Energy-Constrained Private and Quantum Capacities of Quantum Channels

Energy-constrained private and quantum capacities of quantum channels

Mark M. Wilde Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA

Haoyu Qi Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA (Dated: November 22, 2018) This paper establishes a general theory of energy-constrained quantum and private capacities of quantum channels. We begin by defining various energy-constrained communication tasks, including quantum communication with a uniform energy constraint, entanglement transmission with an average energy constraint, private communication with a uniform energy constraint, and secret key transmission with an average energy constraint. We develop several code conversions, which allow us to conclude non-trivial relations between the capacities corresponding to the above tasks. We then show how the regularized, energy-constrained coherent information is equal to the capacity for the first two tasks and is an achievable rate for the latter two tasks, whenever the energy observable satisfies the Gibbs condition of having a well defined thermal state for all temperatures and the channel satisfies a finite output-entropy condition. For degradable channels satisfying these conditions, we find that the single-letter energy-constrained coherent information is equal to all of the capacities. We finally apply our results to degradable quantum Gaussian channels and recover several results already established in the literature (in some cases, we prove new results in this domain). Contrary to what may appear from some statements made in the literature recently, proofs of these results do not require the solution of any kind of minimum output entropy conjecture or entropy photon-number inequality.

I. INTRODUCTION ity is equal to zero but their private capacity is strictly greater than zero [18, 19]. Bosonic Gaussian channels are some of the most im- The capacity of a quantum channel to transmit quan- portant channels to consider, as they model practical tum or private information is a fundamental character- communication links in which the mediators of informa- istic of the channel that guides the design of practical tion are photons (see, e.g., [20, 21] for reviews). Recent communication protocols (see, e.g., [1] for a review). The years have seen advances in the quantum information quantum capacity Q(N ) of a quantum channel N is de- theory of bosonic channels. For example, we now know fined as the maximum rate at which qubits can be trans- the capacity for sending classical information over all mitted faithfully over many independent uses of N , where single-mode phase-insensitive quantum Gaussian chan- the fidelity of transmission tends to one in the limit as nels [22, 23] (and even the strong converse capacity [24]). the number of channel uses tends to infinity [2–4]. Re- The result of this theoretical development is that coher- lated, the private capacity P (N ) of N is defined to be ent states [25] of the light field suffice to achieve classical the maximum rate at which classical bits can be trans- capacity of phase-insensitive bosonic Gaussian channels. mitted over many independent uses of N such that 1) Note that the classical capacity of these channels is non- the receiver can decode the classical bits faithfully and trivial only when there is an energy constraint placed on 2) the environment of the channel cannot learn anything the input signaling states [22, 23]—otherwise, it is equal about the classical bits being transmitted [4,5]. The infinity.

arXiv:1609.01997v2 [quant-ph] 21 Nov 2018 quantum capacity is essential for understanding how fast We have also seen advances related to quantum ca- we will be able to perform distributed quantum computa- pacity of bosonic channels. Important statements, dis- tions between remote locations, and the private capacity cussions, and critical steps concerning quantum capacity is connected to the ability to generate secret key between of single-mode quantum-limited attenuator and ampli- remote locations, as in quantum key distribution (see, fier channels were reported in [26, 27]. In particular, e.g., [6] for a review). Notions from classical information these papers stated a formula for the quantum capacity theory regarding wiretap channels are typically insightful of these channels, whenever infinite energy is available for understanding private communication over quantum at the transmitter. These formulas have been supported channels (see, e.g., [7–14]). In general, there are con- with a proof in [28, Theorem 8] and [29, 30] (see Re- nections between private capacity and quantum capacity mark4 of the present paper for further discussion of this of quantum channels [4] (see also [15]), but the results point). However, in practice, no transmitter could ever of [16–19] demonstrated that these concepts and the ca- use infinite energy to transmit quantum information, and pacities can be very different. In fact, the most striking so the results from [26, 27] have limited applicability to examples are channels for which their quantum capac- realistic scenarios. Given that the notion of quantum ca- 2 pacity itself is already somewhat removed from practice, as argued in [31], it seems that supplanting a sender and receiver with infinite energy in addition to perfect quantum computers and an infinite number of channel uses only serves to push this notion much farther away from 6 practice. One of the main aims of the present paper is to continue the effort of bringing this notion closer to prac- 5 tice, by developing a general theory of energy-constrained quantum and private communication. Considering quan- 4 tum and private capacity with a limited number of channel uses, as was done in [30, 31], in addition to energy 3 constraints, is left for future developments. In light of the above discussion, we are thus motivated 2 to understand both quantum and private communication over quantum channels with realistic energy constraints. Refs. [32, 33] were some of the earlier works to discuss quantum and private communication with energy constraints, in addition to other kinds of communication tasks. The more recent efforts in [28, 34, 35] have considered energy-constrained communication in more general trade-off scenarios, but as special cases, they also fur- FIG. 1. Density plot of the ratio of the unconstrained and nished proofs for energy-constrained quantum and pri- constrained quantum and private capacities of the pure-loss vate capacities of quantum-limited attenuator and am- channel for η ∈ [1/2, 1] and NS ∈ [0, 20]. For lower photon plifier channels (see [28, Theorem 8] and [35]). In more numbers and higher loss η ≈ 0.5, there is a large gap between detail, let Q(N ,NS) and P (N ,NS) denote the respective these capacities. quantum and private capacities of a quantum channel N , such that the mean input photon number for each channel use cannot exceed NS ∈ [0, ∞). Ref. [28, Theorem 8] established that the quantum capacity of a pure-loss channel Lη with transmissivity parameter η ∈ [0, 1] is equal to 1.7 Q(L ,N ) = max{g(ηN ) − g((1 − η)N ), 0}, (1) η S S S 1.6 where g(x) is the entropy of a thermal state with mean 1.5 photon number x, defined as 1.4

g(x) ≡ (x + 1) log2(x + 1) − x log2 x. (2) 1.3 The present paper (see (331)) establishes the private ca- 1.2 pacity formula for Lη: 1.1

P (Lη,NS) = max{g(ηNS) − g((1 − η)NS), 0}. (3) A special case of the results of [35] established that the quantum and private capacities of a quantum-limited am- pliﬁer channel Aκ with gain parameter κ ∈ [1, ∞) are equal to FIG. 2. Density plot of the ratio of the unconstrained and con- Q(Aκ,NS) = P (Aκ,NS) (4) strained quantum and private capacities of the pure-ampliﬁer = g(κN + κ − 1) − g([κ − 1][N + 1]). (5) S S channel for G ∈ [1, 10] and NS ∈ [0, 20]. For lower photon numbers, there is a large gap between these capacities. Taking the limit as NS → ∞, these formulas respectively converge to

