Entanglement-enhanced classical communication
      without a shared frame of reference

M. Skotiniotis1    A. Roy1,2     ...
O UTLINE


  M OTIVATION


  BACKGROUND


  R ESULTS


  S UMMARY
R EFERENCE F RAMES AND I NFORMATION
     Alice and Bob have a conversation but they do not speak a
     common language

 ...
R EFERENCE F RAMES AND I NFORMATION
      Alice and Bob have a conversation but they do not speak a
      common language
...
I NFORMATION AND N OISE
     Alice and Bob share a common language but a noisy
     channel

     Alice and Bob agree on w...
I NFORMATION AND N OISE
      Alice and Bob share a common language but a noisy
      channel

      Alice and Bob agree o...
P OSSIBLE S OLUTIONS
  A LIGNMENT OF R EFERENCE F RAMES
      Alice and Bob can learn a common language

      Alice can s...
P OSSIBLE S OLUTIONS
  A LIGNMENT OF R EFERENCE F RAMES
      Alice and Bob can learn a common language

      Alice can s...
P OSSIBLE S OLUTIONS
  A LIGNMENT OF R EFERENCE F RAMES
      Alice and Bob can learn a common language

      Alice can s...
P OSSIBLE S OLUTIONS
  A LIGNMENT OF R EFERENCE F RAMES
      Alice and Bob can learn a common language

      Alice can s...
P OSSIBLE S OLUTIONS
  A LIGNMENT OF R EFERENCE F RAMES
      Alice and Bob can learn a common language

      Alice can s...
O UTLINE


  M OTIVATION


  BACKGROUND


  R ESULTS


  S UMMARY
R EFERENCE F RAMES AND S YMMETRY G ROUPS




    Alice                                            Bob
      Alice and Bob ...
R EFERENCE F RAMES AND S YMMETRY G ROUPS




    Alice                                            Bob
      Alice and Bob ...
R EFERENCE F RAMES AND S YMMETRY G ROUPS




    Alice                                                Bob
      Alice and ...
R EFERENCE F RAMES AND S YMMETRY G ROUPS




    Alice                                                Bob
      Alice and ...
R EFERENCE F RAMES AND N OISE
                        B(H ) = B(HA ⊗ HB )
       B(HA )                                   ...
R EFERENCE F RAMES AND N OISE
                        B(H ) = B(HA ⊗ HB )
       B(HA )                                   ...
R EFERENCE F RAMES AND N OISE
                        B(H ) = B(HA ⊗ HB )
       B(HA )                                   ...
R EFERENCE F RAMES AND N OISE
                        B(H ) = B(HA ⊗ HB )
       B(HA )                                   ...
R ANDOM U NITARY C HANNEL
                        B(H ) = B(HA ⊗ HB )
       B(HA )                                       ...
R ANDOM U NITARY C HANNEL
                         B(H ) = B(HA ⊗ HB )
       B(HA )                                      ...
R ANDOM U NITARY C HANNEL
                         B(H ) = B(HA ⊗ HB )
       B(HA )                                      ...
R ANDOM U NITARY C HANNEL
                         B(H ) = B(HA ⊗ HB )
       B(HA )                                      ...
P REPARE - AND -M EASURE

                           Φ : B(H ) → B(H )

         A                COMMUNICATION CHANNEL   ...
P REPARE - AND -M EASURE

                           Φ : B(H ) → B(H )

         A                COMMUNICATION CHANNEL   ...
P REPARE - AND -M EASURE

                           Φ : B(H ) → B(H )

         A                COMMUNICATION CHANNEL   ...
P REPARE - AND -M EASURE

                           Φ : B(H ) → B(H )

         A                COMMUNICATION CHANNEL   ...
P REPARE - AND -M EASURE

                           Φ : B(H ) → B(H )

         A                COMMUNICATION CHANNEL   ...
C OMMUNICATION P ROTOCOL
                  Φ : B(HA⊗2 ⊗ HB⊗2 ) → B(HA⊗2 ⊗ HB⊗2 )


                         COMMUNICATION ...
C OMMUNICATION P ROTOCOL
                  Φ : B(HA⊗2 ⊗ HB⊗2 ) → B(HA⊗2 ⊗ HB⊗2 )


                         COMMUNICATION ...
C OMMUNICATION P ROTOCOL
                   Φ : B(HA⊗2 ⊗ HB⊗2 ) → B(HA⊗2 ⊗ HB⊗2 )


                           COMMUNICATI...
O UTLINE


  M OTIVATION


  BACKGROUND


  R ESULTS


  S UMMARY
C OMMUTING R EPRESENTATIONS
  Suppose that [U(h), T(g)] = 0, ∀h ∈ H, ∀g ∈ G
                          1
  Φ[ρA (h) ⊗ IB ] ...
C OMMUTING R EPRESENTATIONS
  Suppose that [U(h), T(g)] = 0, ∀h ∈ H, ∀g ∈ G
                          1
  Φ[ρA (h) ⊗ IB ] ...
C OMMUTING R EPRESENTATIONS
  Suppose that [U(h), T(g)] = 0, ∀h ∈ H, ∀g ∈ G

