0 and what is 1
Alice sends a series of gyroscopes at angles θ from her
z-axis but Bob doesn’t know the direction of Alice’s z-axis

4. R EFERENCE F RAMES AND I NFORMATION
Alice and Bob have a conversation but they do not speak a
common language
Alice sends binary signals to Bob who doesn’t know what
pulse is 0 and what is 1
Alice sends a series of gyroscopes at angles θ from her
z-axis but Bob doesn’t know the direction of Alice’s z-axis
P ROBLEM
How can Alice and Bob communicate speakable information
without prior shared frame of reference

5. I NFORMATION AND N OISE
Alice and Bob share a common language but a noisy
channel
Alice and Bob agree on what voltage pulse denotes 0 and
1, but share a binary symmetric channel
Alice and Bob agree on the z-direction but share a channel
whose action is described by {pθ , T(θ)|θ ∈ [0, 2π]}

6. I NFORMATION AND N OISE
Alice and Bob share a common language but a noisy
channel
Alice and Bob agree on what voltage pulse denotes 0 and
1, but share a binary symmetric channel
Alice and Bob agree on the z-direction but share a channel
whose action is described by {pθ , T(θ)|θ ∈ [0, 2π]}
N OISY C HANNELS
Can think of the lack of a shared reference frame as a kind of
noisy channel. How do Alice and Bob communicate through
such a noisy channel?

7. P OSSIBLE S OLUTIONS
A LIGNMENT OF R EFERENCE F RAMES
Alice and Bob can learn a common language
Alice can send the same voltage pulse several times
Alice can send several gyroscopes pointing in her
z-direction

8. P OSSIBLE S OLUTIONS
A LIGNMENT OF R EFERENCE F RAMES
Alice and Bob can learn a common language
Alice can send the same voltage pulse several times
Alice can send several gyroscopes pointing in her
z-direction
O UR APPROACH
Alignment suffices when g ∈ G is static. What if g ∈ G
changes with time?

9. P OSSIBLE S OLUTIONS
A LIGNMENT OF R EFERENCE F RAMES
Alice and Bob can learn a common language
Alice can send the same voltage pulse several times
Alice can send several gyroscopes pointing in her
z-direction
O UR APPROACH
Alignment suffices when g ∈ G is static. What if g ∈ G
changes with time?
Assuming that all states undergo the same T(g), Alice can
prepare two systems.

10. P OSSIBLE S OLUTIONS
A LIGNMENT OF R EFERENCE F RAMES
Alice and Bob can learn a common language
Alice can send the same voltage pulse several times
Alice can send several gyroscopes pointing in her
z-direction
O UR APPROACH
Alignment suffices when g ∈ G is static. What if g ∈ G
changes with time?
Assuming that all states undergo the same T(g), Alice can
prepare two systems.
Message is the relative parameter transforming first
system into the second

11. P OSSIBLE S OLUTIONS
A LIGNMENT OF R EFERENCE F RAMES
Alice and Bob can learn a common language
Alice can send the same voltage pulse several times
Alice can send several gyroscopes pointing in her
z-direction
O UR APPROACH
Alignment suffices when g ∈ G is static. What if g ∈ G
changes with time?
Assuming that all states undergo the same T(g), Alice can
prepare two systems.
Message is the relative parameter transforming first
system into the second
Alice sends two gyroscopes with relative angle φ to Bob.

12. O UTLINE
M OTIVATION
BACKGROUND
R ESULTS
S UMMARY

13. R EFERENCE F RAMES AND S YMMETRY G ROUPS
Alice Bob
Alice and Bob lack a shared Cartesian reference frame

14. R EFERENCE F RAMES AND S YMMETRY G ROUPS
Alice Bob
Alice and Bob lack a shared Cartesian reference frame
Alice and Bob’s reference frames are related by some
g ∈ SO(3)

15. R EFERENCE F RAMES AND S YMMETRY G ROUPS
Alice Bob
Alice and Bob lack a shared Cartesian reference frame
Alice and Bob’s reference frames are related by some
g ∈ SO(3)
Tg = Rz (α)Rx (β)Rz (γ) is a representation of the group of
rotations of a Cartesian frame

16. R EFERENCE F RAMES AND S YMMETRY G ROUPS
Alice Bob
Alice and Bob lack a shared Cartesian reference frame
Alice and Bob’s reference frames are related by some
g ∈ SO(3)
Tg = Rz (α)Rx (β)Rz (γ) is a representation of the group of
rotations of a Cartesian frame
Alice and Bob’s reference frames are related by some
symmetry group G. The action of G is represented by a set of
matrices {T(g)| T : G → SO(3)}

