GHZ - maximal tripartite entanglement; no remnant bipartite entanglement if one mode .... [4] G. Adesso and F. Illuminati, Continuous variable tangle, monogamy ...
ALICE AND CLARE KEEP THEIR RELATIONSHIP SECRET FROM BOB M. K. Olsen
E. G. Cavalcanti
School of Mathematics and Physics Centre for Quantum Dynamics University of Queensland Griffith University Brisbane AUSTRALIA
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QUANTUM KEY DISTRIBUTION
• QKD is the first mature quantum technology
• Allows for the creation of a secret key between authorised partners connected by quantum and classical channels
• First protocol by Bennett and Brassard – based on ideas presented by Wiesner
• Three different categories with different levels of security and different necessary degrees of entanglement • Standard QKD requires entanglement. One-sided device independent QKD (1SDI-QKD) requires correlations at level of EPR steering. Device-independent QKD requires correlations at level of Bell violations.
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TRIPARTITE ENTANGLEMENT CLASSIFICATIONS
• Two famous tripartite entangled states for discrete variable systems - GHZ and W
• GHZ - maximal tripartite entanglement; no remnant bipartite entanglement if one mode traced over
• W - maximal tripartite entanglement; remnant bipartite entanglement if one mode traced over
• Situation not quite so simple for continuous variable systems - no maximal entanglement ever
• Adesso et al. have divided CV tripartite states into five different classes
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THE FIVE CLASSIFICATIONS
• Class 1 States not separable under all the three possible bipartitions
• Class 2 States separable under only one of the three possible bipartitions
• Class 3 States separable under only two of the three possible bipartitions
• Class 4 States separable under all three possible bipartitions, but impossible to write as a convex sum of tripartite products of pure one-mode states
• Class 5 States that are separable under all the three possible bipartitions, and can be written as a convex sum of tripartite products of pure one-mode states fully separable states.
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THE THREE ENTANGLEMENT TYPES
• Adesso et al. also defined three different types of tripartite entangled CV states, all applicable to optics. • GHZ type. This is fully symmetric and would give perfect violation of entanglement inequalities in the unphysical limit of infinite pumping. It shows no remnant bipartite entanglement. • W type. Can also exhibit maximal tripartite entanglement, but also shows remnant bipartite entanglement. Canonically written as |ψW i = √13 (|1 0 0i + |0 1 0i + |0 0 1i) • T type. A state which exhibits tripartite entanglement only, without being in the GHZ regime.
• In quantum optical systems, the type may vary according to the input fields. A state can be of W type at one power level, and T at another. We show that this flexibility actually has practical uses.
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ENTANGLEMENT AND EPR CORRELATIONS
ˆi ± X ˆ j ) + V (Yˆi ∓ Yˆj ) one less than 4 DSij± = V (X ˆ i)V inf (Yˆi) ΠVij = V inf (X
i inferred from j, less than one
ˆi − X ˆ j ) + V (Yˆi + Yˆj + gk Yˆk ) two less than 4 Vij = V (X
Vijk
ˆj + X ˆk X Yˆj + Yˆk ˆ ˆ = V ( Xi − √ ) + V (Yi + √ ) one less than 4 2 2
• Π(3)Vi and Π(3)Vij are tripartite generalisation of the EPR correlations. • Π(3)Vi < 1 or Π(3)Vij < 4 denote EPR steering.
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A FULLY SYMMETRIC SCHEME
• Triply concurrent downconversion with three pump fields equal gives undepleted pump interaction Hamiltonian H = −i~κ [ˆa1aˆ2 + aˆ1aˆ3 + aˆ2aˆ3] + h.c. • Can solve Heisenberg equations of motion analytically as function of κt • Tripartite EPR and entanglement increase with κt • Bipartite entanglement found for smallish κt - bipartite EPR not found • No pair of Alice, Bob, and Clare can use 1SDI-QKD between themselves. All three are on an equal footing. • Output can be of W or T types, GHZ in unphysical limit.
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VARIOUS CORRELATIONS
5 4.5 4
correlations
3.5
DS-ij
3 2.5
V ijk
2 1.5 1
0
V ij
(3)
0.5
Π V 0
ij
0.5
1
1.5
2
κt
Note T type as interaction increases, W type for smaller interaction. No bipartite EPR.
