Trading resources in quantum Shannon theory

Trading resources in quantum Shannon theory Mark M. Wilde Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana, USA [email protected]

Based on arXiv:1709.01111, 1605.04922, 1206.4886, 1105.0119, 1004.0458, 1001.1732, 0901.3038, 0811.4227 (with Bradler, Guha, Hayden, Hsieh, Leditzky, Qi, Touchette) and Chapter 25 of arXiv:1106.1445

CQuIC Seminar, University of New Mexico, January 31, 2019

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Main message Question: What are the net rates at which a sender and receiver can generate classical communication, quantum communication, and entanglement by using a quantum channel many times?

Many special cases are known, such as the classical capacity theorem [Hol98,, SW97], [Hol98 SW97], quantum capacity theorem [Sch96, SN96 [Sch96, SN96,, BNS98, BKN00, BKN00, Llo97, Sho02, Sho02, Dev05], Dev05], and the entanglement-assisted classical capacity theorem [BSST02] [BSST02] A priori, this question might seem challenging, but there is a













Background — quantum mechanics Quantum channels Quantum channels represent noisy physical evolutions of quantum systems. Mathematically, a quantum channel is a linear, completely positive, trace-preserving map, thus taking an input quantum state to an output quantum state. Quantum channels are usually denoted by , , , etc.

N M P

Isometric extensions of quantum channels Every quantum channel has an isometric extension: There exists an isometry taking an input density operator to the tensor-product Hilbert space of output and environment. Channel is realized by applying isometry













Background — resources Resources [Ben04 [Ben04,, DHW04, DHW08] DHW08]

→ →

Let [c c ] denote a noiseless classical bit channel from Alice (sender) to Bob (receiver), which performs the following mapping on a qubit density operator ρ=

→ →



ρ00 ρ01 ρ10 ρ11

→

ρ00 0 0 ρ11



Let [q q ] denote a noiseless quantum bit channel from Alice to Bob, which perfectly preserves a qubit density operator.













Entanglement distribution













Super-dense coding [BW92 [ BW92]]

Alice and Bob share an ebit. Alice would like to transmit two classical















Teleportation [BBC [BBC+ 93 93]]













Combining protocols [HW10 [HW10]] Think of each protocol as a rate triple (C , Q , E )

−

Entanglement distribution is (0, (0, 1, 1)

− −

Super-dense coding is (2, (2, 1, 1)

−

−

Teleportation is ( 2, 1, 1)













Unit resource capacity region [HW10 [ HW10]]













Trading resources using a quantum channel Main question: What net rates of classical communication, quantum communication, and entanglement generation can we achieve by using a quantum channel many times?

N













Trading resources using a quantum channel















What to do?













Background — entropies













Quantum dynamic capacity theorem (setup) [WH12b [ WH12b]]













Quantum dynamic capacity theorem (statement) [WH12b [ WH12b]]













Example: Qubit dephasing channel







































































































































Example: Quantum-limited amplifier channel

Amplifier channel is defined from the following input-output relation: aˆ

→

ˆ= b

√

√ − 1 e eˆ †, G aˆ + G −

where aˆ is the input annihilation operator for the sender, e ˆ is the input annihilation operator for the environment, and G [1 [ 1, ) is the gain of the channel. The channel is quantum-limited if the environment is prepared in a vacuum state.

∈ ∈ ∞













Example: Amplifier channel [QW17 [ QW17]]

The quantum dynamic capacity region for a quantum-limited amplifier channel with gain G 1 is the union of regions of the form:

≥ ≥ ≤ g (λN ) + g (GN + G ¯) − g (G ¯[λN + 1]), C + 2Q ≤ 1]), 1]), ≤ g (G λN + G ¯) − g (G ¯[λN + 1]), Q + E ≤ ≤ g (GN + G ¯) − g (G ¯[λN + 1]), C + Q + E ≤ 1]), ¯ = G − where G [0,, 1] is a photon-number-sharing parameter − 1 and λ ∈ [0 S S

S S

S S

S S

S S

S S

S S

and g (N ) is the entropy of a thermal state with mean photon number N













Example: Amplifier channel [QW17 [ QW17]]













Recent and other developments Trade-off between secret key, private classical communication, and public classical communication [WH12a [WH12a]] Bounds for triple trade-off capacities using operator space and complex interpolation theory [GJL18] theory [GJL18] Many examples of superadditive effects for the trade-off capacities [ZZS17, ZZHS17]. ZZHS17]. Just because one capacity is single-letter, it doesn’t mean that double- or triple-resource trade-off is single-letter.













Conclusion Summary The quantum dynamic capacity theorem characterizes the net rates at which a sender and a receiver can generate classical communication, quantum communication, and entanglement by using a quantum channel many times The region simplifies for several channels of interest

Open questions Is there a simple characterization for distillation tasks? For progress,















References I [BBC+ 93] Charles Charles H. Benne Bennett, tt, Gilles Brass Brassard ard,, Claude Cr´ epeau, Richard Jozsa, Asher epeau, Peres, and William K. Wootters. Teleporting Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical channels. Physical Review Letters , 70(13):1895–1899, March 1993. [BDH+ 14] Cha Charles rles H. Bennett, Bennett, Igor Devetak Devetak,, Ara Aram m W. Harro Harrow, w, Peter Peter W. Sho Shor, r, and













References II [BSST0 [BS ST02] 2]

Charles H. Benne Charles Bennett, tt, Pete Peterr W. Shor, Shor, John A. Smolin, Smolin, and Ashish Ashish V. Thapliyal. Entanglement-assisted Thapliyal. Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE theorem. IEEE Transactions on Information Theory , 48(10):2637–26 48(10) :2637–2655, 55, October 2002. arX arXiv:qua iv:quant-ph/01 nt-ph/0106052. 06052.

[BW9 [B W92] 2]

Chaarle Ch rless H. Be Benne nnett tt an and d St Step ephe hen n J. Wi Wies esne ner. r. Communication Communication via one- and













References III [GJL [G JL18] 18]

Li Ga Gao, o, Ma Mari rius us Jun Junge ge,, an and d Ni Nich chola olass La LaRac Racue uent nte. e. Capacity Capacity bounds via operator space methods. Journal methods. Journal of Mathematical Physics , 59(12):122202, 2018. arXiv:1509.07294.

[Har04 04]]

Aram Harrow. Coherent communication of classical messages. Physical messages. Physical Review Letters , 92(9):097902, March 2004. arXiv:quant-ph/0307091.













References IV [LKDW18] [LKD W18] Felix Felix Leditzky, Leditzky, Eneet Kaur, Nilanjana Datta, and Mark M. Wilde. Approaches for approximate additivity of the Holevo information of quantum channels. Physical channels. Physical Review A, A, 97(1):012332, January 2018. arXiv:1709.01111. [Llo97]

Seth Lloyd. Capacity of the noisy quantum channel. Physical channel. Physical Review A, A, 55(3):1613–1622, March 1997. arXiv:quant-ph/9604015.













References V [SN9 [S N96] 6]

Benjam Benj amin in Sc Schu huma mach cher er an and d Mi Mich chae aell A. Ni Niel else sen. n. Quantum Quantum data processing and error correction. Physical correction. Physical Review A, A, 54(4):2629–2635, October 1996. arXiv:quant-ph/9604022.

[SW9 [S W97] 7]

Benjam Benj amin in Sc Schum humac ache herr an and d Mi Mich chae aell D. Wes estmo tmore rela land. nd. Sending Sending classical information via noisy quantum channels. Physical channels. Physical Review A, A, 56(1):131–138,















References VI

Trading resources in quantum Shannon theory

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