converse bounds for private communication over quantum …€¦ · meta-converse approach starts by...

Converse bounds for private communication overquantum channels

Mark M. Wilde (LSU), Marco Tomamichel (Univ. Sydney),and Mario Berta (Caltech)

Based on arXiv:1602.08898 [WTB16]

QCrypt 2016, Washington DC, September 12, 2016

Mark M. Wilde (LSU), Marco Tomamichel (Univ. Sydney),and Mario Berta (Caltech) (LSU) 1 / 43

Setting

Given is a quantum channel N and a QKD protocol that uses it n times:

B2’’

B2A’2

NA2’’

B1’’

B1A’1

LOCCLOCC

NA1’’

B’’

NA’’LOCC

Non-asymptotic private capacity: Maximum rate of ε-close secret keyachievable using channel n times and NS mean photons per channel use:

P↔N (n,NS , ε) ≡ sup P : (n,P,NS , ε) is achievable for N using ↔ .

If no photon number constraint, then consider

P↔N (n, ε) = supNS≥0

P↔N (n,NS , ε).

Main question

Practical question: How to characterize P↔N (n,NS , ε) for all n ≥ 1,NS ≥ 0, and ε ∈ (0, 1)?

How to characterize P↔N (n, ε) for all n ≥ 1 and ε ∈ (0, 1)?

The answers give the fundamental limitations of QKD.

Upper bounds on P↔N (n,NS , ε) and P↔N (n, ε) can be used asbenchmarks for quantum repeaters [TGW14].

This talk discusses the tightest known upper bound on P↔N (n, ε) forchannels of practical interest and thus represents the best knownbenchmark for quantum repeaters [WTB16].

What was known before?

Begin by reviewing what is known

Let’s leap back to QCrypt 2014:

Takeoka presented results of [TGW14].

[TGW14] bound with energy constraint

Most interested in the photon loss channel:

Lη : b =√ηa +

√1− ηe

where transmissivity η ∈ [0, 1] and environment in vacuum state.

Practical question is tough, so consider limiting cases. . .

[TGW14] bound: Consider the limit as n→∞ and then ε→ 0:

limε→0

limn→∞

P↔Lη(n,NS , ε) ≤ g((1 + η)NS/2)− g((1− η)NS/2)

where g(x) ≡ (x + 1) log2(x + 1)− x log2 x

is entropy of bosonic thermal state with mean photon number x .

Based on the squashed entanglement measure [CW04].

[TGW14] bound without energy constraint

Optimizing over energy gives the unconstrained [TGW14] bound:

limε→0

limn→∞

P↔Lη(n, ε) ≤ log2

(1 + η

1− η

essentially because supNS≥0 g((1 + η)NS/2)− g((1− η)NS/2) = log2

(1+η1−η

[TGW14] established existence of a fundamental rate-loss trade-off forany possible QKD protocol that uses a photon-loss channel.

Bound is finite for all η ∈ [0, 1) and depends only on η.

Main drawback is that it is an asymptotic statement and thus haslimited applicability in practice.

(Original proof didn’t address issue with unbounded shield systems — now fixed

in [W16])

Fundamental rate-loss trade-off from [TGW14]

Squashed-ent UB

Rev-coh-info LB

Ideal BB84

Imperfect BB84 HdecoyLIdeal CV

Imperfect CV HUCDLImperfect CV HCDL

0 10 20 30 4010-6

Channel loss in dB H=-10log10ΗL

Can translate x-axis to km by assuming fiber has 0.2 dB loss / km

[PLOB15] bound

By a different method, [PLOB15] established the upper bound:

limε→0

limn→∞

1− η

). (?)

In fact, with an infinite number of channel uses, infinite energy, andperfect quantum computers for Alice and Bob, the bound is tight:

limε→0

limn→∞

P↔Lη(n, ε) = log2

1− η

Drawbacks are the same: An asymptotic statement, and thus sayslittle for practical protocols (called a weak converse bound)

Method used in [PLOB15] does not give any improved bound forprotocols using finite energy (Finite-energy SE can be tighter [GEW16])

(Proof of (?) in [PLOB15, Supp. Mat., Sec. III] does not address issue of

unbounded shield systems, & thus their proof gives trivial upper bound of ∞ for

LHS of (?) — this issue is addressed and fixed in [WTB16])

Upper bound for non-asymptotic private capacity [WTB16]

Bound on Non-Asymptotic Private Capacity

One consequence of the meta-converse approach in [WTB16]:

1− η

C (ε)

where C (ε) ≡ log2 6 + 2 log2

(1+ε1−ε

)(other choices possible).