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OUR ASYMMETRIC SCHEME
• A combination of downconversion and sum-frequency generation with pump field ˆb at ω0 • Two downconverted fields at ω1 and ω3, high frequency at ω2 = ω0 + ω3 • Alice has mode 1, Bob has mode 2 and Clare has mode 3 • Again use undepleted pump approximation - advantage is analytic solutions H = i~κ1
• Solutions in terms of ζ =
p
aˆ†1aˆ†3
− aˆ1aˆ3
† † + i~κ2 aˆ3aˆ2 − aˆ3aˆ2
κ21 − κ22 and we use κ2 = 0.6κ1
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ENTANGLEMENT CORRELATIONS
5 4.5 4
correlations
3.5
V 312
3 2.5
-
DS13
2
V 123
1.5 1 0.5
V 13 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
ζt
Note Bob doesn’t share bipartite entanglement with either Alice or Clare
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BIPARTITE EPR STEERING
2 1.8 1.6 1.4
Π V13
Π Vij
1.2 1 0.8 0.6
Π V31
0.4 0.2 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
ζt
Alice and Clare share the ability to steer each other and can potentially use 1SDI-QKD.
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THE BIT RATE −1
2e p Π Vij
Kmin ≤ log
!
1.5
1
K min
0.5
K 13
K 31
0
K sym
-0.5
-1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
ζt
Minimum bit rate larger than zero means 1SDI-QKD is practical.
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THE INTRACVITY SYSTEM ALSO WORKS 1.5
Π V12 Π V21
Π Vij
1
0.5
Π V13
Π V31
0 -10
-5
0
5
10
ω (units of γ 1)
Marginal EPR between Alice and Bob at 80% of threshhold,
Alice can steer Bob but not vice-versa
χ1 = 10−2, χ2 = 0.6χ1
γj6=2 = 1, γ2 = 3
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CHECK THE BIT RATE JUST IN CASE 2
1.5
Alice and Clare
K min
1
0.5
0
Alice and Bob -0.5 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ǫ/ǫ c
Bob is still out of the loop
Positive bit rate for Alice and Clare over most of the range
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CONCLUSIONS AND WARNINGS
• Asymmetric systems offer extra flexibility not available with symmetric ones • Bob cannot participate in any level of QKD with Alice or Clare individually • In the convoluted world of security and cryptography, this may have some practical use • We have not proposed a particular protocol, just depends on homodyne measurements • Three-mode undepleted pump analyses have limited accuracy, but they give an idea • Spectral analysis with intracavity system shows that system still works • Now to interest an experimental group
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RELATED WORKS [1] V. Scarani, H. Bechmann-Pasquinicci, N.J. Cerf, M. Du˘sek, N. Lutkenhaus, and M. Peev, The security of practical quantum key distribution, Rev. Mod. Phys. 81, 1301 (2009). [2] C. Branciard, E.G. Cavalcanti, S.P. Walborn, V. Scarani, and H.M. Wiseman, One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering, Phys. Rev. A 85, 010301 (2012). [3]G. Adesso, A. Serafini, and F. Illuminati, Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence, Phys. Rev. A 73, 032345 (2006). [4] G. Adesso and F. Illuminati, Continuous variable tangle, monogamy inequality, and entanglement sharing in Gaussian states of continuous variable systems, New. J. Phys. 8, 15 (2006). [5]T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, and A. Furusawa, Experimental Creation of a Fully Inseparable Tripartite Continuous-Variable State, Phys. Rev. Lett. 91, 080404 (2003). [6] A.S. Bradley, M.K. Olsen, O. Pfister, and R.C. Pooser, Bright tripartite entanglement in triply concurrent parametric oscillation, Phys. Rev. A 72 053805 (2005). [7] M.E. Smithers and E.Y.C. Lu, Quantum theory of coupled parametric down-conversion and up-conversion with simultaneous phase matching, Phys. Rev. A 10, 1874 (1974). [8] M.K. Olsen and E.G. Cavalcanti, The versatility of continuous-variable tripartite entanglement allows Alice and Clare to keep secrets from Bob, arXiv:1510.01821 ——————————————