17. R EFERENCE F RAMES AND N OISE
B(H ) = B(HA ⊗ HB )
B(HA ) B(HB )
g∈G
Alice Bob
Alice prepares a system in a state ρA ∈ B(HA )

18. R EFERENCE F RAMES AND N OISE
B(H ) = B(HA ⊗ HB )
B(HA ) B(HB )
g∈G
Alice Bob
Alice prepares a system in a state ρA ∈ B(HA )
Bob receives the state σB = T(g)ρA T(g)† ∈ B(HB )

19. R EFERENCE F RAMES AND N OISE
B(H ) = B(HA ⊗ HB )
B(HA ) B(HB )
g∈G
Alice Bob
Alice prepares a system in a state ρA ∈ B(HA )
Bob receives the state σB = T(g)ρA T(g)† ∈ B(HB )
Define Φg : B(H ) → B(H ); Φg [ρA ⊗ |B B|] → |A A| ⊗ σB

20. R EFERENCE F RAMES AND N OISE
B(H ) = B(HA ⊗ HB )
B(HA ) B(HB )
g∈G
Alice Bob
Alice prepares a system in a state ρA ∈ B(HA )
Bob receives the state σB = T(g)ρA T(g)† ∈ B(HB )
Define Φg : B(H ) → B(H ); Φg [ρA ⊗ |B B|] → |A A| ⊗ σB
The lack of a reference frame can be perceived as sharing a
communication channel Φ whose action is described by
operations {T(g); g ∈ G}

21. R ANDOM U NITARY C HANNEL
B(H ) = B(HA ⊗ HB )
B(HA ) B(HB )
{pg , g ∈ G}
Alice Bob
Alice prepares a system in a state ρA ∈ B(HA )

22. R ANDOM U NITARY C HANNEL
B(H ) = B(HA ⊗ HB )
B(HA ) B(HB )
{pg , g ∈ G}
Alice Bob
Alice prepares a system in a state ρA ∈ B(HA )
Bob receives the state σB = pg T(g)ρA T(g)† ∈ B(HB )
g∈G

23. R ANDOM U NITARY C HANNEL
B(H ) = B(HA ⊗ HB )
B(HA ) B(HB )
{pg , g ∈ G}
Alice Bob
Alice prepares a system in a state ρA ∈ B(HA )
Bob receives the state σB = pg T(g)ρA T(g)† ∈ B(HB )
g∈G
If Alice and Bob have no knowledge as to which g ∈ G
relates their reference frames
1/|G|, if G is finite
pg =
dg, if G is compact Lie Group

24. R ANDOM U NITARY C HANNEL
B(H ) = B(HA ⊗ HB )
B(HA ) B(HB )
{pg , g ∈ G}
Alice Bob
Alice prepares a system in a state ρA ∈ B(HA )
Bob receives the state σB = pg T(g)ρA T(g)† ∈ B(HB )
g∈G
If Alice and Bob have no knowledge as to which g ∈ G
relates their reference frames
1/|G|, if G is finite
pg =
dg, if G is compact Lie Group
R ANDOM U NITARY C HANNEL
Φ[ρA ⊗ |B B|] = |A A| ⊗ 1
|G| T(g)ρA T(g)†
g∈G

25. P REPARE - AND -M EASURE
Φ : B(H ) → B(H )
A COMMUNICATION CHANNEL B
(k) (k)† (k)
ρA (h) ⊗ |B B| ∈ B(H ) {|A A| ⊗ MB ; MB MB = IB }
k
Alice prepares message ρA (h) with probability ph

26. P REPARE - AND -M EASURE
Φ : B(H ) → B(H )
A COMMUNICATION CHANNEL B
(k) (k)† (k)
ρA (h) ⊗ |B B| ∈ B(H ) {|A A| ⊗ MB ; MB MB = IB }
k
Alice prepares message ρA (h) with probability ph
Bob receives message Φ[ρA (h) ⊗ |B B|]

27. P REPARE - AND -M EASURE
Φ : B(H ) → B(H )
A COMMUNICATION CHANNEL B
(k) (k)† (k)
ρA (h) ⊗ |B B| ∈ B(H ) {|A A| ⊗ MB ; MB MB = IB }
k
Alice prepares message ρA (h) with probability ph
Bob receives message Φ[ρA (h) ⊗ |B B|]
(k)
Bob performs a measurement {|A A| ⊗ MB } to retrieve
the message.