max{log2(η/ [1 − η]), 0}, (6) the unconstrained to constrained quantum capacity for- log (κ/ [κ − 1]), (7) mulas in (6) and (1), respectively. Figure2 plots the 2 ratios of the unconstrained to constrained quantum ca- which were stated in [26, 27] in the context of quantum pacity formulas in (7) and (5), respectively. capacity, with the latter proved in [29, 30] for both quan- The main purpose of the present paper is to go be- tum and private capacities. Figure1 plots the ratios of yond bosonic channels and establish a general theory of 3 energy-constrained quantum and private communication quantum-limited amplifier channels, and we bound over quantum channels, in a spirit similar to that devel- the capacities in these settings. oped in [36–39] for other communication tasks. We first We conclude in SectionXII with a summary and some recall some preliminary background on quantum infor- open questions. mation in infinite-dimensional, separable Hilbert spaces We would like to suggest that our contribution on this in SectionII. We now summarize the main contributions topic is timely. At the least, we think it should be a use- of our paper: ful resource for the community of researchers working on • In SectionIII, we define several energy-constrained related topics to have such a formalism and associated re- communication tasks, including quantum commu- sults written down explicitly, even though a skeptic might nication with a uniform energy constraint, entan- argue that they have been part of the folklore of quan- glement transmission with an average energy con- tum information theory for many years now. To support straint, private communication with a uniform en- our viewpoint, we note that some statements made in ergy constraint, and secret key transmission with several papers released in the past few years suggest that an average energy constraint. energy-constrained quantum and private capacities have not been sufficiently clarified in the existing literature. • In SectionIV, we develop several code conversions For example, in [41], one of the main results contributed between these various communication tasks, which was a non-tight upper bound on the private capacity of a allow us to conclude non-trivial relations between pure-loss bosonic channel, in spite of the fact that (3) was the capacities corresponding to them, as summa- already part of the folklore of quantum information the- rized in SectionV and Theorem1. ory. In [42], it is stated that the “entropy photon-number inequality turns out to be crucial in the determining the • SectionVI proves that the regularized, energy- classical capacity regions of the quantum bosonic broad- constrained coherent information is an achievable cast and wiretap channels,” in spite of the fact that no rate for all of the tasks, whenever the energy ob- such argument is needed to establish the quantum or pri- servable satisfies the Gibbs condition of having vate capacity of the pure-loss channel. Similarly, it is a well defined thermal state for all temperatures stated in [43] that the entropy photon-number inequal- (Definition3) and the channel satisfies a finite ity “conjecture is of particular significance in quantum output-entropy condition (Condition1). This re- information theory since if it were true then it would al- sult is stated as Theorem2. low one to evaluate classical capacities of various bosonic channels, e.g. the bosonic broadcast channel and the • For degradable channels satisfying the same condi- wiretap channel.” Thus, it seems timely and legitimate tions, we find in SectionVII that the single-letter to confirm that no such entropy photon-number inequal- energy-constrained coherent information is equal to ity or minimum output-entropy conjecture is necessary all of the capacities (stated as Theorem3). in order to establish the results regarding quantum or private capacity of the pure-loss channel—the existing • SectionVIII establishes a regularized converse for literature (specifically, [28, Theorem 8] and now the pre- the energy-constrained private capacity (stated as viously folklore (331)) has established these capacities. Theorem4), and it also establishes that the regu- The same is the case for the quantum-limited amplifier larized, energy-constrained coherent information is channel due to the results of [35]. The entropy photon- equal to the capacity for quantum communication number inequality indeed implies formulas for quantum with a uniform energy constraint and entanglement and private capacities of the quantum-limited attenua- transmission with an average energy constraint, un- tor and amplifier channels, but it appears to be much der the same conditions on the energy observable stronger than what is actually necessary to accomplish and the channel. This latter result is stated as The- this goal. The different proof of these formulas that we orem5. give in the present paper (see SectionX) is based on the • We finally apply our results to quantum Gaussian monotonicity of quantum relative entropy, concavity of channels in SectionX and recover several results coherent information of degradable channels with respect already established in the literature on Gaussian to the input density operator, and covariance of Gaussian quantum information. In some cases, we establish channels with respect to displacement operators. new results, like the formula for private capacity in (3). II. QUANTUM INFORMATION • In SectionXI, we discuss how our general frame- PRELIMINARIES work, along with recent developments in [40], allow for concluding estimates for the energy-constrained A. Quantum states and channels private and quantum capacities of particular non– Gaussian channels. Therein, we also consider al- Background on quantum information in infinite- ternative energy constraints for the pure-loss and dimensional systems is available in [39] (see also [37, 44– 4

48]). We review some aspects here. We use H through- The Stinespring representation theorem also implies that out the paper to denote a separable Hilbert space, unless every quantum channel has a Kraus representation with specified otherwise. Let IH denote the identity operator a countable set {Kl}l of bounded Kraus operators: acting on H. Let B(H) denote the set of bounded linear operators acting on H, and let P(H) denote the subset X † N (τ) = KlτKl , (12) of B(H) that consists of positive semi-definite operators. l Let T (H) denote the set of trace-class operators, those operators A for which the trace norm is finite: kAk ≡ where P K†K = I . The Kraus operators are defined √ 1 l l l HA Tr{|A|} < ∞, where |A| ≡ A†A. The Hilbert-Schmidt by the relation p † norm of A is defined as kAk2 ≡ Tr{A A}. Let D(H) denote the set of density operators (states), which con- hϕ|Kl|ψi = hϕ| ⊗ hl|U|ψi, (13) sists of the positive semi-definite, trace-class operators for |ϕi ∈ H , |ψi ∈ H , and {|li} some orthonormal with trace equal to one. A state ρ ∈ D(H) is pure if B A l basis for H [50]. there exists a unit vector |ψi ∈ H such that ρ = |ψihψ|. E ˆ Every density operator ρ ∈ D(H) has a spectral decom- A complementary channel N : T (HA) → T (HE) of N position in terms of some countable, orthonormal basis is defined for all τ ∈ T (HA) as {|φki}k as Nˆ (τ) = Tr {UτU †}. (14) X B ρ = p(k)|φkihφk|, (8) k Complementary channels are unique up to partial isometries acting on the Hilbert space H . where p(k) is a probability distribution. The tensor E A quantum channel N : T (HA) → T (HB) is degrad- product of two Hilbert spaces HA and HB is denoted able [51] if there exists a quantum channel D : T (HB) → by HA ⊗ HB or HAB. Given a multipartite density op- T (HE), called a degrading channel, such that for some erator ρAB ∈ D(HA ⊗ HB), we unambiguously write complementary channel Nˆ : T (HA) → T (HE) and all ρA = TrHB {ρAB} for the reduced density operator on system A. Every density operator ρ has a purification τ ∈ T (HA): |φρi ∈ H0 ⊗ H, for an auxiliary Hilbert space H0, where ρ ρ ρ Nˆ (τ) = (D ◦ N )(τ). (15) k|φ ik2 = 1 and TrH0 {|φ ihφ |} = ρ. All purifications are related by an isometry acting on the purifying sys- A positive operator-valued measure (POVM) is a set tem. A state ρRA ∈ D(HR ⊗HA) extends ρA ∈ D(HA) if x {Λ }x of positive semi-definite operators acting on a TrHR {ρRA} = ρA. We also say that ρRA is an extension P x Hilbert space H such that x Λ = IH. of ρA. In what follows, we abbreviate notation like TrHR as TrR. For finite-dimensional Hilbert spaces H and H such R S B. Quantum fidelity and trace distance that dim(HR) = dim(HS) ≡ M, we define the maximally entangled state ΦRS ∈ D(HR ⊗ HS) of Schmidt rank M as The fidelity of two quantum states ρ, σ ∈ D(H) is defined as [52] 1 X 0 0 ΦRS ≡ |mihm |R ⊗ |mihm |S, (9) M √ √ 2 m,m0 F (ρ, σ) ≡ ρ σ 1 . (16) where {|mi} is an orthonormal basis for H and H . m R S Uhlmann’s theorem is the statement that the fidelity has We define the maximally correlated state Φ ∈ D(H ⊗ RS R the following alternate expression as a probability over- H ) as S lap [52]: 1 X Φ ≡ |mihm| ⊗ |mihm| , (10) RS M R S F (ρ, σ) = sup |hφρ|U ⊗ I |φσi|2 , (17) m H U which can be understood as arising by applying a com- P where |φρi ∈ H0 ⊗ H and |φσi ∈ H00 ⊗ H are fixed pu- pletely dephasing channel m |mihm|(·)|mihm| to either system R or S of the maximally entangled state Φ . rifications of ρ and σ, respectively, and the optimization RS 00 0 We define the maximally mixed state of system S as is with respect to all partial isometries U : H → H . The fidelity is non-decreasing with respect to a quantum πS ≡ IS/M. channel N : T (H ) → T (H ), in the sense that for all A quantum channel N : T (HA) → T (HB) is a com- A B pletely positive, trace-preserving linear map. The Stine- ρ, σ ∈ D(HA): spring dilation theorem [49] implies that there exists an- F (N (ρ), N (σ)) ≥ F (ρ, σ). (18) other Hilbert space HE and a linear isometry U : HA → H ⊗ H such that for all τ ∈ T (H ) B E A A simple modification of Uhlmann’s theorem, found by † N (τ) = TrE{UτU }. (11) combining (17) with the monotonicity property in (18), 5 implies that for a given extension ρAB of ρA, there exists any countable orthonormal basis of H [55, Definition 2]. an extension σAB of σA such that The quantum entropy is a non-negative, concave, lower semicontinuous function on D(H)[56]. It is also not nec- F (ρAB, σAB) = F (ρA, σA). (19) essarily finite (see, e.g., [57]). When ρA is assigned to a system A, we write H(A)ρ ≡ H(ρA). The trace distance between states ρ and σ is defined as The quantum relative entropy D(ρkσ) of ρ, σ ∈ D(H) kρ − σk1. One can normalize the trace distance by mul- is defined as [58, 59] tiplying it by 1/2 so that the resulting quantity lies in the interval [0, 1]. The trace distance obeys a direct-sum D(ρkσ) property: for an orthonormal basis {|xi}x for an auxil- −1 X 2 p(i) iary Hilbert space HX , probability distributions p(x) and ≡ [ln 2] |hφi|ψji| [p(i) ln + q(j) − p(i)], x x q(j) q(x), and sets {ρ }x and {σ }x of states in D(HB), which i,j realize classical–quantum states (27)