Can be used to assess the performance of any practical quantumrepeater which uses a loss channel n times for desired security ε.

Other variations of this bound are possible if η is not the same foreach channel use, if η is chosen adversarially, etc.

Remaining technical questions: Improve C (ε) to log2

1−ε

Finite-energy bound?

Meta-converse approach from [WTB16]

Building on [Bla74, BDSW96, BK98, HW01, HHHO05, Che05,DJKR06, HHHO09, CKR09, PPV10, BD11, Li14, TH13, TT15,MLDS+13, WWY14, TWW14, DPR15, TBR15, PLOB15]

Meta-converse approach starts by using hypothesis testing relativeentropy to compare the actual state resulting from the protocol to aseparable state, the latter being useless for private comm.

The approach extracts the relevant parameters of the protocol (n,rate P, and ε) and relates them via an information-like quantity.

The meta-converse leads to various other bounds, includingRenyi-entropic strong converse bounds and others in terms of relativeentropy and relative entropy variance.

Result: We get the tightest known upper bounds for non-asymptoticprivate capacity of many channels of practical interest.

Information measures

Hypothesis testing relative entropy defined for a state ρ, positivesemi-definite operator σ, and ε ∈ [0, 1] as

DεH(ρ‖σ) ≡ − log [minTrΛσ : 0 ≤ Λ ≤ I ∧ TrΛρ ≥ 1− ε] .

Has a second-order expansion for i.i.d. states:

DεH(ρ⊗n‖σ⊗n) = nD(ρ‖σ) +

√nV (ρ‖σ)Φ−1(ε) + O(log n).

where D(ρ‖σ) ≡ Trρ[log ρ− log σ],V (ρ‖σ) ≡ Trρ[log ρ− log σ − D(ρ‖σ)]2

Φ(a) ≡ 1√2π

−∞dx exp

(−x2/2

Example: Dephasing channel [TBR15, WTB16]

For the qubit dephasing channel

Zγ : ρ 7→ (1− γ) ρ+ γZρZ ,

with γ ∈ (0, 1), the non-asymptotic private capacity P↔(n, ε) satisfies

P↔(n, ε) = 1− h(γ) +

√v(γ)

nΦ−1(ε) +

where Φ is the cumulative standard Gaussian distribution, h(γ) denotesthe binary entropy and v(γ) the corresponding variance, defined as

h(γ) ≡ −γ log γ − (1− γ) log(1− γ),

v(γ) ≡ γ(log γ + h(γ))2 + (1− γ)(log(1− γ) + h(γ))2.

Example: Dephasing channel [TBR15, WTB16]

(γ = 0.1, plot taken from [TBR15])Mark M. Wilde (LSU), Marco Tomamichel (Univ. Sydney),and Mario Berta (Caltech) (LSU) 13 / 43

Example: Erasure channel [TBR15, WTB16]

For the qubit erasure channel

EpA′→B : ρA′ 7→ (1− p)ρB + p|e〉〈e|B

with p ∈ (0, 1), the non-asymptotic private capacity P↔Ep (n, ε) satisfies

ε =n∑

l=n−k+1

)pl(1− p)n−l

(1− 2n(1−P↔Ep (n,ε))−l

Moreover, the following expansion holds

P↔Ep (n, ε) = 1− p +

√p(1− p)

nΦ−1(ε) + O

Example: Erasure channel [TBR15, WTB16]

(p = 0.25, plot taken from [TBR15])Mark M. Wilde (LSU), Marco Tomamichel (Univ. Sydney),and Mario Berta (Caltech) (LSU) 15 / 43

Second-order expansions of converse bounds [WTB16]

Theorem

If a finite-dim. quantum channel NA′→B is covariant, then

P↔N (n, ε) ≤ ER(A;B)ρ +

√V εER

(A;B)ρn

Φ−1(ε) + O

(log n

where ρAB = NA′→B(ΦAA′), ER(A;B)ρ is the relative entropy ofentanglement,

and V εER

(A;B)ρ ≡

maxσAB′∈ΠS V (ρAB‖σAB) for ε < 1/2minσAB∈ΠS V (ρAB‖σAB) for ε ≥ 1/2

with ΠS ⊆ S(A :B) the set of states achieving minimum in ER(A;B)ρ

Application: Quantum Gaussian channels [WTB16]

Definitions of quantum Gaussian channels

Thermal channel Lη,NB: b =

√ηa +

√1− ηe,

Amplifier channel AG ,NB: b =

√Ga +

√G − 1e†,

Additive-noise channel Wξ : b = a + (x + ip) /√

Thermal channel has transmissivity η ∈ [0, 1] and environmentprepared in thermal state of mean photon number NB .