28. P REPARE - AND -M EASURE
Φ : B(H ) → B(H )
A COMMUNICATION CHANNEL B
(k) (k)† (k)
ρA (h) ⊗ |B B| ∈ B(H ) {|A A| ⊗ MB ; MB MB = IB }
k
Alice prepares message ρA (h) with probability ph
Bob receives message Φ[ρA (h) ⊗ |B B|]
(k)
Bob performs a measurement {|A A| ⊗ MB } to retrieve
the message.
(h)
p(h|h) = Tr Φ (ρA (h) ⊗ |B B|) |A A| ⊗ MB

29. P REPARE - AND -M EASURE
Φ : B(H ) → B(H )
A COMMUNICATION CHANNEL B
(k) (k)† (k)
ρA (h) ⊗ |B B| ∈ B(H ) {|A A| ⊗ MB ; MB MB = IB }
k
Alice prepares message ρA (h) with probability ph
Bob receives message Φ[ρA (h) ⊗ |B B|]
(k)
Bob performs a measurement {|A A| ⊗ MB } to retrieve
the message.
(h)
p(h|h) = Tr Φ (ρA (h) ⊗ |B B|) |A A| ⊗ MB
S UCCESS C RITERION
A prepare-and-measure procedure is succesful if it maximizes
¯
∆= ph p(h|h)
h

30. C OMMUNICATION P ROTOCOL
Φ : B(HA⊗2 ⊗ HB⊗2 ) → B(HA⊗2 ⊗ HB⊗2 )
COMMUNICATION CHANNEL B
(k) (k)† (k)
ρA (h) ⊗ |B B| {|A A| ⊗ MB ; MB MB = IB }
k
Alice prepares two systems in state
ρA (h) = I ⊗ U(h)[ρA ]I ⊗ U(h)† , U : H → SU(HA )
Going through the channel both systems experience the
same transformation
1
Φ[ρA (h) ⊗ |B B|] = |A A| ⊗ T(g)⊗2 ρA (h)T(g)⊗2†
|G|
g∈G
Bob performs joint measurements in order to read the
message.

31. C OMMUNICATION P ROTOCOL
Φ : B(HA⊗2 ⊗ HB⊗2 ) → B(HA⊗2 ⊗ HB⊗2 )
COMMUNICATION CHANNEL B
(k) (k)† (k)
ρA (h) ⊗ |B B| {|A A| ⊗ MB ; MB MB = IB }
k
1
Φ[ρA (h) ⊗ |B B|] = |A A| ⊗ T(g)⊗2 ρA (h)T(g)⊗2†
|G|
g∈G
Using Schur’s lemmas
Φ[ρA (h) ⊗ |B B|] = |A A| ⊗ (DMq ⊗ INq ) ◦ Pq [ρA (h)]
q

32. C OMMUNICATION P ROTOCOL
Φ : B(HA⊗2 ⊗ HB⊗2 ) → B(HA⊗2 ⊗ HB⊗2 )
COMMUNICATION CHANNEL B
(k) (k)† (k)
ρA (h) ⊗ |B B| {|A A| ⊗ MB ; MB MB = IB }
k
(k)
Find ρA and {MB } that maximize
¯ 1
∆= p(k|h) f (U(h), U(k))
|H|
h,k∈H
1. f (U(gh), U(gk)) = f (U(h), U(k))
2. f (T(g)U(h), T(g)U(k)) = f (U(h), U(k))

33. O UTLINE
M OTIVATION
BACKGROUND
R ESULTS
S UMMARY

34. C OMMUTING R EPRESENTATIONS
Suppose that [U(h), T(g)] = 0, ∀h ∈ H, ∀g ∈ G
1
Φ[ρA (h) ⊗ IB ] = IA ⊗ T(g)⊗2 I ⊗ U(h)[ρA ]I ⊗ U(h)† T(g)⊗2†
|G|
g∈G
= IA ⊗ I ⊗ U(h)[σB ]I ⊗ U(h)†
= IA ⊗ σB (h)

35. C OMMUTING R EPRESENTATIONS
Suppose that [U(h), T(g)] = 0, ∀h ∈ H, ∀g ∈ G
1
Φ[ρA (h) ⊗ IB ] = IA ⊗ T(g)⊗2 I ⊗ U(h)[ρA ]I ⊗ U(h)† T(g)⊗2†
|G|
g∈G
= IA ⊗ I ⊗ U(h)[σB ]I ⊗ U(h)†
= IA ⊗ σB (h)
H OLEVO
If the set of states to be distinguished are covariant, the
minimum average error is achieved by a covariant
measurement
(k) †
(0)
MB = Vk MB Vk V : H → SU(HB⊗2 )