X x P P ρXB ≡ p(x)|xihx|X ⊗ ρB, (20) where ρ = i p(i)|φiihφi| and σ = j q(j)|ψjihψj| are x spectral decompositions of ρ and σ with {|φii}i and −1 X x {|ψji}j orthonormal bases. The prefactor [ln 2] is σXB ≡ q(x)|xihx|X ⊗ σB, (21) x there to ensure that the units of the quantum relative entropy are bits. We take the convention in (27) that the following holds 0 c 0 ln 0 = 0 ln 0 = 0 but ln 0 = +∞ for c > 0. Each term in the sum in (27) is non-negative due to the in- X x x kρXB − σXBk1 = kp(x)ρB − q(x)σBk1 . (22) equality x x ln(x/y) + y − x ≥ 0 (28) The trace distance is monotone non-increasing with respect to a quantum channel N : T (HA) → T (HB), in holding for all x, y ≥ 0 [58]. Thus, by Tonelli’s theorem, the sense that for all ρ, σ ∈ D(HA): the sums in (27) may be taken in either order as discussed in [58, 59], and it follows that D(ρkσ) ≥ 0 for all ρ, σ ∈ kN (ρ) − N (σ)k1 ≤ kρ − σk1 . (23) D(H), with equality holding if and only if ρ = σ [58]. If The following equality holds for any two pure states the support of ρ is not contained in the support of σ, then φ, ψ ∈ D(H): D(ρkσ) = +∞. The converse statement need not hold in general: there exist ρ, σ ∈ D(H) with the support of ρ 1 contained in the support of σ such that D(ρkσ) = +∞. kφ − ψk = p1 − F (φ, ψ). (24) 2 1 For example, take ρ and σ diagonal in the same basis with the eigenvalues of ρ as in [57, Eq. (7)] and those of For any two arbitrary states ρ, σ ∈ D(H), the following σ as ∝ 1/n2 for n ≥ dee. inequalities hold One of the most important properties of the quantum relative entropy D(ρkσ) is that it is monotone with re- p 1 p 1 − F (ρ, σ) ≤ kρ − σk ≤ 1 − F (ρ, σ). (25) spect to a quantum channel N : T (HA) → T (HB)[60]: 2 1 The inequality on the left is a consequence of the Powers- D(ρkσ) ≥ D(N (ρ)kN (σ)). (29) Stormer inequality [53, Lemma 4.1], which states that The quantum mutual information I(A; B)ρ of a bipar- 1/2 1/2 2 kP − Qk1 ≥ P − Q 2 for P,Q ∈ P(H). The in- tite state ρAB ∈ D(HA ⊗ HB) is deﬁned as [59] equality on the right follows from the monotonicity of trace distance with respect to quantum channels, the I(A; B)ρ = D(ρABkρA ⊗ ρB), (30) identity in (24), and Uhlmann’s theorem in (17). These inequalities are called Fuchs-van-de-Graaf inequalities, as and obeys the bound [59] they were established in [54] for ﬁnite-dimensional states. I(A; B)ρ ≤ 2 min{H(A)ρ,H(B)ρ}. (31)

C. Quantum entropies and information The coherent information I(AiB)ρ of ρAB is deﬁned as [45, 61]

The quantum entropy of a state ρ ∈ D(H) is deﬁned I(AiB)ρ ≡ I(A; B)ρ − H(A)ρ, (32) as when H(A)ρ < ∞. This expression reduces to H(ρ) ≡ Tr{η(ρ)}, (26) I(AiB)ρ = H(B)ρ − H(AB)ρ (33) where η(x) = −x log2 x if x > 0 and η(0) = 0. The trace in the above equation can be taken with respect to if H(B)ρ < ∞ [45, 61]. 6

The mutual information of a quantum channel N : This quantity is non-negative from data processing of T (HA) → T (HB) with respect to a state ρ ∈ D(HA) is mutual information because we can apply the degrading deﬁned as [45] channel DB→E to system B of ρUB and recover σUE:

I(ρ, N ) ≡ I(R; B)ω, (34) σUE = DB→E(ρUB). (44)

ρ ρ where ωRB ≡ (idR ⊗NA→B)(ψRA) and ψRA ∈ D(HR ⊗ This concludes the proof. HA) is a purification of ρ, with HR 'HA. The coherent The conditional quantum mutual information (CQMI) information of a quantum channel N : T (H ) → T (H ) A B of a finite-dimensional tripartite state ρABC is defined as with respect to a state ρ ∈ D(HA) is defined as [45]

I(A; B|C)ρ ≡ H(AC)ρ + H(BC)ρ − H(ABC)ρ − H(C)ρ. Ic(ρ, N ) ≡ I(RiB)ω, (35) (45) In the general case, it is defined as [47, 48] with ωRB defined as above. These quantities obey a data processing inequality, which is that for a quantum chan- I(A; B|C) ≡ nel M : T (HB) → T (HC ) and ρ and N as before, the ρ following holds [45] sup {I(A; BC)QρQ − I(A; C)QρQ : Q = PA ⊗ IBC } , PA I(ρ, N ) ≥ I(ρ, M ◦ N ), (36) (46) I (ρ, N ) ≥ I (ρ, M ◦ N ). (37) c c where the supremum is with respect to all finite-rank We require the following proposition for some of the projections PA ∈ B(HA) and we take the convention developments in this paper: as in [47, 48] that I(A; BC)QρQ = λI(A; BC)QρQ/λ where λ = Tr{QρABC Q}. The above definition guar- Proposition 1 Let N be a degradable quantum channel antees that many properties of CQMI in finite dimen- and Nˆ a complementary channel for it. Let ρ and ρ be sions carry over to the general case [47, 48]. In particu- 0 1 lar, the following chain rule holds for a four-party state states and let ρλ = λρ0 +(1−λ)ρ1 for λ ∈ [0, 1]. Suppose ρABCD ∈ D(HABCD): that the entropies H(ρλ) and H(N (ρλ)) are finite for all λ ∈ [0, 1]. Then the coherent information of N is concave with respect to these inputs, in the sense that I(A; BC|D)ρ = I(A; C|D)ρ + I(A; B|CD)ρ. (47)

