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Maximally entangled mixed states with fixed marginals Giuseppe Baio SUPA & University of Strathclyde, Glasgow, UK 51 Symposium of Mathematical Physics, Toru, Poland 17 th June 2019 17 th June 2019 Giuseppe Baio 51 SMP Toru 1 / 22 My

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Maximally entangled mixed states with fixed marginals

Giuseppe Baio SUPA & University of Strathclyde, Glasgow, UK 51 Symposium of Mathematical Physics, Toruń, Poland

17th June 2019

Giuseppe Baio 51 SMP Toruń 17th June 2019 1 / 22

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My research activity @ Strathclyde

Computational Nonlinear and Quantum Optics

Cold Atoms, Nanophotonics, Quantum Information and Many-Body Physics, Structured Light etc. http://cnqo.phys.strath.ac.uk

ColOpt ITN

Collective effects and optomechanics in ultra cold matter https://www.colopt.eu/

Giuseppe Baio 51 SMP Toruń 17th June 2019 2 / 22

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My research activity @ Strathclyde

Computational Nonlinear and Quantum Optics

Cold Atoms, Nanophotonics, Quantum Information and Many-Body Physics, Structured Light etc. http://cnqo.phys.strath.ac.uk

ColOpt ITN

Collective effects and optomechanics in ultra cold matter https://www.colopt.eu/

Talk based on recent paper: Phys. Rev. A 99, 062312 (2019)

Joint work with: D. Chruściński, G. Sarbicki (Toruń, Poland), P. Horodecki (Gdańsk, Poland), A. Messina (Palermo, Italy)

Giuseppe Baio 51 SMP Toruń 17th June 2019 2 / 22

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Outline

Maximally entangled mixed states (MEMS)

Giuseppe Baio 51 SMP Toruń 17th June 2019 3 / 22

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Outline

Maximally entangled mixed states (MEMS) Fixing marginals: reconstructing states from local information

Giuseppe Baio 51 SMP Toruń 17th June 2019 3 / 22

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Outline

Maximally entangled mixed states (MEMS) Fixing marginals: reconstructing states from local information What is known: Two qubit case

Giuseppe Baio 51 SMP Toruń 17th June 2019 3 / 22

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Outline

Maximally entangled mixed states (MEMS) Fixing marginals: reconstructing states from local information What is known: Two qubit case Higher dimensions: Two qutrit case and quasidistillation

Giuseppe Baio 51 SMP Toruń 17th June 2019 3 / 22

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Outline

Maximally entangled mixed states (MEMS) Fixing marginals: reconstructing states from local information What is known: Two qubit case Higher dimensions: Two qutrit case and quasidistillation Future directions

Giuseppe Baio 51 SMP Toruń 17th June 2019 3 / 22

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Preliminaries: Mixed bipartite entanglement and measures

von Neumann entropy for pure states |ΨABΨAB|: E(ΨAB) = S(ρA) = −Tr(ρA log ρA) (1)

  • 1M. B. Plenio and S. Virmani, Quant. Inf. Comput. 7, 1 (2007).

Giuseppe Baio 51 SMP Toruń 17th June 2019 4 / 22

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Preliminaries: Mixed bipartite entanglement and measures

von Neumann entropy for pure states |ΨABΨAB|: E(ΨAB) = S(ρA) = −Tr(ρA log ρA) (1) For mixed states, i.e. Tr(ρ2

AB) < 1: convex roof construction, e.g.:

EOF(ρAB) = min

pk,Ψk

  • k

pkE(Ψk) (2)

  • 1M. B. Plenio and S. Virmani, Quant. Inf. Comput. 7, 1 (2007).

Giuseppe Baio 51 SMP Toruń 17th June 2019 4 / 22

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Preliminaries: Mixed bipartite entanglement and measures

von Neumann entropy for pure states |ΨABΨAB|: E(ΨAB) = S(ρA) = −Tr(ρA log ρA) (1) For mixed states, i.e. Tr(ρ2

AB) < 1: convex roof construction, e.g.:

EOF(ρAB) = min

pk,Ψk

  • k

pkE(Ψk) (2) Several tools adopted: concurrence and negativity1: N(ρAB) ≡ 1 2 (ρτ

AB1 − 1)

(3) Partial transpose: ρτ

AB = (I ⊗ τ) ρAB

  • 1M. B. Plenio and S. Virmani, Quant. Inf. Comput. 7, 1 (2007).