Amplifier channel has gain G ∈ [1,∞) and environment prepared inthermal state of mean photon number NB .

If NB = 0, then channels are quantum-limited.

Additive noise channel has x and p be zero-mean Gaussian randomvariables with variance ξ ≥ 0.

Unconstrained rel. entropies of entanglement [PLOB15]

For the thermal channel Lη,NB, ER evaluates to

− log2

((1− η) ηNB

)− g(NB).

For the amplifier channel AG ,NB, ER evaluates to

(GNB+1

G − 1

)− g(NB).

For the additive noise channel Wξ, ER evaluates to

ξ − 1

ln 2− log2 ξ.

Unconstrained relative entropy variances [WTB16]

Let VLη,NB , VAG ,NB, and VWξ

be the unconstrained relative entropyvariances of the thermalizing, amplifier, and additive-noise channels,respectively:

VLη,NB ≡ NB(NB + 1) log22(η [NB + 1] /NB),

VAG ,NB≡ NB(NB + 1) log2

2(G−1 [NB + 1] /NB),

VWξ≡ (1− ξ)2 / ln2 2.

Can compute these from a general formula for relative entropy variance oftwo Gaussian states [WTLB16].

Strong converse bounds for Gaussian channels [WTB16]

Theorem

The following strong converse bounds hold for ε ∈ (0, 1):

P↔Lη,NB(n, ε) ≤ − log2

((1− η) ηNB

)− g(NB) +

√2VLη,NBn(1− ε)

+ C (ε)/n,

P↔AG ,NB(n, ε) ≤ log2

(GNB+1

G − 1

)− g(NB) +

√2VAG ,NB

n(1− ε)+ C (ε)/n,

P↔Wξ(n, ε) ≤ ξ − 1

ln 2− log2 ξ +

√2VWξ

n(1− ε)+ C (ε)/n.

Strong converse for quantum-limited channels [WTB16]

Corollary

For the pure-loss channel Lη and quantum-limited amplifier channel AG ,the following bounds hold

1− η

C (ε)

P↔AG(n, ε) ≤ log

1− 1/G

C (ε)

Summary

We have established bounds for QKD protocols conducted overquantum channels that are unassisted by quantum repeaters.

Meta-converse has several applications, including strong conversebounds and second-order characterizations of private communication

The bounds are related to the relative entropy of entanglement andsharpen known upper bounds on rates of QKD protocols

We establish the strong converse property for the two-way assistedprivate capacity of the pure-loss and quantum-limited amplifierchannels. We also get strong converse rates for other quantumGaussian channels.

We have generalized these results to broadcast channels with a singlesender and multiple receivers [TSW16]

Squashed entanglement technique applied more generally in [GEW16]

Methods

As said before, we build on a variety of techniques and approachesgiven in previous literature:

Meta-converse approach for hypothesis testing [Li14, TH13, DPR15],classical communication [TT15], and quantum communication[TWW14, TBR15]

Private states [HHHO05, HHHO09], a privacy test[HHH+08b, HHH+08a], and relative entropy of entanglement as anupper bound on distillable key [HHHO05, HHHO09]

Gaussian states and channels [HW01] and formulas for relativeentropy for Gaussian states [Che05, PLOB15]

ε-relative entropy of entanglement [BD11] and sandwiched Renyirelative entropy [MLDS+13, WWY14]

Reduction of adaptive protocols to non-adaptive ones via simulationof channels by teleportation [BDSW96, PLOB15]

Private states

Tripartite key state

A tripartite key state γABE contains logK bits of secret key if there existsa state σE and measurement channels MA and MB such that

(MA ⊗MB)(γABE ) =1

|i〉〈i |A ⊗ |i〉〈i |B ⊗ σE .

Bipartite private state

A bipartite private state γABA′B′ has the following form:

γABA′B′ = UABA′B′(ΦAB ⊗ θA′B′)U†ABA′B′ ,

where UABA′B′ is a “twisting” unitary of the formUABA′B′ =

∑i ,j |i〉〈i |A ⊗ |j〉〈j |B ⊗ U ij

A′B′ , with each U ijA′B′ a unitary, and

θA′B′ a state.

