Single-shot Quantum State Merging

We consider an unknown quantum state shared between two parties, Alice and Bob, and ask how much quantum communication is needed to transfer the full state to Bob. This problem is known as state merging and was introduced in [Horodecki et al., Nature, 436, 673 (2005)]. It has been shown that for free classical communication the minimal number of quantum bits that need to be sent from Alice to Bob is given by the conditional von Neumann entropy. However this result only holds asymptotically (in the sense that Alice and Bob share initially many identical copies of the state) and it was unclear how much quantum communication is necessary to merge a single copy. We show that the minimal amount of quantum communication needed to achieve this single-shot state merging is given by minus the smooth conditional min-entropy of Alice conditioned on the environment. This gives an operational meaning to the smooth conditional min-entropy.

[1]  W. Stinespring Positive functions on *-algebras , 1955 .

[2]  N. S. Barnett,et al.  Private communication , 1969 .

[3]  Donald E. Knuth,et al.  Big Omicron and big Omega and big Theta , 1976, SIGA.

[4]  A. Uhlmann The "transition probability" in the state space of a ∗-algebra , 1976 .

[5]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[6]  R. Jozsa Fidelity for Mixed Quantum States , 1994 .

[7]  Jeroen van de Graaf,et al.  Cryptographic Distinguishability Measures for Quantum-Mechanical States , 1997, IEEE Trans. Inf. Theory.

[8]  Masahito Hayashi,et al.  General formulas for capacity of classical-quantum channels , 2003, IEEE Transactions on Information Theory.

[9]  Renato Renner,et al.  Smooth Renyi entropy and applications , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[10]  Renato Renner,et al.  Security of quantum key distribution , 2005, Ausgezeichnete Informatikdissertationen.

[11]  Robert König,et al.  Universally Composable Privacy Amplification Against Quantum Adversaries , 2004, TCC.

[12]  Andreas Winter,et al.  Partial quantum information , 2005, Nature.

[13]  M. Horodecki,et al.  Quantum State Merging and Negative Information , 2005, quant-ph/0512247.

[14]  Nilanjana Datta,et al.  Beyond i.i.d. in Quantum Information Theory , 2006, 2006 IEEE International Symposium on Information Theory.

[15]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[16]  Nilanjana Datta,et al.  Smooth Entropies and the Quantum Information Spectrum , 2009, IEEE Transactions on Information Theory.