Giuseppe Baio 51 SMP Toruń 17th June 2019 4 / 22

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Maximally entangled mixed states (MEMS)

Relation between entanglement and purity Tr(ρ2

AB): 2

  • 2W. J. Munro et al., Phys. Rev. A 64, 030302(R) (2001).

Giuseppe Baio 51 SMP Toruń 17th June 2019 5 / 22

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Maximally entangled mixed states (MEMS)

Relation between entanglement and purity Tr(ρ2

AB): 2

  • 2W. J. Munro et al., Phys. Rev. A 64, 030302(R) (2001).

Giuseppe Baio 51 SMP Toruń 17th June 2019 5 / 22

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Maximally entangled mixed states (MEMS)

Relation between entanglement and purity Tr(ρ2

AB): 2

  • 2W. J. Munro et al., Phys. Rev. A 64, 030302(R) (2001).

Giuseppe Baio 51 SMP Toruń 17th June 2019 5 / 22

Werner

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Maximally entangled mixed states (MEMS)

Relation between entanglement and purity Tr(ρ2

AB): 2

MEMS: states ρ∗ such that any measure E(ρ∗) ≥ E(Uρ∗U †), ∀ U.

  • 2W. J. Munro et al., Phys. Rev. A 64, 030302(R) (2001).

Giuseppe Baio 51 SMP Toruń 17th June 2019 5 / 22

Werner

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Maximally entangled mixed states (MEMS)

Two qubit MEMS found solving the spectral constrained analogue3

  • 3F. Verstraete et al., Phys. Rev. A 64, 012316 (2001).
  • 4T. C. Wei et al., Phys. Rev. A 67, 022110 (2003).

Giuseppe Baio 51 SMP Toruń 17th June 2019 6 / 22

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Maximally entangled mixed states (MEMS)

Two qubit MEMS found solving the spectral constrained analogue3 Theorem (Verstraete, 2001) Given a state ρ = ΦΛΦ†, the unitary maximising EOF and negativity is: U = (U1 ⊗ U2)     1 1/ √ 2 1/ √ 2 1/ √ 2 −1/ √ 2 1     DφΦ† (4) MEMS depend on the entanglement measure considered4

  • 3F. Verstraete et al., Phys. Rev. A 64, 012316 (2001).
  • 4T. C. Wei et al., Phys. Rev. A 67, 022110 (2003).

Giuseppe Baio 51 SMP Toruń 17th June 2019 6 / 22

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Maximally entangled mixed states (MEMS)

Two qubit MEMS found solving the spectral constrained analogue3 Theorem (Verstraete, 2001) Given a state ρ = ΦΛΦ†, the unitary maximising EOF and negativity is: U = (U1 ⊗ U2)     1 1/ √ 2 1/ √ 2 1/ √ 2 −1/ √ 2 1     DφΦ† (4) MEMS depend on the entanglement measure considered4 For negativity: ρMEMS = 1−r

4 I2 ⊗ I2 + rP + 2 ,

P +

2 = 1 2

2

i,j=1 |iijj|

  • 3F. Verstraete et al., Phys. Rev. A 64, 012316 (2001).
  • 4T. C. Wei et al., Phys. Rev. A 67, 022110 (2003).

Giuseppe Baio 51 SMP Toruń 17th June 2019 6 / 22

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Fixing marginals: reconstructing states from local info

Entanglement characterization:5

  • 5G. Adesso et al., Phys. Rev. A 68, 062318 (2003)

Giuseppe Baio 51 SMP Toruń 17th June 2019 7 / 22

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Fixing marginals: reconstructing states from local info

Entanglement characterization:5 → Maximally entangled marginally mixed states (MEMMS), i.e. MEMS with respect to local purities

  • 5G. Adesso et al., Phys. Rev. A 68, 062318 (2003)

Giuseppe Baio 51 SMP Toruń 17th June 2019 7 / 22

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Fixing marginals: reconstructing states from local info

Entanglement characterization:5 → Maximally entangled marginally mixed states (MEMMS), i.e. MEMS with respect to local purities What is the upper bound Emax(ρ) on bipartite entanglement when only marginals are known? MEMS with respect to fixed marginals Given ρA, ρB, find Emax(ρ) : TrB(ρ) = ρA, TrA(ρ) = ρB