λIc(ρ0, N ) + (1 − λ)Ic(ρ1, N ) ≤ Ic(ρλ, N ). (38) Fano’s inequality [63] is the statement that for random variables X and Y with alphabets X and Y, respectively, Proof. This was established for the finite-dimensional the following inequality holds case in [62]. We follow the proof given in [1, Theo- rem 13.5.2]. First note that H(ρλ) and H(N (ρλ)) be- H(X|Y ) ≤ ε log2(|X | − 1) + h2(ε), (48) ing finite for all λ ∈ [0, 1] imply that H(Nˆ (ρλ)) is finite, by an application of the isometric invariance of the en- where tropy, the Stinespring dilation theorem, and the entropy triangle inequality from [55, Theorem 2], allowing us to ε ≡ Pr{X 6= Y }, (49) conclude that h2(ε) ≡ −ε log2 ε − (1 − ε) log2(1 − ε). (50) ˆ H(N (ρλ)) ≤ H(ρλ) + H(N (ρλ)). (39) Observe that limε→0 h2(ε) = 0. Let ρAB, σAB ∈ D(HA ⊗ HB) with dim(HA) < ∞, ε ∈ [0, 1], and suppose Set λ ≡ 1 − λ. Consider that that kρAB − σABk1 /2 ≤ ε. The Alicki–Fannes–Winter (AFW) inequality is as follows [64, 65]: Ic(ρλ, N ) − λIc(ρ0, N ) − λIc(ρ1, N ) |H(A|B)ρ − H(A|B)σ| ≤ 2ε log2 dim(HA) + g(ε), (51) = H(N (ρλ)) − H(Nˆ (ρλ)) − λH(N (ρ0))

+ λH(Nˆ (ρ0)) − λH(N (ρ1)) + λH(Nˆ (ρ1)). (40) where

Deﬁning the states g(ε) ≡ (ε + 1) log2 (ε + 1) − ε log2 ε. (52) ρ = λ|0ih0| ⊗ N (ρ ) + λ|1ih1| ⊗ N (ρ ), (41) UB U 0 U 1 Observe that limε→0 g(ε) = 0. If the states are classical σUE = λ|0ih0|U ⊗ Nˆ (ρ0) + λ|1ih1|U ⊗ Nˆ (ρ1), (42) on the ﬁrst system, as in (20)–(21), and dim(HX ) < ∞ and kρXB − σXBk1 /2 ≤ ε, then the inequality can be we can then rewrite the last line above as strengthened to [1, Theorem 11.10.3]

I(U; B)ρ − I(U; E)σ. (43) |H(X|B)ρ − H(X|B)σ| ≤ ε log2 dim(HX ) + g(ε). (53) 7

III. ENERGY-CONSTRAINED QUANTUM A. Quantum communication with a uniform AND PRIVATE CAPACITIES energy constraint

In this section, we define various notions of energy- An (n, M, G, P, ε) code for quantum communication constrained quantum and private capacity of quantum with uniform energy constraint consists of an encoding n ⊗n channels. We start by defining an energy observable (see channel E : T (HS) → T (HA ) and a decoding channel n ⊗n [39, Definition 11.3]): D : T (HB ) → T (HS), where M = dim(HS). The energy constraint is uniform, in the sense that the following bound is required to hold for all states resulting from the Definition 1 (Energy observable) Let G be a posi- output of the encoding channel En: tive semi-definite operator, i.e., G ∈ P(HA). Through- out, we refer to G as an energy observable. In more de- n Tr GnE (ρS) ≤ P, (57) tail, we define G as follows: let {|eji}j be an orthonormal basis for a Hilbert space H, and let {gj}j be a sequence where ρS ∈ D(HS). Note that of non-negative real numbers bounded from below. Then n the following formula Tr GnE (ρS) = Tr {Gρn} , (58)

∞ where X G|ψi = gj|ejihej|ψi (54) n 1 X n j=1 ρ ≡ Tr n {E (ρ )}. (59) n n A \Ai S i=1 deﬁnes a self-adjoint operator G on the dense domain P∞ 2 2 due to the i.i.d. nature of the observable Gn. Further- {|ψi : g |hej|ψi| < ∞}, for which |eji is an eigen- j=1 j more, the encoding and decoding channels are good for vector with corresponding eigenvalue gj. quantum communication, in the sense that for all pure states φRS ∈ D(HR ⊗ HS), where HR is isomorphic For a state ρ ∈ D(HA), we follow the convention [38] that to HS, the following entanglement ﬁdelity criterion holds

n ⊗n n Tr{Gρ} ≡ sup Tr{ΠnGΠnρ}, (55) F (φRS, (idR ⊗[D ◦ N ◦ E ])(φRS)) ≥ 1 − ε. (60) n A rate R is achievable for quantum communication where Πn denotes the spectral projection of G corre- over N subject to the uniform energy constraint P if sponding to the interval [0, n]. for all ε ∈ (0, 1), δ > 0, and sufficiently large n, there exists an (n, 2n[R−δ], G, P, ε) quantum communication code with uniform energy constraint. The quantum capacity Definition 2 The nth extension Gn of an energy observable G is defined as Q(N , G, P ) of N with uniform energy constraint is equal to the supremum of all achievable rates. 1 Gn ≡ [G ⊗ I ⊗ · · · ⊗ I + ··· + I ⊗ · · · ⊗ I ⊗ G] , (56) n B. Entanglement transmission with an average energy constraint where n is the number of factors in each tensor product above. An (n, M, G, P, ε) code for entanglement transmission with average energy constraint is defined very similarly In the subsections that follow, let N : T (HA) → as above, except that the requirements are less stringent. T (HB) denote a quantum channel, and let G be an en- The energy constraint holds on average, in the sense that ergy observable. Let n ∈ denote the number of chan- N it need only hold for the maximally mixed state πS input nel uses, M ∈ N the size of a code, P ∈ [0, ∞) an en- to the encoding channel En: ergy parameter, and ε ∈ [0, 1] an error parameter. In n what follows, we discuss four different notions of capac- Tr GnE (πS) ≤ P. (61) ity: quantum communication with a uniform energy constraint, entanglement transmission with an average en- Furthermore, we only demand that the particular maxi- ergy constraint, private communication with a uniform mally entangled state ΦRS ∈ D(HR ⊗ HS), defined as energy constraint, and secret key transmission with an M average energy constraint. Note that it is possible to 1 X Φ ≡ |mihm0| ⊗ |mihm0| , (62) consider other combinations, such as quantum commu- RS R S M 0 nication with an average energy constraint, or secret key m,m =1 transmission with a uniform energy constraint, but we is preserved with good fidelity: have decided to focus on the above four scenarios for n ⊗n n simplicity. F (ΦRS, (idR ⊗[D ◦ N ◦ E ])(ΦRS)) ≥ 1 − ε. (63) 8

A rate R is achievable for entanglement transmission Furthermore, we only demand that the conditions in over N subject to the average energy constraint P if (66)–(67) hold on average: for all ε ∈ (0, 1), δ > 0, and sufficiently large n, there exists an (n, 2n[R−δ], G, P, ε) entanglement transmission M 1 X m ⊗n m Tr{Λ n N (ρ n )} ≥ 1 − ε, (71) code with average energy constraint. The entanglement M B A transmission capacity E(N , G, P ) of N with average en- m=1 ergy constraint is equal to the supremum of all achievable M 1 X 1 ⊗n m ˆ n rates. N (ρAn ) − ωE ≤ ε, (72) M 2 1 From definitions, it immediately follows that quantum m=1 capacity with uniform energy constraint can never exceed ⊗n entanglement transmission capacity with average energy with ωEn some fixed state in D(HE ). constraint: A rate R is achievable for secret key transmission over N subject to the average energy constraint P if for all Q(N , G, P ) ≤ E(N , G, P ). (64) ε ∈ (0, 1), δ > 0, and sufficiently large n, there exists an (n, 2n[R−δ], G, P, ε) secret key transmission code with In SectionV, we establish the opposite inequality. average energy constraint. The secret key transmission capacity K(N , G, P ) of N with average energy constraint C. Private communication with a uniform energy is equal to the supremum of all achievable rates. constraint From definitions, it immediately follows that private capacity with uniform energy constraint can never exceed secret key transmission capacity with average en- An (n, M, G, P, ε) code for private communication con- m M ergy constraint sists of a set {ρAn }m=1 of quantum states, each in ⊗n m M D(H ), and a POVM {Λ n } such that A B m=1 P (N , G, P ) ≤ K(N , G, P ). (73) m Tr G ρ n ≤ P, (65) n A In SectionV, we establish the opposite inequality. m ⊗n m Tr{ΛBn N (ρAn )} ≥ 1 − ε, (66)