      The measurements Bob performs must be i...
C OMMUTING R EPRESENTATIONS
  Suppose that [U(h), T(g)] = 0, ∀h ∈ H, ∀g ∈ G


  T HEOREM (1)
  G-invariance and H-covarian...
C OMMUTING R EPRESENTATIONS
  Suppose that [U(h), T(g)] = 0, ∀h ∈ H, ∀g ∈ G

  T HEOREM (2)
                              ...
E XAMPLE —C3
                  1 0
  U(g) = T(g) =          ,   ω = exp(2π/3)
                  0 ωg
E XAMPLE —C3
                   1 0
  U(g) = T(g) =             ,   ω = exp(2π/3)
                   0 ωg
      T ⊗2 = 1 ⊕...
E XAMPLE —C3
                   1 0
  U(g) = T(g) =             ,   ω = exp(2π/3)
                   0 ωg
      T ⊗2 = 1 ⊕...
E XAMPLE —C3
                   1 0
  U(g) = T(g) =             ,   ω = exp(2π/3)
                   0 ωg
      T ⊗2 = 1 ⊕...
E XAMPLE —C3
                     1 0
  U(g) = T(g) =               ,   ω = exp(2π/3)
                     0 ωg
      T ⊗2...
E XAMPLE —C3
                     1 0
  U(g) = T(g) =               ,   ω = exp(2π/3)
                     0 ωg
      T ⊗2...
P ERFECT COMMUNICATION
                    1 0                      1   0
  G = C3 , T(g) =          H = C2 , U(h) =
     ...
P ERFECT COMMUNICATION
                      1 0                       1   0
  G = C3 , T(g) =             H = C2 , U(h) =...
P ERFECT COMMUNICATION
                      1 0                       1   0
  G = C3 , T(g) =             H = C2 , U(h) =...
P ERFECT COMMUNICATION
                      1 0                           1   0
  G = C3 , T(g) =                H = C2 ,...
P ERFECT COMMUNICATION
                      1 0                           1   0
  G = C3 , T(g) =                H = C2 ,...
P ERFECT COMMUNICATION
                      1 0                           1   0
  G = C3 , T(g) =                H = C2 ,...
N ON - COMMUTING REPRESENTATIONS
  Suppose that [U(h), T(g)] = 0, ∀h ∈ H, g ∈ G

      Alice’s messages are no longer cova...
N ON - COMMUTING REPRESENTATIONS
  Suppose that [U(h), T(g)] = 0, ∀h ∈ H, g ∈ G

                         ¯
              ...
N ON - COMMUTING REPRESENTATIONS
  Suppose that [U(h), T(g)] = 0, ∀h ∈ H, g ∈ G

      Use semi-definite programming to find...
E XAMPLE —C3
           1 0                    cos(2gπ/3) − sin(2gπ/3)
  T(g) =             ,   U(g) =
           0 ωg    ...
E XAMPLE —C3
            1 0                    cos(2gπ/3) − sin(2gπ/3)
  T(g) =              ,   U(g) =
            0 ωg ...
O UTLINE


  M OTIVATION


  BACKGROUND


  R ESULTS


  S UMMARY
C ONCLUSIONS
     How to communicate messages between parties that lack
     a requisite shared frame of references
     P...
ACKNOWLEDGEMENTS


         Pacific Institute for the
         Mathematical Sciences
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Cptp Nv.1

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Entanglement-enhanced classical communication
without a shared frame of reference

1st CPTPN conference, Lethbridge, 25-26 August 2010

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Cptp Nv.1

  1. 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. 2. O UTLINE M OTIVATION BACKGROUND R ESULTS S UMMARY
  3. 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. 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. 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. 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. 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. 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. 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. 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. 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. 12. O UTLINE M OTIVATION BACKGROUND R ESULTS S UMMARY
  13. 13. R EFERENCE F RAMES AND S YMMETRY G ROUPS Alice Bob Alice and Bob lack a shared Cartesian reference frame
  14. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 33. O UTLINE M OTIVATION BACKGROUND R ESULTS S UMMARY
  34. 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. 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. 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. 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. 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. 39. E XAMPLE —C3 1 0 U(g) = T(g) = , ω = exp(2π/3) 0 ωg
  40. 40. E XAMPLE —C3 1 0 U(g) = T(g) = , ω = exp(2π/3) 0 ωg T ⊗2 = 1 ⊕ 2ω ⊕ ω 2
  41. 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. 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. 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. 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. 45. P ERFECT COMMUNICATION 1 0 1 0 G = C3 , T(g) = H = C2 , U(h) = 0 ωg 0 (−1)h
  46. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 56. O UTLINE M OTIVATION BACKGROUND R ESULTS S UMMARY
  57. 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. 58. ACKNOWLEDGEMENTS Pacific Institute for the Mathematical Sciences

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