  • 5G. Adesso et al., Phys. Rev. A 68, 062318 (2003)

Giuseppe Baio 51 SMP Toruń 17th June 2019 7 / 22

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Fixing marginals: reconstructing states from local info

Entanglement characterization:5 → Maximally entangled marginally mixed states (MEMMS), i.e. MEMS with respect to local purities What is the upper bound Emax(ρ) on bipartite entanglement when only marginals are known? MEMS with respect to fixed marginals Given ρA, ρB, find Emax(ρ) : TrB(ρ) = ρA, TrA(ρ) = ρB Characterizing states from local measurements: Quantum marginal constraints (Klyachko), Quantum tomography etc.

  • 5G. Adesso et al., Phys. Rev. A 68, 062318 (2003)

Giuseppe Baio 51 SMP Toruń 17th June 2019 7 / 22

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What is known: Two qubits

Let ρA = diag{1 − λA, λA}, ρB = diag{1 − λB, λB} be two qubit states. λA, λB ∈

  • 0, 1

2

  • , λA ≥ λB

Giuseppe Baio 51 SMP Toruń 17th June 2019 8 / 22

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What is known: Two qubits

Let ρA = diag{1 − λA, λA}, ρB = diag{1 − λB, λB} be two qubit states. λA, λB ∈

  • 0, 1

2

  • , λA ≥ λB

C(ρA, ρB) set of two-qubit states with fixed marginals: ρAB = ρA ⊗ ρB +     ǫ ∆12 ∆13 ∆14 −ǫ ∆23 −∆13 −ǫ −∆12 (c.c) ǫ     (5)

Giuseppe Baio 51 SMP Toruń 17th June 2019 8 / 22

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What is known: Two qubits

Let ρA = diag{1 − λA, λA}, ρB = diag{1 − λB, λB} be two qubit states. λA, λB ∈

  • 0, 1

2

  • , λA ≥ λB

C(ρA, ρB) set of two-qubit states with fixed marginals 6: ρAB = ρA ⊗ ρB +     ǫ · · ∆14 −ǫ ∆23 · −ǫ · (c.c) ǫ     (5)

  • 6F. Verstraete et al., Phys. Rev. A 64, 012316 (2001).

Giuseppe Baio 51 SMP Toruń 17th June 2019 8 / 22

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What is known: Two qubits

Let ρA = diag{1 − λA, λA}, ρB = diag{1 − λB, λB} be two qubit states. λA, λB ∈

  • 0, 1

2

  • , λA ≥ λB

C(ρA, ρB) set of two-qubit states with fixed marginals: ˜ ρAB =     1 − λA · ·

  • (1 − λA)λB

· · · · · λA − λB ·

  • (1 − λA)λB

· · λB     (5) Maximal neg: N(˜ ρAB) = 1

2

  • λA − λB −
  • (λA − λB)2 + 4λB(1 − λA)
  • Giuseppe Baio

51 SMP Toruń 17th June 2019 8 / 22

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What is known: Two qubits

Let ρA = diag{1 − λA, λA}, ρB = diag{1 − λB, λB} be two qubit states. λA, λB ∈

  • 0, 1

2

  • , λA ≥ λB

C(ρA, ρB) set of two-qubit states with fixed marginals: ˜ ρAB =     1 − λA · ·

  • (1 − λA)λB

· · · · · λA − λB ·

  • (1 − λA)λB

· · λB     (5) Maximal neg: N(˜ ρAB) = 1

2

  • λA − λB −
  • (λA − λB)2 + 4λB(1 − λA)
  • 2-qubit MEMS with respect to ρA and ρB

˜ ρAB = (1 − η)|ΨmcΨmc| + η|1010|, σmc =

d−1

  • i,j=0

αij|iijj| (6)

Giuseppe Baio 51 SMP Toruń 17th June 2019 8 / 22

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What is known: Two qubits

Random two qubit states with fixed marginals (N − P plot):

Giuseppe Baio 51 SMP Toruń 17th June 2019 9 / 22

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What is known: Two qubits

Random two qubit states with fixed marginals (N − P plot): Both P and N are maximised by the same state ˜ ρAB

Giuseppe Baio 51 SMP Toruń 17th June 2019 9 / 22

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Interlude: Extremal states

N(ρAB) convex function and C(ρA, ρB) also convex: maximised by extremal point. How to characterize extemals in C(ρA, ρB)6,7?

