Potential Capacities of Quantum Channels

We introduce potential capacities of quantum channels in an operational way and provide upper bounds for these quantities, which quantify the ultimate limit of usefulness of a channel for a given task in the best possible context. Unfortunately, except for a few isolated cases, potential capacities seem to be as hard to compute as their plain analogues. We thus study upper bounds on some potential capacities. For the classical capacity, we give an upper bound in terms of the entanglement of formation. To establish a bound for the quantum and private capacity, we first lift the channel to a Hadamard channel and then prove that the quantum and private capacity of a Hadamard channel is strongly additive, implying that for these channels, potential and plain capacity are equal. Employing these upper bounds, we show that if a channel is noisy, however close it is to the noiseless channel, then it cannot be activated into the noiseless channel by any other contextual channel; this conclusion holds for all the three capacities. We also discuss the so-called environment-assisted quantum capacity, because we are able to characterize its potential version.

[1]  A. Winter,et al.  Remarks on Additivity of the Holevo Channel Capacity and of the Entanglement of Formation , 2002, quant-ph/0206148.

[2]  Mark M. Wilde,et al.  The quantum dynamic capacity formula of a quantum channel , 2010, Quantum Inf. Process..

[3]  M. Horodecki,et al.  Irreversibility for all bound entangled states. , 2005, Physical Review Letters.

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

[5]  Ning Cai,et al.  Quantum privacy and quantum wiretap channels , 2004, Probl. Inf. Transm..

[6]  Howard Barnum,et al.  On quantum fidelities and channel capacities , 2000, IEEE Trans. Inf. Theory.

[7]  M. Ruskai,et al.  Entanglement Breaking Channels , 2003, quant-ph/0302031.

[8]  A. Winter,et al.  Quantum privacy and quantum wiretap channels , 2004 .

[9]  Alexander S. Holevo,et al.  The Capacity of the Quantum Channel with General Signal States , 1996, IEEE Trans. Inf. Theory.

[10]  Peter W. Shor,et al.  Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem , 2001, IEEE Trans. Inf. Theory.

[11]  J Eisert,et al.  Entangled inputs cannot make imperfect quantum channels perfect. , 2010, Physical review letters.

[12]  S. Lloyd.  Capacity of the noisy quantum channel , 1996, quant-ph/9604015.

[13]  Schumacher,et al.  Sending entanglement through noisy quantum channels. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[14]  Runyao Duan,et al.  A new property of the Lovász number and duality relations between graph parameters , 2015, Discret. Appl. Math..

[15]  David Elkouss,et al.  Unbounded number of channel uses may be required to detect quantum capacity , 2014, Nature Communications.

[16]  David Elkouss,et al.  Superadditivity of private information for any number of uses of the channel , 2015, Physical review letters.

[17]  Graeme Smith,et al.  Extensive nonadditivity of privacy. , 2009, Physical review letters.

[18]  Christopher King,et al.  Properties of Conjugate Channels with Applications to Additivity and Multiplicativity , 2005 .

[19]  Michael D. Westmoreland,et al.  Entanglement and perfect quantum error correction , 2002, quant-ph/0201061.

[20]  Andreas Winter.  On environment-assisted capacities of quantum channels , 2005 .

[21]  Graeme Smith.  Private classical capacity with a symmetric side channel and its application to quantum cryptography , 2007, 0705.3838.

[22]  M. Hastings.  Superadditivity of communication capacity using entangled inputs , 2009 .

[23]  Schumacher,et al.  Quantum data processing and error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

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

[25]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[26]  David Elkouss,et al.  qu an tph ] 2 1 A ug 2 01 4 Unbounded number of channel uses are required to see quantum capacity , 2014 .

[27]  Dave Touchette,et al.  Trade-off capacities of the quantum Hadamard channels , 2010, ArXiv.

[28]  Andreas J. Winter,et al.  The Quantum Capacity With Symmetric Side Channels , 2008, IEEE Transactions on Information Theory.

[29]  A. Winter,et al.  Communications in Mathematical Physics Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality , 2022 .

[30]  C. Adami,et al.  VON NEUMANN CAPACITY OF NOISY QUANTUM CHANNELS , 1996 .

[31]  A. Winter,et al.  Private capacity of quantum channels is not additive. , 2009, Physical review letters.

[32]  Axthonv G. Oettinger,et al.  IEEE Transactions on Information Theory , 1998 .

[33]  John A. Smolin,et al.  Entanglement of assistance and multipartite state distillation , 2005 .

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

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

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

[37]  M. Nielsen,et al.  Information transmission through a noisy quantum channel , 1997, quant-ph/9702049.

[38]  Graeme Smith,et al.  Quantum Communication with Zero-Capacity Channels , 2008, Science.

[39]  Michael D. Westmoreland,et al.  Sending classical information via noisy quantum channels , 1997 .

[40]  Graeme Smith,et al.  Additive extensions of a quantum channel , 2007, 2008 IEEE Information Theory Workshop.

[41]  Mario Berta,et al.  Entanglement cost of quantum channels , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.