Common Information in Secrecy Transformation
Min-Hsiu Hsieh
University of Technology Sydney
[1] Chitambar, Fortescue, MH. Quantum Versus Classical Advantages in Secret Key Distillation. IEEE Transactions on Information Theory (accepted in June 2018).
[2] Chitambar and MH. Round Complexity in the Local Transformations of Quantum and Classical States. Nature Communications 8, no. 2086 (2017).
[3] Chitambar, Fortescue, and MH. A classical analog to entanglement reversibility. Physical Review Letters, vol. 115, p. 090501 (2015)
[4] Chitambar, Fortescue, and MH. Distributions attaining secret key at a rate of the conditional mutual information. Advances in Cryptology - CRYPTO 2015 - 35th Annual Cryptology Conference, pp. 443- 462, (2015).
Gacs-Korner Common Information
[1] Gacs and Korner, “Common information is far less than mutual information,” Problems of Control and Information Theory, vol. 2, no. 2, p. 149, 1973.
JXY:=argmaxK{H(K):0=H(K∣X)=H(K∣Y)}
J_{XY}:=\arg\max_K\{H(K):0=H(K|X)=H(K|Y)\}
Wyner Common Information
[1] Wyner, “The common information of two dependent random variables,” IEEE Transactions on Information Theory, vol. 21, no. 2, pp. 163–179, Mar 1975.
C(X:Y)=minV:X−V−YI(XY:V)
C(X:Y)=\min_{V:X-V-Y} I(XY:V)
[2] Winter, “Secret, public and quantum correlation cost of triples of random variables,” in Proceedings of International Symposium on Information Theory, 2005., Sept 2005, pp. 2270–2274.
[3] Chitambar, MH, and Winter, “The private and public correlation cost of three random variables with collaboration,” IEEE Transactions on Information Theory, vol. 62, no. 4, pp. 2034–2043, April 2016.
Extension to Tripartite
[1] Chitambar, Fortescue, and MH. A classical analog to entanglement reversibility. Physical Review Letters, vol. 115, p. 090501 (2015)
Maximal conditional common variable
:JXY∣Z={JXY∣Z=z}z∈Z
:J_{XY|Z} = \{J_{XY|Z=z}\}_{z\in\mathcal{Z}}
Conditional Common Information
:H(JXY∣Z∣Z)
:H(J_{XY|Z}|Z)
coarse-grained
Conditional Common Information
:H(JXY∣Z)
:H(J_{XY}|Z)
Classes of Tripartite Distributions
Block Independent (BI) Distributions
: I(X:Y∣JXY∣ZZ)=0
:\ I(X:Y|J_{XY|Z}Z)=0
Classes of Tripartite Distributions
Uniform Block Independent (UBI) Distributions
: H(JXY∣Z∣Z)=H(JXY∣Z)
:\ H(J_{XY|Z}|Z)=H(J_{XY}|Z)
Classes of Tripartite Distributions
Uniform block independent under public discussion (UBI-PD)
:
:
∃M s.t. P(MX)(MY)(MZ) is UBI
I(M:JXY∣Z∣Z)=0
Classes of Tripartite Distributions
Uniform block independent under public discussion and eavesdropper’s local processing (UBI-PD↓)
:
:
∃M,Zˉ∣Z s.t. PXY∣Zˉ is UBI
I(Z:JXY∣Zˉ∣MZˉ)=0
Classes of Tripartite Distributions
Semi-unambiguous
:
:
H(Z∣XY)=0
[1] Christandl, Ekert, Horodecki, Horodecki, Oppenheim, and Renner, “Unifying classical and quantum key distillation,” in Theory of Cryptography, vol. 4392, pp. 456–478.
Unambiguous
:
:
H(Z∣XY)=0 & H(XY∣JXY∣ZZ)=0
[2] Ozols, Smith, and Smolin, “Bound entangled states with a private key and their classical counterpart,” Phys. Rev. Lett., vol. 112, p. 110502, Mar 2014.
Classes of Tripartite Distributions
Reversible Secrecy
Determine which pXYZ yielding KD(X:Y∣Z)=KC(X:Y∣Z)
-
KD(X:Y∣Z) is Open!
-
KC(X:Y∣Z)=
min{I(XY:V∣U):XY-Z-U,X-UV-Y}
[1] Winter, “Secret, public and quantum correlation cost of triples of random variables,” in Proceedings of International Symposium on Information Theory, 2005., Sept 2005, pp. 2270–2274.
KD(X:Y∣Z)≤I(X:Y↓Z)≤KC(X:Y∣Z)
where I(X:Y↓Z)=minZˉ∣ZI(X:Y∣Zˉ)
[2] Renner and Wolf, in Advances in Cryptology, EUROCRYPT 2003, pp. 562–577.
If min{∣X∣,∣Y∣}=2, then KD(X:Y∣Z)=KC(X:Y∣Z) iff pXYZ is UBI-PD↓.
[1] Chitambar, Fortescue, and MH. A classical analog to entanglement reversibility. Physical Review Letters, vol. 115, p. 090501 (2015)
If ∣X∣=∣Y∣=2, then KD(X:Y∣Z)=KC(X:Y∣Z) iff pXYZ is UBI.
Open Question:
What state ρAB yields ED(ρ)=EC(ρ)?
Classical and Quantum Key Distillation
[1] Chitambar, Fortescue, MH. Quantum Versus Classical Advantages in Secret Key Distillation. IEEE Transactions on Information Theory (accepted in June 2018).
