Typical entanglement of stabilizer states

How entangled is a randomly chosen bipartite stabilizer state? We show that if the number of qubits each party holds is large, the state will be close to maximally entangled with probability exponentially close to 1. We provide a similar tight characterization of the entanglement present in the maximally mixed state of a randomly chosen stabilizer code. Finally, we show that typically very few Greenberger-Horne-Zeilinger states can be extracted from a random multipartite stabilizer state via local unitary operations. Our main tool is a concentration inequality which bounds deviations from the mean of random variables which are naturally defined on the Clifford group.

[1]  E. Lubkin.  Entropy of an n‐system from its correlation with a k‐reservoir , 1978 .

[2]  M. Talagrand.  Concentration of measure and isoperimetric inequalities in product spaces , 1994, math/9406212.

[3]  M. Talagrand.  Majorizing measures: the generic chaining , 1996 .

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

[5]  P. Shor,et al.  Quantum Error-Correcting Codes Need Not Completely Reveal the Error Syndrome , 1996, quant-ph/9604006.

[6]  M. Talagrand.  A new look at independence , 1996 .

[7]  E. Rains.  Entanglement purification via separable superoperators , 1997, quant-ph/9707002.

[8]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[9]  W. Wootters.  Entanglement of Formation of an Arbitrary State of Two Qubits , 1997, quant-ph/9709029.

[10]  P. Shor,et al.  QUANTUM-CHANNEL CAPACITY OF VERY NOISY CHANNELS , 1997, quant-ph/9706061.

[11]  H. Sommers,et al.  Induced measures in the space of mixed quantum states , 2000, quant-ph/0012101.

[12]  Terhal,et al.  Entanglement of formation for isotropic states , 2000, Physical review letters.

[13]  H. Briegel,et al.  Quantum computing via measurements only , 2000, quant-ph/0010033.

[14]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

[15]  M. Ledoux.  The concentration of measure phenomenon , 2001 .

[16]  A. Winter.  ‘‘Extrinsic’’ and ‘‘Intrinsic’’ Data in Quantum Measurements: Asymptotic Convex Decomposition of Positive Operator Valued Measures , 2001, quant-ph/0109050.

[17]  Debbie W. Leung,et al.  Quantum data hiding , 2002, IEEE Trans. Inf. Theory.

[18]  P. Shor.  Equivalence of Additivity Questions in Quantum Information Theory , 2003, quant-ph/0305035.

[19]  B. Moor,et al.  Clifford group, stabilizer states, and linear and quadratic operations over GF(2) , 2003, quant-ph/0304125.

[20]  P. Shor,et al.  The Capacity of a Quantum Channel for Simultaneous Transmission of Classical and Quantum Information , 2003, quant-ph/0311131.

[21]  Seth Lloyd,et al.  Pseudo-Random Unitary Operators for Quantum Information Processing , 2003, Science.

[22]  A. Winter,et al.  Randomizing Quantum States: Constructions and Applications , 2003, quant-ph/0307104.

[23]  Andreas J. Winter,et al.  A family of quantum protocols , 2004, ISIT.

[24]  Isaac L. Chuang,et al.  Entanglement in the stabilizer formalism , 2004 .

[25]  A. Harrow,et al.  Superdense coding of quantum states. , 2003, Physical review letters.

[26]  Seth Lloyd,et al.  Convergence conditions for random quantum circuits , 2005, quant-ph/0503210.

[27]  P. Hayden,et al.  On the (Im)Possibility of Quantum String Commitment , 2005 .

[28]  Martin B. Plenio,et al.  Entanglement on mixed stabilizer states: normal forms and reduction procedures , 2005, quant-ph/0505036.

[29]  Igor Devetak.  The private classical capacity and quantum capacity of a quantum channel , 2005, IEEE Transactions on Information Theory.

[30]  Patrick Hayden,et al.  Multiparty data hiding of quantum information , 2005 .

[31]  John A. Smolin,et al.  Degenerate coding for Pauli channels , 2006 .

[32]  Oscar C. O. Dahlsten,et al.  Entanglement probability distribution of bi-partite randomised stabilizer states , 2006, Quantum Inf. Comput..

[33]  A. Winter,et al.  Aspects of Generic Entanglement , 2004, quant-ph/0407049.

[34]  D. Gottesman,et al.  GHZ extraction yield for multipartite stabilizer states , 2005, quant-ph/0504208.

[35]  M B Plenio,et al.  Generic entanglement can be generated efficiently. , 2007, Physical review letters.