1 ˆ ⊗n m N (ρAn ) − ωEn ≤ ε, (67) 2 1 IV. CODE CONVERSIONS for all m ∈ {1,...,M}, with ωEn some ﬁxed state in In this section, we establish several code conversions, D(H⊗n). In the above, Nˆ is a channel complementary E which allow for converting one type of code into another to N . Observe that type of code along with some loss in the code parameters. m m Tr GnρAn = Tr {GρA } , (68) In particular, in the forthcoming subsections, we show how to convert where n 1. an entanglement transmission code with an aver- m 1 X m ρ ≡ Tr n {ρ n }. (69) A n A \Ai A age energy constraint to a quantum communication i=1 code with a uniform energy constraint, A rate R is achievable for private communication over N subject to uniform energy constraint P if for all 2. a quantum communication code with a uniform en- ε ∈ (0, 1), δ > 0, and suﬃciently large n, there exists ergy constraint to a private communication code an (n, 2n[R−δ], G, P, ε) private communication code. The with a uniform energy constraint, private capacity P (N , G, P ) of N with uniform energy 3. and a secret key transmission code with an average constraint is equal to the supremum of all achievable energy constraint to a private communication code rates. with a uniform energy constraint.

These code conversions then allow us to establish several D. Secret key transmission with an average energy non-trivial relations between the corresponding capaci- constraint ties, which we do in SectionV.

An (n, M, G, P, ε) code for secret key transmission with average energy constraint is deﬁned very similarly as A. Entanglement transmission with an average above, except that the requirements are less stringent. energy constraint to quantum communication with a The energy constraint holds on average, in the sense that uniform energy constraint it need only hold for the average input state:

M In this subsection, we show how an entanglement 1 X m transmission code with an average energy constraint im- Tr G ρ n ≤ P. (70) M n A m=1 plies the existence of a quantum communication code 9 with a uniform energy constraint, such that there is a and set the fidelity Fk and energy Ek of |φki as loss in performance in the resulting code with respect to follows: several code parameters. n A result like this was first established in [66] and re- Fk ≡ hφk|C (|φkihφk|)|φki (81) n viewed in [67–69], under the assumption that there is no Ek ≡ max Tr{GnE (|φihφ|)} (82) energy constraint. Here we follow the proof approach |φi∈Hk n available in [68, 69], but we make several modifications = Tr{GnE (|φkihφk|)}. (83) in order to deal with going from an average energy constraint to a uniform energy constraint. 6. Set

Proposition 2 For all δ ∈ (1/M, 1/2), the existence Hk−1 ≡ span{|ψi ∈ Hk : |hψ|φki| = 0}. (84) of an (n, M, G, P, ε) entanglement transmission code with average energy constraint implies the existence Set k := k − 1. of an (n, bδMc , G, P/ (1 − 2δ) , min{1, 2pε/[δ − 1/M]}) 7. Repeat steps 5-6 until k = 0 after step 6. quantum communication code with uniform energy constraint. The idea behind this algorithm is to successively remove minimum ﬁdelity states from HS until k = Proof. Suppose that an (n, M, G, P, ε) entanglement (1 − δ) M. By the structure of the algorithm and some transmission code with average energy constraint exists. analysis given below, we are then guaranteed for this k This implies that the conditions in (61) and (63) hold. and lower that n Let C : T (HS) → T (HS) denote the ﬁnite-dimensional channel consisting of the encoding, communication chan- 1 − min hφ|Cn(|φihφ|)|φi ≤ ε/δ. (85) nel, and decoding: |φi∈Hk

n n ⊗n n That is, the subspace Hk is good for quantum communi- C ≡ D ◦ N ◦ E . (74) cation with ﬁdelity at least 1−ε/δ. After this k, we then successively remove maximum energy states from H un- We proceed with the following algorithm: k til the algorithm terminates. Furthermore, the algorithm implies that 1. Set k = M, HM = HS, and δ ∈ (1/M, 1/2). Sup- pose for now that δM is a positive integer. FM ≤ FM−1 ≤ · · · ≤ F(1−δ)M+1, (86)

2. Set |φki ∈ Hk to be a state vector such that the E(1−δ)M ≥ E(1−δ)M−1 ≥ · · · ≥ E1, (87) input-output ﬁdelity is minimized: HM ⊇ HM−1 ⊇ · · · ⊇ H1. (88)

n |φki ≡ arg min hφ|C (|φihφ|)|φi, (75) Also, {|φ i}l is an orthonormal basis for H , where |φi∈H k k=1 l k l ∈ {1,...,M}. We now analyze the result of this algorithm by em- and set the ﬁdelity Fk and energy Ek of |φki as follows: ploying Markov’s inequality and some other tools. From the condition in (63) that the original code is good for n entanglement transmission, we have that Fk ≡ min hφ|C (|φihφ|)|φi (76) |φi∈Hk n n F (ΦRS, (idR ⊗C )(ΦRS)) ≥ 1 − ε. (89) = hφk|C (|φkihφk|)|φki, (77) n M Ek ≡ Tr{GnE (|φkihφk|)}. (78) Since {|φki}k=1 is an orthonormal basis for HM , we can write

3. Set M 1 X ∗ |ΦiRS = √ |φkiR ⊗ |φkiS, (90) Hk−1 ≡ span{|ψi ∈ Hk : |hψ|φki| = 0}. (79) M k=1

That is, Hk−1 is set to the orthogonal complement where ∗ denotes complex conjugate with respect to the of |φki in Hk, so that Hk = Hk−1 ⊕ span{|φki}. basis in (62), and the reduced state can be written as 1 PM Set k := k − 1. ΦS = M k=1 |φkihφk|S. A consequence of [1, Exer- cise 9.5.1] is that 4. Repeat steps 2-3 until k = (1 − δ) M after step 3. n 1 X n F (ΦRS, (idR ⊗C )(ΦRS)) ≤ hφk|C (|φkihφk|)|φki 5. Let |φki ∈ Hk be a state vector such that the input M energy is maximized: k 1 X = F . (91) |φ i ≡ arg max Tr{G En(|φihφ|)}, (80) M k k n k |φi∈Hk 10

So this means that We can then conclude that the subspace HδM is such that 1 X 1 X Fk ≥ 1 − ε ⇔ (1 − Fk) ≤ ε. (92) M M dim(H ) = δM, (104) k k δM min hφ|Cn(|φihφ|)|φi ≥ 1 − ε/δ, (105) Now taking K as a uniform random variable with realiza- |φi∈HδM n tions k ∈ {1,...,M} and applying Markov’s inequality, max Tr{GnE (|φihφ|)} ≤ P/ (1 − 2δ) . (106) we ﬁnd that |φi∈HδM