Quantum Embedding
Ψqqq:=∣ΨABE⟩=∑xyzp(xyz)∣xyz⟩
\Psi_{qqq}:=|\Psi_{ABE}\rangle = \sum_{xyz}\sqrt{p(xyz)}|xyz\rangle
ρAB=Tr∣ΨABE⟩⟨ΨABE∣
\rho_{AB} = \text{Tr}|\Psi_{ABE}\rangle \langle \Psi_{ABE}|
Ψqqq⟶(1)ρcqq⟶(2)ρccq⟶(3)ρccc
\Psi_{qqq}\overset{(1)}{\longrightarrow}\rho_{cqq}\overset{(2)}{\longrightarrow}\rho_{ccq}\overset{(3)}{\longrightarrow}\rho_{ccc}
How does KD(Ψqqq) compare with KD(pXYZ)/KD(ρccc)?
KD(pXYZ)=KD(ρccc)
Show that every quantum LOCC protocol can be transformed into a LOPC protocol.
KD(Ψqqq) ? KD(pXYZ)
The gap can be made arbitrarily large by increasing the number of copies.
Quantum Embedding
∣ΨABE⟩=∑xyzp(xyz)∣xyz⟩
|\Psi_{ABE}\rangle = \sum_{xyz}\sqrt{p(xyz)}|xyz\rangle
- KD(pXYZ): Classical Key Capacity
- KD(ΨABBE): Quantum Key Capacity
- ED(ρ): Distillable Entanglement
- EC(ρ): Entanglement Cost
- EF(ρ): Entanglement Formation
- Er(ρ): Relative entropy of entanglement
- Esq(ρ): squash entanglement
ρAB=Tr∣ΨABE⟩⟨ΨABE∣
\rho_{AB} = \text{Tr}|\Psi_{ABE}\rangle \langle \Psi_{ABE}|
{ED(ρAB)KD(Ψqqq)≤{Er(ρAB)Esq(ρAB)≤EC(ρAB)≤EF(ρAB)
\begin{cases}E_D(\rho^{AB})\\K_D(\Psi_{qqq})\end{cases}\leq\begin{cases} E_{r}(\rho^{AB})\\E_{sq}(\rho^{AB})\end{cases}\leq E_C(\rho^{AB})\leq E_F(\rho^{AB})
If pXYZ is reversible, then KD(pXYZ)≥Esq(ρAB)
If reversible, then ∃ Zˉ∣Z such that pXYZˉ is BI.
{ED(ρAB)KD(Ψqqq)≤{Er(ρAB)Esq(ρAB)≤EC(ρAB)≤EF(ρAB)
\begin{cases}E_D(\rho^{AB})\\K_D(\Psi_{qqq})\end{cases}\leq\begin{cases} E_{r}(\rho^{AB})\\E_{sq}(\rho^{AB})\end{cases}\leq E_C(\rho^{AB})\leq E_F(\rho^{AB})
If pXYZ is UBI-PD, then KD(pXYZ)≥EF(ρAB)
KD(pXYZ)=H(JXY∣Z∣Z)
{ED(ρAB)KD(Ψqqq)≤{Er(ρAB)Esq(ρAB)≤EC(ρAB)≤EF(ρAB)
\begin{cases}E_D(\rho^{AB})\\K_D(\Psi_{qqq})\end{cases}\leq\begin{cases} E_{r}(\rho^{AB})\\E_{sq}(\rho^{AB})\end{cases}\leq E_C(\rho^{AB})\leq E_F(\rho^{AB})
If pXYZ is reversible and Semi-unambiguous ,
then KD(pXYZ)=Esq(⋅)=KD(Ψqqq)
If semi-unambiguous, then Esq(ρAB)≥KD(Ψqqq)≥KD(pXYZ)
[2] Ozols, Smith, and Smolin, “Bound entangled states with a private key and their classical counterpart,” Phys. Rev. Lett., vol. 112, p. 110502, Mar 2014.
{ED(ρAB)KD(Ψqqq)≤{Er(ρAB)Esq(ρAB)≤EC(ρAB)≤EF(ρAB)
\begin{cases}E_D(\rho^{AB})\\K_D(\Psi_{qqq})\end{cases}\leq\begin{cases} E_{r}(\rho^{AB})\\E_{sq}(\rho^{AB})\end{cases}\leq E_C(\rho^{AB})\leq E_F(\rho^{AB})
If pXYZ is UBI-PD and Semi-unambiguous,
then all quantities are equal to H(JXY∣Z∣Z)
Round Complexity
What Tasks Demonstrate a Separation between r−1 and r round LOCC/LOPC?
An Example that Fails to Separate the Rounds
[1] Lo and Popescu, "Concentrating entanglement by local actions: Beyond mean values", Phys. Rev. A 63, 022301
For every r, there exists ρr needing r-round LOCC to achieve ρr→∣ϕ⟩
Construction
Construction
- Bob announces whether Y belongs to {0,1} or {2,3}.
- Eve learns nothing from this announcement.
- Alice knows exactly which Y Bob has
Construction
Bob Starts:
Thank you for your attention!
Common Information in Secrecy Transformation Min-Hsiu Hsieh University of Technology Sydney
Common Information In Quantum Information Science
By Lawrence Min-Hsiu Hsieh
Common Information In Quantum Information Science
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