36. C OMMUTING R EPRESENTATIONS
Suppose that [U(h), T(g)] = 0, ∀h ∈ H, ∀g ∈ G
The measurements Bob performs must be independent of
referene frame
(k) (k)
MB = T(g)⊗2 MB T(g)⊗2† , ∀k ∈ H
g∈G
G-invariance
(k)
[MB , T(g)⊗2 ] = 0 ∀k ∈ H, ∀g ∈ G
By Holevo, we need only search over H-covariant
measurements
(k) † (0)
MB = Vk MB Vk V : G → SU(HB⊗2 )

37. C OMMUTING R EPRESENTATIONS
Suppose that [U(h), T(g)] = 0, ∀h ∈ H, ∀g ∈ G
T HEOREM (1)
G-invariance and H-covariance imply
Vk = Πk ◦ ⊕q IMq ,k ⊗ VNq ,k
where Πk is a block permutation matrix mapping q → π(q) such
that dim(Mπ(q) ) = dim(Mq ) and dim(Nπ(q) ) = dim(Nq )

38. C OMMUTING R EPRESENTATIONS
Suppose that [U(h), T(g)] = 0, ∀h ∈ H, ∀g ∈ G
T HEOREM (2)
(0) (0)
MB = ⊕q IMq ⊗ MNq
(0)
MNq = |e Nq e|
dr
dim(mr ) (r) (r)
|e Nq = ζi ξi
r
|H|
i=1
where mr is the space on which VNq ,k acts irreducibly,
(r) (r)
ζi ∈ mr , ξi ∈ νr

39. E XAMPLE —C3
1 0
U(g) = T(g) = , ω = exp(2π/3)
0 ωg

40. E XAMPLE —C3
1 0
U(g) = T(g) = , ω = exp(2π/3)
0 ωg
T ⊗2 = 1 ⊕ 2ω ⊕ ω 2

41. E XAMPLE —C3
1 0
U(g) = T(g) = , ω = exp(2π/3)
0 ωg
T ⊗2 = 1 ⊕ 2ω ⊕ ω 2
(1) (ω) (ω 2 )
Φ[ρA (h) ⊗ |B B|] = |A A| ⊗ ρA ⊕ U(h)[ρA ]U(h)† ⊕ ρA

42. E XAMPLE —C3
1 0
U(g) = T(g) = , ω = exp(2π/3)
0 ωg
T ⊗2 = 1 ⊕ 2ω ⊕ ω 2
(1) (ω) (ω 2 )
Φ[ρA (h) ⊗ |B B|] = |A A| ⊗ ρA ⊕ U(h)[ρA ]U(h)† ⊕ ρA
(0) 11
MB = 1/3(1 ⊕ 11 ⊕ 1)

43. E XAMPLE —C3
1 0
U(g) = T(g) = , ω = exp(2π/3)
0 ωg
T ⊗2 = 1 ⊕ 2ω ⊕ ω 2
(1) (ω) (ω 2 )
Φ[ρA (h) ⊗ |B B|] = |A A| ⊗ ρA ⊕ U(h)[ρA ]U(h)† ⊕ ρA
(0) 11
MB = 1/3(1 ⊕ 11 ⊕ 1)
1 0
Vk = 1 ⊕ 0 ωk
⊕1
f (U(h), U(k)) = δhk

44. E XAMPLE —C3
1 0
U(g) = T(g) = , ω = exp(2π/3)
0 ωg
T ⊗2 = 1 ⊕ 2ω ⊕ ω 2
(1) (ω) (ω 2 )
Φ[ρA (h) ⊗ |B B|] = |A A| ⊗ ρA ⊕ U(h)[ρA ]U(h)† ⊕ ρA
(0) 11
MB = 1/3(1 ⊕ 11 ⊕ 1)
1 0
Vk = 1 ⊕ 0 ωk
⊕1
f (U(h), U(k)) = δhk
O PTIMAL P ROTOCOL
ρA = |Ψ+ Ψ+ | maximizes the probability of success 2/3

45. P ERFECT COMMUNICATION
1 0 1 0
G = C3 , T(g) = H = C2 , U(h) =
0 ωg 0 (−1)h

46. P ERFECT COMMUNICATION
1 0 1 0
G = C3 , T(g) = H = C2 , U(h) =
0 ωg 0 (−1)h
T ⊗2 = 1 ⊕ 2ω ⊕ ω 2

47. P ERFECT COMMUNICATION
1 0 1 0
G = C3 , T(g) = H = C2 , U(h) =
0 ωg 0 (−1)h
T ⊗2 = 1 ⊕ 2ω ⊕ ω 2
I ⊗ U = 2(1 ⊕ −1)

48. P ERFECT COMMUNICATION
1 0 1 0
G = C3 , T(g) = H = C2 , U(h) =
0 ωg 0 (−1)h
T ⊗2 = 1 ⊕ 2ω ⊕ ω 2
I ⊗ U = 2(1 ⊕ −1)
U is the regular representation of C2
Let Alice pick |Ψ+ Ψ+ |. Then {Φ[ρA (h) ⊗ |B B|]} are
orthogonal

