Dual entanglement measures based on no local cloning and no local deleting (12 pages)

The impossibility of cloning and deleting of unknown states constitute important restrictions on processing of information in the quantum world. On the other hand, a known quantum state can always be cloned or deleted. However, if we restrict the class of allowed operations, there will arise restrictions on the ability of cloning and deleting machines. We have shown that cloning and deleting of known states is in general not possible by local operations. This impossibility hints at quantum correlation in the state. We propose dual measures of quantum correlation based on the dual restrictions of no local cloning and no local deleting. The measures are relative entropy distances of the desired states in a (generally impossible) perfect local cloning or local deleting process from the best approximate state that is actually obtained by imperfect local cloning or deleting machines. Just like the dual measures of entanglement cost and distillable entanglement, the proposed measures are based on important processes in quantum information. We discuss their properties. For the case of pure states, estimations of these two measures are also provided. Interestingly, the entanglement of cloning for a maximally entangled state of two two-level systems is not unity.

[1]  A. Winter,et al.  “Squashed entanglement”: An additive entanglement measure , 2003, quant-ph/0308088.

[2]  M. Horodecki,et al.  The uniqueness theorem for entanglement measures , 2001, quant-ph/0105017.

[3]  M. Horodecki,et al.  The asymptotic entanglement cost of preparing a quantum state , 2000, quant-ph/0008134.

[4]  S. Braunstein,et al.  Impossibility of deleting an unknown quantum state , 1999, Nature.

[5]  M. Horodecki,et al.  Limits for entanglement measures. , 1999, Physical review letters.

[6]  M. Plenio,et al.  MINIMAL CONDITIONS FOR LOCAL PURE-STATE ENTANGLEMENT MANIPULATION , 1999, quant-ph/9903054.

[7]  L. Hardy.  METHOD OF AREAS FOR MANIPULATING THE ENTANGLEMENT PROPERTIES OF ONE COPY OF A TWO-PARTICLE PURE ENTANGLED STATE , 1999, quant-ph/9903001.

[8]  M. Nielsen.  Conditions for a Class of Entanglement Transformations , 1998, quant-ph/9811053.

[9]  E. Rains.  RIGOROUS TREATMENT OF DISTILLABLE ENTANGLEMENT , 1998, quant-ph/9809078.

[10]  M. Horodecki,et al.  General teleportation channel, singlet fraction and quasi-distillation , 1998, quant-ph/9807091.

[11]  G. Vidal.  On the characterization of entanglement , 1998 .

[12]  M. Horodecki,et al.  Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature? , 1998, quant-ph/9801069.

[13]  M. Plenio,et al.  Entanglement measures and purification procedures , 1997, quant-ph/9707035.

[14]  D. Bruß,et al.  Optimal universal and state-dependent quantum cloning , 1997, quant-ph/9705038.

[15]  V. Buzek,et al.  Broadcasting of entanglement via local copying , 1997, quant-ph/9701028.

[16]  M. Hillery,et al.  Quantum copying: Beyond the no-cloning theorem. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[17]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[18]  Charles H. Bennett,et al.  Concentrating partial entanglement by local operations. , 1995, Physical review. A, Atomic, molecular, and optical physics.

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

[20]  Horace P. Yuen,et al.  Amplification of quantum states and noiseless photon amplifiers , 1986 .

[21]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[22]  Michal Horodecki,et al.  Entanglement measures , 2001, Quantum Inf. Comput..