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.
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)=\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
:J_{XY|Z} = \{J_{XY|Z=z}\}_{z\in\mathcal{Z}}
Conditional Common Information
:H(J_{XY|Z}|Z)
coarse-grained
Conditional Common Information
:H(J_{XY}|Z)
Classes of Tripartite Distributions
Block Independent (BI) Distributions
:\ I(X:Y|J_{XY|Z}Z)=0
Classes of Tripartite Distributions
Uniform Block Independent (UBI) Distributions
:\ H(J_{XY|Z}|Z)=H(J_{XY}|Z)
Classes of Tripartite Distributions
Uniform block independent under public discussion (UBI-PD)
:
\(\exist M\) s.t. \(P_{(MX)(MY)(MZ)}\) is UBI
\(I(M:J_{XY|Z}|Z)=0\)
Classes of Tripartite Distributions
Uniform block independent under public discussion and eavesdropper’s local processing (UBI-PD\(\downarrow\))
:
\(\exist M, \bar{Z}|Z\) s.t. \(P_{XY|\bar{Z}}\) is UBI
\(I(Z:J_{XY|\bar{Z}}|M\bar{Z})=0\)
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
:
\(H(Z|XY)=0\) & \(H(XY|J_{XY|Z}Z)=0\)
[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 \(p_{XYZ}\) yielding \(K_D(X:Y|Z)= K_C(X:Y|Z)\)
-
\(K_D(X:Y|Z)\) is Open!
-
\(K_C(X:Y|Z)=\)
\(\min\{I(XY:V|U):\text{XY-Z-U},\text{X-UV-Y}\}\)
[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.
\(K_D(X:Y|Z)\leq I(X:Y\downarrow Z)\leq K_C(X:Y|Z)\)
where \(I(X:Y\downarrow Z)=\min_{\bar{Z}|Z} I(X:Y|\bar{Z})\)
[2] Renner and Wolf, in Advances in Cryptology, EUROCRYPT 2003, pp. 562–577.
If \(\min\{|\mathcal{X}|,|\mathcal{Y}|\}=2\), then \(K_D(X:Y|Z)= K_C(X:Y|Z)\) iff \(p_{XYZ}\) is UBI-PD\(\downarrow\).
[1] Chitambar, Fortescue, and MH. A classical analog to entanglement reversibility. Physical Review Letters, vol. 115, p. 090501 (2015)
If \(|\mathcal{X}|=|\mathcal{Y}|=2\), then \(K_D(X:Y|Z)= K_C(X:Y|Z)\) iff \(p_{XYZ}\) is UBI.
Open Question:
What state \(\rho_{AB}\) yields \(E_D(\rho)=E_C(\rho)\)?
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
\Psi_{qqq}:=|\Psi_{ABE}\rangle = \sum_{xyz}\sqrt{p(xyz)}|xyz\rangle
\rho_{AB} = \text{Tr}|\Psi_{ABE}\rangle \langle \Psi_{ABE}|
\Psi_{qqq}\overset{(1)}{\longrightarrow}\rho_{cqq}\overset{(2)}{\longrightarrow}\rho_{ccq}\overset{(3)}{\longrightarrow}\rho_{ccc}
How does \(K_D(\Psi_{qqq})\) compare with \(K_D(p_{XYZ})\)/\(K_D(\rho_{ccc})\)?
\(K_D(p_{XYZ})\)=\(K_D(\rho_{ccc})\)
Show that every quantum LOCC protocol can be transformed into a LOPC protocol.
\(K_D(\Psi_{qqq})\) ? \(K_D(p_{XYZ})\)
The gap can be made arbitrarily large by increasing the number of copies.
Quantum Embedding
|\Psi_{ABE}\rangle = \sum_{xyz}\sqrt{p(xyz)}|xyz\rangle
- \(K_D(p_{XYZ})\): Classical Key Capacity
- \(K_D(\Psi_{ABBE})\): Quantum Key Capacity
- \(E_D(\rho)\): Distillable Entanglement
- \(E_C(\rho)\): Entanglement Cost
- \(E_F(\rho)\): Entanglement Formation
- \(E_r(\rho)\): Relative entropy of entanglement
- \(E_{sq}(\rho)\): squash entanglement
\rho_{AB} = \text{Tr}|\Psi_{ABE}\rangle \langle \Psi_{ABE}|
\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 \(p_{XYZ}\) is reversible, then \(K_D(p_{XYZ})\geq E_{sq}(\rho^{AB})\)
If reversible, then \(\exist \ \bar{Z}|Z\) such that \(p_{XY\bar{Z}}\) is BI.
\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 \(p_{XYZ}\) is UBI-PD, then \(K_D(p_{XYZ})\geq E_F(\rho^{AB})\)
\(K_D(p_{XYZ})=H(J_{XY|Z}|Z)\)
\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 \(p_{XYZ}\) is reversible and Semi-unambiguous ,
then \(K_D(p_{XYZ})= E_{sq}(\cdot)=K_D(\Psi_{qqq})\)
If semi-unambiguous, then \(E_{sq}(\rho^{AB})\geq K_D(\Psi_{qqq})\geq K_D(p_{XYZ})\)
[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.
\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 \(p_{XYZ}\) is UBI-PD and Semi-unambiguous,
then all quantities are equal to \(H(J_{XY|Z}|Z)\)
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 \(\rho_r\) needing \(r\)-round LOCC to achieve \(\rho_r\to|\phi\rangle\)
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 Quantum Information Science
By Lawrence Min-Hsiu Hsieh