EK {1 − FK } ε Now applying Proposition5 (in the appendix) to (105), Pr{1 − FK ≥ ε/δ} ≤ ≤ = δ. (93) K ε/δ ε/δ we can conclude that the minimum entanglement fidelity obeys the following bound: So this implies that (1 − δ) M of the F values are such k n p min hψ|(id 0 ⊗C )(|ψihψ|)|ψi ≥ 1 − 2 ε/δ. that Fk ≥ 1 − ε/δ. Since they are ordered as given in 0 HδM |ψi∈HδM ⊗HδM (86), we can conclude that H(1−δ)M is a subspace good (107) for quantum communication in the following sense: To finish off the proof, suppose that δM is not an integer. Then there exists a δ0 < δ such that δ0M = bδMc min hφ|Cn(|φihφ|)|φi ≥ 1 − ε/δ. (94) |φi∈H(1−δ)M is a positive integer. By the above reasoning, there exists a code with parameters as given in (104)–(107), Now consider from the average energy constraint in except with δ replaced by δ0. Then the code dimen- (61) that sion is equal to bδMc. Using that δ0M = bδMc > δM − 1, we find that δ0 > δ − 1/M, which implies that n p p P ≥ Tr GnE (πS) (95) 1 − 2 ε/δ0 > 1 − 2 ε/[δ − 1/M]. We also have that 0 M P/ (1 − 2δ ) < P/ (1 − 2δ). This concludes the proof. 1 X = Tr G En(|φ ihφ | ) (96) M n k k S k=1 M B. Quantum communication with a uniform energy 1 X = E (97) constraint implies private communication with a M k uniform energy constraint k=1 (1−δ)M 1 − δ X ≥ E , (98) This subsection establishes that a quantum communi- (1 − δ) M k cation code with uniform energy constraint can always k=1 be converted to one for private communication with uni- which we can rewrite as form energy constraint, such that there is negligible loss with respect to code parameters. (1−δ)M 1 X E ≤ P/ (1 − δ) . (99) Proposition 3 The existence of an (n, M, G, P, ε) quan- (1 − δ) M k k=1 tum communication code with uniform energy constraint√ implies the existence of an (n, bM/2c , G, P, min{1, 2 ε}) Taking K0 as a uniform random variable with realizations code for private communication with uniform energy con- k ∈ {1,..., (1 − δ) M} and applying Markov’s inequality, straint. we find that Proof. Starting from an (n, M, G, P, ε) quantum com- P/ (1 − δ) munication code with uniform energy constraint, we can Pr {EK0 ≥ P/ (1 − 2δ)} ≤ (100) K0 P/ (1 − 2δ) use it to transmit a maximally entangled state 1 − 2δ = . (101) M 1 − δ 1 X Φ ≡ |mihm0| ⊗ |mihm0| (108) RS M R S Rewriting this, we find that m,m0=1

1 − 2δ of Schmidt rank M faithfully, by applying (60): Pr {EK0 ≤ P/ (1 − 2δ)} ≥ 1 − (102) K0 1 − δ F (Φ , (id ⊗Dn ◦ N ⊗n ◦ En)(Φ )) ≥ 1 − ε. (109) δ RS R RS = . (103) 1 − δ Consider that the state

n N ⊗n n Thus, a fraction δ/ (1 − δ) of the remaining (1 − δ) M σRSEn ≡ (idR ⊗D ◦ [U ] ◦ E )(ΦRS) (110) state vectors |φki are such that Ek ≤ P/ (1 − 2δ). Since they are ordered as in (87), this means that extends the state output from the actual protocol. By {|φδM i,..., |φ1i} have this property. Uhlmann’s theorem (see (19)), there exists an extension 11 of ΦRS such that the fidelity between this extension and (117) with respect to system S, we find that the following 0 the state σRSEn is equal to the fidelity in (109). However, condition holds for all m ∈ M : the maximally entangled state Φ is “unextendible” in RS 1 ⊗n m √ n ˆ the sense that the only possible extension is a tensor- ωE − N (ρAn ) ≤ 2 ε, (120) 2 1 product state ΦRS ⊗ωEn for some state ωEn . So, putting these statements together, we find that which gives the desired security condition in (67). Ap- plying monotonicity of partial trace to (117) with respect n N ⊗n n n F (ΦRS ⊗ ωEn , (idR ⊗D ◦ [U ] ◦ E )(ΦRS)) ≥ 1 − ε. to system E gives that (111) 1 n ⊗n m √ |mihm|S − (D ◦ N )(ρAn ) ≤ 2 ε, (121) Furthermore, measuring the R and S systems locally in 2 1 the Schmidt basis of Φ only increases the fidelity, so RS 0 m0 n† 0 0 that for all m ∈ M . Abbreviating ΓBn ≡ D (|m ihm |), consider then that for all m ∈ M0 n N ⊗n n F (ΦRS ⊗ ωEn , (idR ⊗D ◦ [U ] ◦ E )(ΦRS)) ≥ 1 − ε, 1 n ⊗n m |mihm|S − (D ◦ N )(ρAn ) (112) 1 n 2 where D denotes the concatenation of the original de- M n 1 X m0 ⊗n m 0 0 coder D followed by the local measurement: = |mihm| − Tr{Γ n N (ρ n )}|m ihm | 2 S B A m0=1 n X 1 D (·) ≡ |mihm|Dn(·)|mihm| (113)

m 1 X m0 ⊗n m 0 0 = pe|mihm|S − Tr{ΓBn N (ρAn )}|m ihm | X n† 2 = Tr{D [|mihm|](·)}|mihm|. (114) m06=m 1 m   1 X m0 ⊗n m n† = p + Tr{Γ n N (ρ n )} Observe that {D [|mihm|]}m is a valid POVM. Employ- 2  e B A  ing the inequalities in (25), we can conclude that m06=m m ⊗n m = 1 − Tr{ΛBn N (ρAn )}, (122) 1 n N ⊗n n √ ΦRS ⊗ ωEn − (idR ⊗D ◦ [U ] ◦ E )(ΦRS) ≤ ε. m ⊗n m 2 1 where pe ≡ 1 − Tr{ΛBn N (ρAn )}. Combining this (115) equality with (121) gives the desired reliable decoding Using the direct sum property of the trace distance from condition in (66) for all m ∈ M0 m n √ (22) and defining ρAn ≡ E (|mihm|S), we can then m ⊗n m Tr{Λ n N (ρ n )} ≥ 1 − 2 ε. (123) rewrite this as B A Thus, we have shown that from an (n, M, G, P, ε) quan- M 1 X n N ⊗n m √ tum communication code with uniform energy constraint, |mihm|S ⊗ ωEn − (D ◦ [U ] )(ρAn ) ≤ ε. √ 2M 1 one can realize an (n, bM/2c , G, P, 2 ε) code for private m=1 (116) communication with uniform energy constraint. Markov’s inequality then guarantees that there exists a Remark 1 That a quantum communication code can be subset M0 of [M] of size bM/2c such that the following easily converted to a private communication code is part condition holds for all m ∈ M0: of the folklore of quantum information theory. Ref. [4] proved that the unconstrained quantum capacity never 1 n N ⊗n m √ |mihm|S ⊗ ωEn − (D ◦ [U ] )(ρAn ) ≤ 2 ε. exceeds the unconstrained private capacity, but we are 2 1 (117) not aware of an explicit code conversion statement of the We now define the private communication code to con- form given in Proposition3. m n sist of codewords {ρAn ≡ E (|mihm|S)}m∈M0 and the decoding POVM to be C. Secret key transmission with an average energy m n† constraint implies private communication with a {ΛBn ≡ D (|mihm|)}m∈M0 uniform energy constraint     0 n† X  ∪ ΛBn ≡ D  |mihm| . (118) We finally establish that a secret key transmission code  m6∈M0  with average energy constraint can be converted to a private communication code with uniform energy con- Note that the energy constraint holds for all codewords straint. m Proposition 4 For δ ∈ (1/M, 1/3), the existence of Tr{GnρAn } ≤ P, (119) an (n, M, G, P, ε) secret key transmission code with av- due to the assumption that we start from a quantum erage energy constraint implies the existence of an communication code with uniform energy constraint as (n, bδMc , G, P/(1 − 3δ), min{1, ε/[δ − 1/M]}) private given in (57). Applying monotonicity of partial trace to communication code with uniform energy constraint. 12