Private states

The systems A′ and B ′ are called the “shield” systems because they,along with the twisting unitary, can help to protect the key in systemsAB from any party possessing a purification of γABA′B′ .

Such bipartite private states are in one-to-one correspondence withtripartite key states. That is, for every tripartite key state γABE , wecan find a bipartite private state and vice versa.

This correspondence takes on a more physical form: any tripartiteprotocol whose aim it is to extract tripartite key states is in 1-to-1correspondence with a bipartite protocol whose aim it is to extractbipartite private states.

Private communication protocols

Unassisted private communication

Given is a quantum channel NA′→B . Let UNA′→BE be an isometricextension of NA′→B .

A secret-key generation protocol for n channel uses consists of a triple|K | , E ,D, where |K | is the size of the secret key to be generated,EK ′→A′n is the encoder, and DBn→K is the decoder.

EveAlice

Private communication protocols

Unassisted private communication

A triple (n,P, ε) consists of the number n of channel uses, the rate Pof secret-key generation, and the error ε ∈ [0, 1].

Such a triple is achievable on NA′→B if there exists a secret-keygeneration protocol |K | , E ,D and some state ωEn such that1n log |K | ≥ P and

F (ΦKK ⊗ ωEn , ρKKEn) ≥ 1− ε,

where ρKKEn ≡ (DBn→K (UNA′→BE )⊗n EK ′→A′n)(ΦKK ′) and

ΦKK ′ ≡1

|K |−1∑i=0

|i〉〈i |K ⊗ |i〉〈i |K ′ .

Equivalent bipartite protocol

Can reformulate such a protocol in the bipartite picture: perform everystep coherently, with the goal to produce a bipartite private state

Alice Bob

A’’

B’’

Due to equivalence between tripartite and bipartite pictures

F (γKAKBSASB , ρKKMA′′B′′) ≥ 1− ε,

for some private state γKAKBSASB , where we identify KA ≡ K , KB ≡ K ,SA ≡ MA′′, and SB ≡ B ′′, and

ρKKMA′′B′′ ≡ (UDBn→KB′′

(UNA′→BE )⊗n UEK ′→A′nA′′)(ΦGHZKK ′M).

Non-asymptotic achievable region

Non-asymptotic fundamental limit

Boundary of the achievable region:

PN (n, ε) ≡ max P : (n,P, ε) is achievable for N , .

Interpretation

Boundary PN (n, ε) identifies how rate can change as a function of nfor fixed error ε, and 2nd-order coding rates can characterize it

LOCC-assisted private communication protocols

LOCC-assisted protocols are defined similarly, but allow for rounds ofLOCC between channel uses (like in QKD)

B2’’

B2A’2

NA2’’

B1’’

B1A’1

LOCCLOCC

NA1’’

B’’

NA’’LOCC

Define boundary of non-asymptotic achievable region similarly as

P↔N (n, ε) ≡ max P : (n,P, ε) is achievable for N using ↔ .

Information measures

Can use hypothesis testing relative entropy to define the ε-relativeentropy of entanglement:

E εR(A;B)ρ ≡ infσAB∈S(A:B)

DεH(ρAB‖σAB).

where S(A :B) is the set of separable states

Can also define a channel’s ε-relative entropy of entanglement:

E εR(N ) ≡ sup|ψ〉AA′∈HAA′

E εR(A;B)ρ,

where ρAB ≡ NA′→B(ψAA′)

Standard relative entropies of entanglement defined by replacing DεH

with quantum relative entropy D

Privacy test

Can test whether a given state is a γ-private state by “untwisting”and projecting onto the maximally entangled state:

ΠABA′B′ , IABA′B′ − ΠABA′B′ ,

where ΠABA′B′ ≡ UABA′B′ (ΦAB ⊗ IA′B′)U†ABA′B′ .

Let ε ∈ [0, 1] and let ρABA′B′ be an ε-approximate γ-private state.The probability for ρABA′B′ to pass the γ-privacy test satisfies

TrΠABA′B′ρABA′B′ ≥ 1− ε,

For a separable state σABA′B′ ∈ S(AA′ :BB ′), the probability ofpassing any γ-privacy test is never larger than 1/K :

TrΠABA′B′σABA′B′ ≤1

where K is the number of values that the secret key can take.