49. P ERFECT COMMUNICATION
1 0 1 0
G = C3 , T(g) = H = C2 , U(h) =
0 ωg 0 (−1)h
T ⊗2 = 1 ⊕ 2ω ⊕ ω 2
I ⊗ U = 2(1 ⊕ −1)
U is the regular representation of C2
Let Alice pick |Ψ+ Ψ+ |. Then {Φ[ρA (h) ⊗ |B B|]} are
orthogonal
There exist G-invariant measurements that perfectly
distinguish the messages

50. P ERFECT COMMUNICATION
1 0 1 0
G = C3 , T(g) = H = C2 , U(h) =
0 ωg 0 (−1)h
T ⊗2 = 1 ⊕ 2ω ⊕ ω 2
I ⊗ U = 2(1 ⊕ −1)
U is the regular representation of C2
Let Alice pick |Ψ+ Ψ+ |. Then {Φ[ρA (h) ⊗ |B B|]} are
orthogonal
There exist G-invariant measurements that perfectly
distinguish the messages
P ERFECT C OMMUNICATION
Perfect transmission of message iff
1. U contains the regular representation of H
2. ∃ G-invariant measurements distinguishing
{Φ[ρA (h) ⊗ |B B|]}

51. N ON - COMMUTING REPRESENTATIONS
Suppose that [U(h), T(g)] = 0, ∀h ∈ H, g ∈ G
Alice’s messages are no longer covariant
Bob’s measurements still obey
[M (k) , T(g)⊗2 ] = 0 ∀k ∈ H, ∀g ∈ G
but Holevo’s result no longer applies.
 
1
¯
∆ = ph Tr  T(g)⊗2 ρA (h)T(g)⊗2† M (k)  f (U(h), U(k))
|G|
h,k∈H g∈G
= Tr Wk M (k)
k∈H

52. N ON - COMMUTING REPRESENTATIONS
Suppose that [U(h), T(g)] = 0, ∀h ∈ H, g ∈ G
¯
∆ = Tr Wk M (k)
k∈H
(k)
MB = IH ⊗2
B
k
For a given ρA (h) the optimal measurements are given by
(k) (k)
(Wk − Γ) MB = MB (Wk − Γ) = 0
Wk − Γ ≥ 0
(l)
where Γ = MB Wl
l∈H
Maximum given by Tr[Γ]

53. N ON - COMMUTING REPRESENTATIONS
Suppose that [U(h), T(g)] = 0, ∀h ∈ H, g ∈ G
Use semi-definite programming to find optimal Γ
Optimal measurements belong to the kernel of Wk − Γ
If kernel is 1-dimensional then measurements are unique
If kernel is multi-dimensional then there may be several
measurements that yield the optimum.

54. E XAMPLE —C3
1 0 cos(2gπ/3) − sin(2gπ/3)
T(g) = , U(g) =
0 ωg sin(2gπ/3) cos(2gπ/3)
T =1⊕ω
U = ω ⊕ ω2
f (U(h), U(k)) = δhk

55. E XAMPLE —C3
1 0 cos(2gπ/3) − sin(2gπ/3)
T(g) = , U(g) =
0 ωg sin(2gπ/3) cos(2gπ/3)
T =1⊕ω
U = ω ⊕ ω2
f (U(h), U(k)) = δhk
O PTIMAL M EASUREMENTS :
(0)
MB = |0 0|, M (1) = M (2) = 1/2(I − M (0) )
O PTIMAL S TATES : |Ψ+ Ψ+ |, |0 0|
7
S UCCESS : 12

56. O UTLINE
M OTIVATION
BACKGROUND
R ESULTS
S UMMARY

57. C ONCLUSIONS
How to communicate messages between parties that lack
a requisite shared frame of references
Protocol that circumvents the restriction by encoding
information in relational parameters
Our criterion of success f (U(h), U(k)) enables the
construction of protocols where the message group and
reference frame group are different
Closed-form expressions for optimal states and
measurements when [U(h), T(g)] = 0 ∀h ∈ H, ∀g ∈ G
Numerical solutions for optimal states and measurements
when [U(h), T(g)] = 0 ∀h ∈ H, ∀g ∈ G
Trade-off between better success rate vs. entanglement.

58. ACKNOWLEDGEMENTS
Pacific Institute for the
Mathematical Sciences