Proof. To begin with, suppose that δM is an integer. From (124), we get that The existence of an (n, M, G, P, ε) secret key transmission code with average energy constraint implies that the M (1−2δ)M 1 X 1 − 2δ X following three conditions hold: P ≥ E 00 ≥ E 00 , (137) M m (1 − 2δ)M m m00=1 m00=1 M M 1 X 1 X E ≤ P, T ≥ 1 − ε , (124) which can be rewritten as M m M m m=1 m=1 (1−2δ)M 1 M 1 X P X E 00 ≤ . (138) Dm ≤ ε , (125) (1 − 2δ)M m 1 − 2δ M 00 m=1 m =1 where Taking Mˆ 00 as a uniform random variable with realiza- 00 m tions m ∈ {1, ..., (1 − 2δ)M} and applying Markov’s Em ≡ Tr{GnρAn } , (126) inequality, we ﬁnd that m ⊗n m Tm ≡ Tr{ΛBn N (ρAn )} , (127) {E } 1 EMˆ 00 Mˆ 00 ⊗n m Pr E 00 ≥ P/(1 − 3δ) ≤ (139) ˆ n Mˆ Dm ≡ N (ρAn ) − ωE . (128) ˆ 00 P/(1 − δ) 2 1 M P/(1 − 2δ) ˆ ≤ (140) Now taking M as a uniform random variable with realiza- P/(1 − 3δ) tions m ∈ {1,...,M} and applying Markov’s inequality, 1 − 3δ we have for δ ∈ (0, 1/3) that = . (141) 1 − 2δ

EMˆ {1 − TMˆ } ε Pr{1 − TMˆ ≥ ε/δ} ≤ ≤ . (129) Thus a fraction 1 − (1 − 3δ)/(1 − 2δ) = δ/(1 − 2δ) of Mˆ ε/δ ε/δ the first (1 − 2δ)M variables Em00 satisfy the condition This implies that (1−δ)M of the T values are such that EMˆ 00 ≤ P/(1 − 3δ). We can finally relabel Tm00 , Dm00 , m 000 and E 00 with a label m such that the first δM of them Tm ≥ 1 − ε/δ. We then rearrange the order of Tm, Dm, m 0 satisfy and Em using a label m such that the first (1 − δ)M of the Tm0 variables satisfy the condition Tm0 ≥ 1 − ε/δ. Now from (124), we have that Em000 ≤ P/(1 − 3δ) , (142) Tm000 ≥ 1 − ε/δ , (143) M (1−δ)M 1 X 1 − δ X Dm000 ≤ ε/δ . (144) ε ≥ D 0 ≥ D 0 , (130) M m (1 − δ)M m m0=1 m0=1 The corresponding codewords then constitute an which can be rewritten as (n, δM, G, P/(1 − 3δ), ε/δ) private communication code with uniform energy constraint. (1−δ)M 1 X ε To finish off the proof, suppose that δM is not an in- Dm0 ≤ . (131) teger. Then there exists a δ0 < δ such that δ0M = bδMc (1 − δ)M 1 − δ m=1 is a positive integer. By the above reasoning, there ex-

0 ists a code with parameters as given in (142)–(144), ex- Now taking Mˆ as a uniform random variable with real- 0 0 cept with δ replaced by δ . Then the code size is equal izations m ∈ {1,..., (1 − δ)M} and applying Markov’s to bδMc. Using that δ0M = bδMc > δM − 1, we inequality, we ﬁnd that ﬁnd that δ0 > δ − 1/M, which implies that 1 − ε/δ0 > 1 − ε/[δ − 1/M] and ε/δ0 < ε/ [δ − 1/M]. We also EMˆ 0 {DMˆ 0 } 0 Pr DMˆ 0 ≥ ε/δ ≤ (132) have that P/ (1 − 3δ ) < P/ (1 − 3δ). This concludes the ˆ 0 ε/δ M proof. ε/(1 − δ) ≤ (133) ε/δ δ V. IMPLICATIONS OF CODE CONVERSIONS = . (134) 1 − δ FOR CAPACITIES

Thus a fraction 1 − [δ/(1 − δ)] = (1 − 2δ)/(1 − δ) of the In this brief section, we show how the various code first (1 − δ)M variables Dm0 satisfy DMˆ 0 ≤ ε/δ. Now conversions from SectionIV have implications for the ca- 00 rearrange the order of Tm0 , Dm0 , and Em0 with label m pacities defined in SectionIII. The main result is the such that the first (1 − 2δ)M of them satisfy following theorem:

Tm00 ≥ 1 − ε/δ , (135) Theorem 1 Let N : T (HA) → T (HB) be a quan- D 00 ≤ ε/δ . (136) m tum channel, G ∈ P(HA) an energy observable, and 13

P ∈ [0, ∞). Then the following relations hold for the VI. ACHIEVABILITY OF REGULARIZED, capacities defined in Section III: ENERGY-CONSTRAINED COHERENT INFORMATION FOR ENERGY-CONSTRAINED QUANTUM COMMUNICATION Q(N , G, P ) = E(N , G, P ) ≤ P (N , G, P ) = K(N , G, P ). (145) The main result of this section is Theorem2, which shows that the regularized energy-constrained coherent Proof. As a consequence of the definitions of these ca- information is achievable for energy-constrained quan- pacities and as remarked in (64) and (73), we have that tum communication. In order to do so, we need to restrict the energy observables and channels that we consider. Q(N , G, P ) ≤ E(N , G, P ), (146) We impose two arguably natural constraints: that the energy observable be a Gibbs observable as given in Def- P (N , G, P ) ≤ K(N , G, P ). (147) inition3 and that the channel have finite output entropy as given in Condition1. Gibbs observables have been So it suffices to prove the following three inequalities: considered in several prior works [36, 37, 39, 65, 70, 71] as well as finite output-entropy channels [36, 37, 39]. Q(N , G, P ) ≥ E(N , G, P ), (148) When defining a Gibbs observable, we follow [39, Q(N , G, P ) ≤ P (N , G, P ), (149) Lemma 11.8] and [65, Section IV]: P (N , G, P ) ≥ K(N , G, P ). (150) Definition 3 (Gibbs observable) Let G be an energy observable as given in Definition1. Such an operator G These follow from Propositions2,3, and4, respectively. is a Gibbs observable if for all β > 0, the following holds Let us establish (148). Fix a constant δ ∈ (0, 1/2). Sup- pose that R is an achievable rate for entanglement trans- Tr{exp(−βG)} < ∞. (156) mission with an average energy constraint P (1−2δ). This implies the existence of a sequence of (n, Mn, G, P (1 − The above condition implies that a Gibbs observable G 2δ), εn) codes such that always has a finite value of the partition function Tr{exp(−βG)} for all β > 0 and thus a well defined ther- 1 mal state for all β > 0, given by e−βG/ Tr{e−βG}. lim inf log Mn = R, (151) n→∞ n lim εn = 0. (152) Condition 1 (Finite output entropy) Let G be a n→∞ Gibbs observable and P ∈ [0, ∞). A quantum channel N satisfies the finite-output entropy condition with respect Suppose that the sequence is such that Mn is to G and P if non-decreasing with n (if it is not the case, then pick out a subsequence for which it is the sup H(N (ρ)) < ∞, (157) case). Now pick n large enough such that ρ:Tr{Gρ}≤P δ ≥ 1/Mn. Invoking Proposition2, there exists p an (n, bδMnc , G, P, min[1, 2 εn/ [δ − 1/Mn]]) quantum Lemma 1 Let N denote a quantum channel satisfying communication code with uniform energy constraint. Condition1, G a Gibbs observable, and P ∈ [0, ∞). Then From the facts that any complementary channel Nˆ of N satisfies the finite- entropy condition 1 1 lim inf log (bδMnc) = lim inf log Mn (153) n→∞ n n→∞ n sup H(Nˆ (ρ)) < ∞. (158) = R, (154) ρ:Tr{Gρ}≤P p lim sup 2 εn/ [δ − 1/Mn] = 0, (155) Proof. Let ρ be a density operator satisfying Tr{Gρ} ≤ n→∞ P P , and let i pi|iihi| be a spectral decomposition of ρ. Let we can conclude that R is an achievable rate for quantum communication with uniform energy constraint P . −βG −βG θβ ≡ e / Tr{e } (159) So this implies that Q(N , G, P ) ≥ E(N , G, P (1 − 2δ)). However, since we have shown this inequality to be denote a thermal state of G with inverse temperature true for all δ ∈ (0, 1/2), we can then take a supre- β > 0. Consider that H(ρ) is finite because a rewriting mum over δ ∈ (0, 1/2) to conclude that Q(N , G, P ) ≥ of D(ρkθβ) ≥ 0 implies that supδ∈(0,1/2) E(N , G, P (1 − 2δ)) = E(N , G, P ). So we conclude (148). We can argue the other inequalities in H(ρ) ≤ β Tr{Gρ} + log Tr{e−βG} (160) (149) and (150) similarly, by applying Propositions3 −βG and4, respectively. ≤ βP + log Tr{e } < ∞, (161) 14