Meta-converse bound for private communication

Theorem

For any fixed ε ∈ (0, 1), the achievable region satisfies

PN (1, ε) ≤ E εR(N ).

“One-shot ε-private capacity ≤ channel’s ε-relative entropy ofentanglement.”The same bound holds when allowing for a round of LOCC before andafter the channel use.

Proof idea: use monotonicity of E εR with respect to LOCC and use thebounds on the previous slide.

Meta-converse bound for private communication

Corollary

The following bound holds for n channel uses:

PN (n, ε) ≤ 1

nE εR(N⊗n).

The same bound holds when allowing for rounds of LOCC before and afterall n channel uses. The same bound holds for P↔N (n, ε) if the channel N isteleportation simulable.

The previous theorem and this corollary then imply all of our previousresults, with some extra work needed to establish a formula for the relativeentropy variance of Gaussian states.

Application: Second-order expansions of converse bounds

Theorem

If a quantum channel NA′→B is teleportation-simulable with associatedstate ωAB , then

P↔N (n, ε) ≤ ER(A;B)ω +

√V εER

(A;B)ωn

Φ−1(ε) + O

(log n

where V εER

(A;B)ρ ≡

maxσAB′∈ΠS V (ρAB‖σAB) for ε < 1/2minσAB∈ΠS V (ρAB‖σAB) for ε ≥ 1/2

with ΠS ⊆ S(A :B) the set of states achieving minimum in ER(A;B)ρ

Tool: Relative entropy variance for Gaussian states

Writing zero-mean Gaussian states in exponential form as

ρ = Z−1/2ρ exp

2xTGρx

, σ = Z−1/2

σ exp

2xTGσ x

Zρ ≡ det(V ρ + iΩ/2), Zσ ≡ det(V σ + iΩ/2),

Gρ ≡ 2iΩ arcoth(2V ρiΩ), Gσ ≡ 2iΩ arcoth(2V σ iΩ),

and V ρ and V σ are Wigner function covariance matrices for ρ and σ.

Theorem

For zero-mean Gaussian states ρ and σ, the relative entropy variance is

V (ρ‖σ) =1

2Tr∆V ρ∆V ρ+

8Tr∆Ω∆Ω,

where ∆ ≡ Gρ − Gσ.

References I

[BD11] Fernando G. S. L. Brandao and Nilanjana Datta. One-shot rates forentanglement manipulation under non-entangling maps. IEEE Transactionson Information Theory, 57(3):1754–1760, March 2011. arXiv:0905.2673.

[BDSW96] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K.Wootters. Mixed-state entanglement and quantum error correction.Physical Review A, 54(5):3824–3851, November 1996.arXiv:quant-ph/9604024.

[BK98] Samuel L. Braunstein and H. J. Kimble. Teleportation of continuousquantum variables. Physical Review Letters, 80(4):869–872, January 1998.

[Bla74] Richard Blahut. Hypothesis testing and information theory. IEEETransactions on Information Theory, 20(4):405–417, July 1974.

[Che05] Xiao-yu Chen. Gaussian relative entropy of entanglement. Physical ReviewA, 71(6):062320, June 2005. arXiv:quant-ph/0402109.

References II

[CKR09] Matthias Christandl, Robert Konig, and Renato Renner. Postselectiontechnique for quantum channels with applications to quantumcryptography. Physical Review Letters, 102(2):020504, January 2009.arXiv:0809.3019.

[CW04] Matthias Christandl and Andreas Winter. “Squashed entanglement”: Anadditive entanglement measure. Journal of Mathematical Physics,45(3):829–840, March 2004. arXiv:quant-ph/0308088.

[DJKR06] Igor Devetak, Marius Junge, Christopher King, and Mary Beth Ruskai.Multiplicativity of completely bounded p-norms implies a new additivityresult. Communications in Mathematical Physics, 266(1):37–63, August2006. arXiv:quant-ph/0506196.

[DPR15] Nilanjana Datta, Yan Pautrat, and Cambyse Rouze. Second-orderasymptotics for quantum hypothesis testing in settings beyond i.i.d. -quantum lattice systems and more. now published in Journal ofMathematical Physics, October 2015. arXiv:1510.04682.

References III

[GEW16] Kenneth Goodenough, David Elkouss, and Stephanie Wehner. Assessingthe performance of quantum repeaters for all phase-insensitive Gaussianbosonic channels. New Journal of Physics, 18(6):063005, June 2016.arXiv:1511.08710.