  • 6K. R. Parthasarathy, Ann. l’Inst. H. Poincaré (B) Prob. Stat. 41, 257 (2005)
  • 7O. Rudolph, J. Math. Phys. 45(11), 4035 (2004).

Giuseppe Baio 51 SMP Toruń 17th June 2019 10 / 22

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Interlude: Extremal states

N(ρAB) convex function and C(ρA, ρB) also convex: maximised by extremal point. How to characterize extemals in C(ρA, ρB)6,7? States/CP maps duality (Choi-Jamiołkowski): ρΛ = [idd ⊗ Λ](P +

d )

  • 6K. R. Parthasarathy, Ann. l’Inst. H. Poincaré (B) Prob. Stat. 41, 257 (2005)
  • 7O. Rudolph, J. Math. Phys. 45(11), 4035 (2004).

Giuseppe Baio 51 SMP Toruń 17th June 2019 10 / 22

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Interlude: Extremal states

N(ρAB) convex function and C(ρA, ρB) also convex: maximised by extremal point. How to characterize extemals in C(ρA, ρB)6,7? States/CP maps duality (Choi-Jamiołkowski): ρΛ = [idd ⊗ Λ](P +

d )

Given Λρ[σ] =

α KασK† α, extremal iff {K† αKβ}α,β=1,...,d2 is L.I.

  • 6K. R. Parthasarathy, Ann. l’Inst. H. Poincaré (B) Prob. Stat. 41, 257 (2005)
  • 7O. Rudolph, J. Math. Phys. 45(11), 4035 (2004).

Giuseppe Baio 51 SMP Toruń 17th June 2019 10 / 22

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Interlude: Extremal states

N(ρAB) convex function and C(ρA, ρB) also convex: maximised by extremal point. How to characterize extemals in C(ρA, ρB)6,7? States/CP maps duality (Choi-Jamiołkowski): ρΛ = [idd ⊗ Λ](P +

d )

Given Λρ[σ] =

α KασK† α, extremal iff {K† αKβ}α,β=1,...,d2 is L.I.

Extremal points in C(ρA, ρB) 1 dΛρ(Id) = 1 d

  • α

KαK†

α = ρA

1 dΛ∗

ρ(Id) = 1

d

  • α

K†

αKα = ρB

{K†

αKβ ⊕ KβK† α}α,β=1,...,d2 L.I

(7)

  • 6K. R. Parthasarathy, Ann. l’Inst. H. Poincaré (B) Prob. Stat. 41, 257 (2005)
  • 7O. Rudolph, J. Math. Phys. 45(11), 4035 (2004).

Giuseppe Baio 51 SMP Toruń 17th June 2019 10 / 22

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Higher dimensions

The search for higher dimensional MEMS is still open8

  • 8P. E. M. F. Mendonca et al., Phys. Rev. A 95, 022324, 2017.

Giuseppe Baio 51 SMP Toruń 17th June 2019 11 / 22

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Higher dimensions

The search for higher dimensional MEMS is still open8 We generalize the two qubit results to Cd ⊗ Cd ˜ ρ = (1 − η)σmc +

  • i=j

pij|ijij| (8)

  • 8P. E. M. F. Mendonca et al., Phys. Rev. A 95, 022324, 2017.

Giuseppe Baio 51 SMP Toruń 17th June 2019 11 / 22

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Higher dimensions

The search for higher dimensional MEMS is still open8 We generalize the two qubit results to Cd ⊗ Cd ˜ ρ = (1 − η)σmc +

  • i=j

pij|ijij| (8) Example : Two qutrits

˜ ρ =

           

ρ11 · · · ∆15 · · · ∆19 ρ22 · · · · · · · ρ33 · · · · · · ρ44 · · · · · ρ55 · · · ∆59 ρ66 · · · ρ77 · · ρ88 · (c.c.) ρ99

           

(9)

  • 8P. E. M. F. Mendonca et al., Phys. Rev. A 95, 022324, 2017.

Giuseppe Baio 51 SMP Toruń 17th June 2019 11 / 22

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Higher dimensions

The search for higher dimensional MEMS is still open8 We generalize the two qubit results to Cd ⊗ Cd ˜ ρ = (1 − η)σmc +

  • i=j

pij|ijij| (8) Example : Two qutrits

˜ ρ =

           

ρ11 · · · ∆15 · · · ∆19 ρ22 · · · · · · · ρ33 · · · · · · ρ44 · · · · · ρ55 · · · ∆59 ρ66 · · · ρ77 · · ρ88 · (c.c.) ρ99

           

(9)

We identify in this family candidates for MEMS with fixed marginals

  • 8P. E. M. F. Mendonca et al., Phys. Rev. A 95, 022324, 2017.

Giuseppe Baio 51 SMP Toruń 17th June 2019 11 / 22

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Higher dimensions

Marginals ρA = diag {1 − λ1 − λ2, λ1, λ2}, ρB = diag {1 − µ1 − µ2, µ1, µ2} λ1 ≥ λ2, µ1 ≥ µ2

Giuseppe Baio 51 SMP Toruń 17th June 2019 12 / 22

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Higher dimensions

Marginals ρA = diag {1 − λ1 − λ2, λ1, λ2}, ρB = diag {1 − µ1 − µ2, µ1, µ2} λ1 ≥ λ2, µ1 ≥ µ2 λ2, µ2 ≤ 1

  • 3. We assume λ1 + λ2 ≥ µ1 + µ2

Giuseppe Baio 51 SMP Toruń 17th June 2019 12 / 22

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Higher dimensions

Marginals ρA = diag {1 − λ1 − λ2, λ1, λ2}, ρB = diag {1 − µ1 − µ2, µ1, µ2} λ1 ≥ λ2, µ1 ≥ µ2 λ2, µ2 ≤ 1

  • 3. We assume λ1 + λ2 ≥ µ1 + µ2

Partial transpose: ˜ ρτ =               ρ11 · · · · · · · · ρ22 · ∆15 · · · · · ρ33 · · · ∆19 · · ρ44 · · · · · ρ55 · · · · ρ66 · ∆59 · ρ77 · · ρ88 · (c.c.) ρ99               (10) Guideline for max negativity: lowest possible number of ρii = 0 (2)

Giuseppe Baio 51 SMP Toruń 17th June 2019 12 / 22

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Higher dimensions

Our candidates as MEMS with respect to marginals:

Giuseppe Baio 51 SMP Toruń 17th June 2019 13 / 22

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Higher dimensions

Our candidates as MEMS with respect to marginals: ˜ ρ(1)

AB = (1 − p10 − p20) |Ψ(1) mcΨ(1) mc| + p10 |1010| + p20 |2020|

p10 = λ1 − µ1, p20 = λ2 − µ2 (11) valid when λ1 > µ1 and λ2 > µ2,

Giuseppe Baio 51 SMP Toruń 17th June 2019 13 / 22

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Higher dimensions

Our candidates as MEMS with respect to marginals: ˜ ρ(1)

AB = (1 − p10 − p20) |Ψ(1) mcΨ(1) mc| + p10 |1010| + p20 |2020|

p10 = λ1 − µ1, p20 = λ2 − µ2 (11) valid when λ1 > µ1 and λ2 > µ2, ˜ ρ(2)

AB = (1 − p10 − p12) |Ψ(2) mcΨ(2) mc| + p10 |1010| + p12 |1212|

p10 = λ1 + λ1 − (µ1 + µ2), p12 = µ2 − λ2 (12) when λ1 > µ1 and λ2 < µ2

Giuseppe Baio 51 SMP Toruń 17th June 2019 13 / 22

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Higher dimensions

Our candidates as MEMS with respect to marginals: ˜ ρ(1)

AB = (1 − p10 − p20) |Ψ(1) mcΨ(1) mc| + p10 |1010| + p20 |2020|

p10 = λ1 − µ1, p20 = λ2 − µ2 (11) valid when λ1 > µ1 and λ2 > µ2, ˜ ρ(2)

AB = (1 − p10 − p12) |Ψ(2) mcΨ(2) mc| + p10 |1010| + p12 |1212|

p10 = λ1 + λ1 − (µ1 + µ2), p12 = µ2 − λ2 (12) when λ1 > µ1 and λ2 < µ2 , and finally (λ1 < µ1 and λ2 > µ2) ˜ ρ(3)