[HHH+08a] Karol Horodecki, Micha l Horodecki, Pawe l Horodecki, Debbie Leung, andJonathan Oppenheim. Quantum key distribution based on private states:Unconditional security over untrusted channels with zero quantum capacity.IEEE Transactions on Information Theory, 54(6):2604–2620, June 2008.arXiv:quant-ph/0608195.

[HHH+08b] Karol Horodecki, Micha l Horodecki, Pawe l Horodecki, Debbie Leung, andJonathan Oppenheim. Unconditional privacy over channels which cannotconvey quantum information. Physical Review Letters, 100(11):110502,March 2008. arXiv:quant-ph/0702077.

[HHHO05] Karol Horodecki, Micha l Horodecki, Pawe l Horodecki, and JonathanOppenheim. Secure key from bound entanglement. Physical ReviewLetters, 94(16):160502, April 2005. arXiv:quant-ph/0309110.

References IV

[HHHO09] Karol Horodecki, Michal Horodecki, Pawel Horodecki, and JonathanOppenheim. General paradigm for distilling classical key from quantumstates. IEEE Transactions on Information Theory, 55(4):1898–1929, April2009. arXiv:quant-ph/0506189.

[HW01] Alexander S. Holevo and Reinhard F. Werner. Evaluating capacities ofbosonic Gaussian channels. Physical Review A, 63(3):032312, February2001. arXiv:quant-ph/9912067.

[Li14] Ke Li. Second order asymptotics for quantum hypothesis testing. Annals ofStatistics, 42(1):171–189, February 2014. arXiv:1208.1400.

[MLDS+13] Martin Muller-Lennert, Frederic Dupuis, Oleg Szehr, Serge Fehr, andMarco Tomamichel. On quantum Renyi entropies: a new generalizationand some properties. Journal of Mathematical Physics, 54(12):122203,December 2013. arXiv:1306.3142.

[PLOB15] Stefano Pirandola, Riccardo Laurenza, Carlo Ottaviani, and LeonardoBanchi. Fundamental limits of repeaterless quantum communications.2015. arXiv:1510.08863v6.

References V

[PPV10] Yury Polyanskiy, H. Vincent Poor, and Sergio Verdu. Channel coding ratein the finite blocklength regime. IEEE Transactions on Information Theory,56(5):2307–2359, May 2010.

[TBR15] Marco Tomamichel, Mario Berta, and Joseph M. Renes. Quantum codingwith finite resources. now published in Nature Communications, April 2015.arXiv:1504.04617.

[TGW14] Masahiro Takeoka, Saikat Guha, and Mark M. Wilde. Fundamentalrate-loss tradeoff for optical quantum key distribution. NatureCommunications, 5:5235, October 2014. arXiv:1504.06390.

[TH13] Marco Tomamichel and Masahito Hayashi. A hierarchy of informationquantities for finite block length analysis of quantum tasks. IEEETransactions on Information Theory, 59(11):7693–7710, November 2013.arXiv:1208.1478.

References VI

[TSW16] Masahiro Takeoka, Kaushik P. Seshadreesan, and Mark M. Wilde.Unconstrained distillation capacities of a pure-loss bosonic broadcastchannel. Proceedings of the 2016 IEEE International Symposium onInformation Theory, pages 2484–2488, July 2016. Barcelona, Spain.arXiv:1601.05563.

[TT15] Marco Tomamichel and Vincent Y. F. Tan. Second-order asymptotics forthe classical capacity of image-additive quantum channels. Communicationsin Mathematical Physics, 338(1):103–137, August 2015. arXiv:1308.6503.

[TWW14] Marco Tomamichel, Mark M. Wilde, and Andreas Winter. Strong converserates for quantum communication. June 2014. arXiv:1406.2946.

[WTB16] Mark M. Wilde, Marco Tomamichel, and Mario Berta. Converse bounds forprivate communication over quantum channels. February 2016.arXiv:1602.08898.

[WTLB16] Mark M. Wilde, Marco Tomamichel, Seth Lloyd, and Mario Berta.Gaussian hypothesis testing and quantum illumination. August 2016.arXiv:1608.06991.

References VII

[WWY14] Mark M. Wilde, Andreas Winter, and Dong Yang. Strong converse for theclassical capacity of entanglement-breaking and Hadamard channels via asandwiched Renyi relative entropy. Communications in MathematicalPhysics, 331(2):593–622, October 2014. arXiv:1306.1586.

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