AB = (1 − p20 − p21) |Ψ(3) mcΨ(3) mc| + p20 |2020| + p21 |2121|

p20 = λ1 + λ1 − (µ1 + µ2), p21 = µ1 − λ1 (13)

Giuseppe Baio 51 SMP Toruń 17th June 2019 13 / 22

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Higher dimensions

Our candidates as MEMS with respect to marginals: ˜ ρ(1)

AB = (1 − p10 − p20) |Ψ(1) mcΨ(1) mc| + p10 |1010| + p20 |2020|

p10 = λ1 − µ1, p20 = λ2 − µ2 (11) valid when λ1 > µ1 and λ2 > µ2, ˜ ρ(2)

AB = (1 − p10 − p12) |Ψ(2) mcΨ(2) mc| + p10 |1010| + p12 |1212|

p10 = λ1 + λ1 − (µ1 + µ2), p12 = µ2 − λ2 (12) when λ1 > µ1 and λ2 < µ2 , and finally (λ1 < µ1 and λ2 > µ2) ˜ ρ(3)

AB = (1 − p20 − p21) |Ψ(3) mcΨ(3) mc| + p20 |2020| + p21 |2121|

p20 = λ1 + λ1 − (µ1 + µ2), p21 = µ1 − λ1 (13) Preposition 1 States (11), (12), (13) are extremal in C(ρA, ρB).

Giuseppe Baio 51 SMP Toruń 17th June 2019 13 / 22

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Quasidistillation

Both the 2-qubit state and our candidates are quasidistillable9

  • 9M. Horodecki, P. Horodecki & R. Horodecki, Phys. Rev. A 60, 1888 (1999).

Giuseppe Baio 51 SMP Toruń 17th June 2019 14 / 22

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Quasidistillation

Both the 2-qubit state and our candidates are quasidistillable9

  • Def. 1: Non-collective distillation

ρ is distillable iff there exist A and B (filtering) such that: (A ⊗ B)ρ(A† ⊗ B†) Tr [(A ⊗ B)ρ(A† ⊗ B†)] = P +

d

(14)

  • 9M. Horodecki, P. Horodecki & R. Horodecki, Phys. Rev. A 60, 1888 (1999).

Giuseppe Baio 51 SMP Toruń 17th June 2019 14 / 22

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Quasidistillation

Both the 2-qubit state and our candidates are quasidistillable9

  • Def. 1: Non-collective distillation

ρ is distillable iff there exist A and B (filtering) such that: (A ⊗ B)ρ(A† ⊗ B†) Tr [(A ⊗ B)ρ(A† ⊗ B†)] = P +

d

(14) No mixed state is non-collectively distillable!

  • Def. 2: Quasidistillation

ρ is quasidistillable iff there exist {An} and {Bn} such that: (An ⊗ Bn)ρ(A†

n ⊗ B† n)

Tr

  • (An ⊗ Bn)ρ(A†

n ⊗ B† n)

− − − − →

n→∞ P + d ,

pn = Tr[Λ(n)(ρ)] → 0 (15)

  • 9M. Horodecki, P. Horodecki & R. Horodecki, Phys. Rev. A 60, 1888 (1999).

Giuseppe Baio 51 SMP Toruń 17th June 2019 14 / 22

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Quasidistillation

We characterize quasidistillable states within the family: ˜ ρ = (1 − η)σmc +

  • i=j

pij|ijij| (16)

Giuseppe Baio 51 SMP Toruń 17th June 2019 15 / 22

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Quasidistillation

We characterize quasidistillable states within the family: ˜ ρ = (1 − η)σmc +

  • i=j

pij|ijij| (16) Theorem 1 σmc is quasidistillable iff it is rank-1, i.e. σmc = |ΨmcΨmc|, and s − rank(|Ψk) = d.

Giuseppe Baio 51 SMP Toruń 17th June 2019 15 / 22

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SLIDE 51

Quasidistillation

We characterize quasidistillable states within the family: ˜ ρ = (1 − η)σmc +

  • i=j

pij|ijij| (16) Theorem 1 σmc is quasidistillable iff it is rank-1, i.e. σmc = |ΨmcΨmc|, and s − rank(|Ψk) = d. Adding some extra noise.. Theorem 2 ρ of the form (16) quasidistillable iff among pij = 0, pijpjk . . . pli = 0. (No looping indices)

Giuseppe Baio 51 SMP Toruń 17th June 2019 15 / 22

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SLIDE 52

Quasidistillation

We characterize quasidistillable states within the family: ˜ ρ = (1 − η)σmc +

  • i=j

pij|ijij| (16) Theorem 1 σmc is quasidistillable iff it is rank-1, i.e. σmc = |ΨmcΨmc|, and s − rank(|Ψk) = d. Adding some extra noise.. Theorem 2 ρ of the form (16) quasidistillable iff among pij = 0, pijpjk . . . pli = 0. (No looping indices) At most d

2

  • non-zero pij

Giuseppe Baio 51 SMP Toruń 17th June 2019 15 / 22

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SLIDE 53

Quasidistillation

Our candidate states are quasidistillable: ˜ ρ(1)

AB = (1 − p10 − p20) |Ψ(1) mcΨ(1) mc| + p10 |1010| + p20 |2020|

˜ ρ(2)

AB = (1 − p10 − p12) |Ψ(2) mcΨ(2) mc| + p10 |1010| + p12 |1212|

˜ ρ(3)

AB = (1 − p20 − p21) |Ψ(3) mcΨ(3) mc| + p20 |2020| + p21 |2121|

  • 10M. Horodecki, P. Horodecki & R. Horodecki, Phys. Rev. A 60, 1888 (1999).

Giuseppe Baio 51 SMP Toruń 17th June 2019 16 / 22

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SLIDE 54

Quasidistillation

Our candidate states are quasidistillable: ˜ ρ(1)

AB = (1 − p10 − p20) |Ψ(1) mcΨ(1) mc| + p10 |1010| + p20 |2020|

˜ ρ(2)

AB = (1 − p10 − p12) |Ψ(2) mcΨ(2) mc| + p10 |1010| + p12 |1212|

˜ ρ(3)

AB = (1 − p20 − p21) |Ψ(3) mcΨ(3) mc| + p20 |2020| + p21 |2121|

Another two-qutrit example10: ρ = ηP +

3 + (1 − η)

3 (|0101| + |1212| + |2020|) (17)

  • 10M. Horodecki, P. Horodecki & R. Horodecki, Phys. Rev. A 60, 1888 (1999).

Giuseppe Baio 51 SMP Toruń 17th June 2019 16 / 22

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SLIDE 55

Numerical results

Constrained optimization in C(ρA, ρB). Rank-4 two-qutrit states11

  • 11K. R. Parthasarathy, Ann. l’Inst. H. Poincaré (B) Prob. Stat. 41, 257 (2005)

Giuseppe Baio 51 SMP Toruń 17th June 2019 17 / 22

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SLIDE 56

Numerical results

Constrained optimization in C(ρA, ρB). Rank-4 two-qutrit states11

  • 11K. R. Parthasarathy, Ann. l’Inst. H. Poincaré (B) Prob. Stat. 41, 257 (2005)

Giuseppe Baio 51 SMP Toruń 17th June 2019 17 / 22

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SLIDE 57

Numerical results

Constrained optimization in C(ρA, ρB). Rank-4 two-qutrit states

Giuseppe Baio 51 SMP Toruń 17th June 2019 18 / 22

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SLIDE 58

Numerical results

Constrained optimization in C(ρA, ρB). Rank-4 two-qutrit states

Giuseppe Baio 51 SMP Toruń 17th June 2019 19 / 22

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SLIDE 59

Conclusions

We identify a family of candidate states as MEMS with fixed marginals Generalization of 2-qubit results + extremal states Quasidistillation and MEMS: a deep connection Numerical results strengthen our conjecture

Giuseppe Baio 51 SMP Toruń 17th June 2019 20 / 22

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SLIDE 60

Future directions

Quasidistillation may help in finding higher dimensional MEMS Fixed marginal purities/other non convex sets Generalization to many body/multipartite entanglement Upper bounds on other measures of interest

Giuseppe Baio 51 SMP Toruń 17th June 2019 21 / 22

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SLIDE 61

Thank you for your attention

Giuseppe Baio 51 SMP Toruń 17th June 2019 22 / 22