† n,δ where the last inequality follows from (156) and from Tr{ΠV KyKxΠV } = 0 for x 6= y. Let TY denote the ρ the assumption that P < ∞. Consider that |ψ i = δ-entropy-typical set for pY , defined as P √ i pi|ii ⊗ |ii is a purification of ρ and satisfies n,δ n n TY ≡ {y : |− [log pY n (y )] /n − H(Y )| ≤ δ} , (166) H(Nˆ (ρ)) = H((id ⊗N )(|ψρihψρ|)) (162) n for integer n ≥ 1 and real δ > 0, where p n (y ) ≡ ≤ H(ρ) + H(N (ρ)) < ∞. (163) Y n pY (y1)pY (y2) ··· pY (yn). Let Ky ≡ Ky1 ⊗ Ky2 ⊗ · · · ⊗

The equality follows because the marginals of a pure bi- Kyn . Now define the (trace-non-increasing) quantum op- partite state have the same entropy. The first inequality eration Ln,δ to be a map consisting of only the entropy- n n,δ follows from subadditivity of entropy, and the last from typical Kraus operators Kyn such that y ∈ TY . (161) and the assumption that Condition1 holds. We The number of such Kraus operators is no larger than n[H(Y )+δ] ˆ have shown that the entropy H(Nˆ (ρ)) is finite for all 2 , and one can show that H(Y ) = H(M(πV )), ˆ states satisfying Tr{Gρ} ≤ P , and so (158) holds. where M is a channel complementary to M and πV ≡ ΠV /L denotes the maximally mixed state on V [68]. Theorem 2 Let N : T (HA) → T (HB) denote a quan- One can then further reduce the quantum operation tum channel satisfying Condition1, G a Gibbs observ- Ln,δ to another one Len,δ defined by projecting the output able, and P ∈ [0, ∞). Then the energy-constrained en- of Ln,δ to the entropy-typical subspace of the density tanglement transmission capacity E(N , G, P ) is bounded operator L(πV ) = M(πV ). The entropy-typical subspace from below by the regularized energy-constrained coherent of a density operator σ with spectral decomposition σ = information of the channel N : P z pZ (z)|zihz| is defined as

1 ⊗k n,δ n n E(N , G, P ) ≥ lim Ic(N , Gk,P ), T ≡ span{|z i : |− [log pZn (z )] /n − H(σ)| ≤ δ}, k→∞ k σ (167) where the energy-constrained coherent information of N for integer n ≥ 1 and real δ > 0. The resulting quantum is defined as operation Len,δ is thus finite-dimensional and has a finite number of Kraus operators. We then have the following ˆ Ic(N , G, P ) ≡ sup H(N (ρ)) − H(N (ρ)), (164) bounds argued in [68]: ρ:Tr{Gρ}≤P ˆ n,δ n[H(M(πV ))+δ] and Nˆ denotes a complementary channel of N . Le ≤ 2 , (168) n,δ Tr{L (π ⊗n )} ≥ 1 − ε , (169) Proof. The main challenge in proving this theorem is e V 1 2 to have codes achieving the coherent information while n,δ −n[H(M(πV ))−3δ] Le (πV ⊗n ) ≤ 2 , (170) meeting the average energy constraint. We prove the the- 2 ⊗n n,δ orem by combining Klesse’s technique for constructing Fe(Cn, L ) ≥ Fe(Cn, Le ), (171) entanglement transmission codes [68, 72] with an adap- tation of Holevo’s technique of approximation and con- where Len,δ denotes the number of Kraus operators for structing codes meeting an energy constraint [36, 37]. n,δ Le and the second inequality inequality holds for all We follow their arguments very closely and show how to ε1 ∈ (0, 1) and sufficiently large n. Note that for this lat- combine the techniques to achieve the desired result. ter estimate, we require the law of large numbers to hold First, we recall what Klesse accomplished in [68] (see when we only know that the entropy is finite (this can be also the companion paper [72]). Let M : T (HA) → accomplished using the technique discussed in [73]). In T (HB) denote a quantum channel satisfying Condition1 the last line, we have written the entanglement fidelity of for some Gibbs observable and energy constraint, so that ⊗n a code Cn (some subspace of V ), which is defined as the receiver entropy is finite, as well as the environment entropy by Lemma1. This implies that entropy-typical ⊗n n ⊗n Fe(Cn, L ) ≡ suphΦCn |(id ⊗[R ◦ L ])(ΦCn )|ΦCn i, subspaces and sequences corresponding to these entropies Rn are well defined and finite, a fact of which we make use. (172) where |Φ i denotes a maximally entangled state built Let V denote a finite-dimensional linear subspace of HA. Cn Set L ≡ dim(V ), and let L denote a channel defined to from an orthonormal basis of Cn and the optimization is n be the restriction of M to states with support contained with respect to recovery channels R . Let Kn ≡ dim Cn. From the developments in [68], the following bound holds in V . Let {Ky}y be a set of Kraus operators for M and define the probability pY (y) by n,δ ⊗n EUK (V ){Fe(UKn Cn, Le )} 1 n † r pY (y) ≡ Tr{ΠV KyKyΠV }, (165) 2 L n,δ n,δ n,δ ≥ Tr{Le (πV ⊗n )} − KLe Le (πV ⊗n ) , (173) 2 where ΠV is a projection onto V . As discussed in [68], there is unitary freedom in the choice of the Kraus op- where ⊗n denotes the expected entanglement fi- EUKn (V ) erators, and they can be chosen “diagonal,” so that delity when we apply a randomly selected unitary UKn 15

m m to the codespace Cn, taking it to some diﬀerent subspace where |j i ≡ |j1i ⊗ · · · ⊗ |jmi and N(j|j ) denotes the ⊗n of V . The unitary UK is selected according to the number of appearances of the symbol j in the sequence unitarily invariant measure on the group U(V ⊗n) of uni- jm. Let taries acting on the subspace V ⊗n. Combining with the m,δ m,δ m,δ inequalities in (168)–(171), we ﬁnd that πd ≡ Πd / Tr{Πd } (182)

⊗n ⊗n {F (U C , L )} denote the maximally mixed state on the strongly typi- EUKn (V ) e Kn n 1 cal subspace. We then ﬁnd that for positive integers m h ˆ i 2 −n[H(M(πV ))−M(πV ))−R−4δ] ≥ 1 − ε1 − 2 , (174) and n,

⊗n where the rate R of entanglement transmission is defined h m,δi ⊗mn Tr Gmn πd − ρd as R ≡ [log Kn] /n. Thus, if we choose h i⊗n ˆ m,δ ⊗mn R = H(M(πV )) − M(πV )) − 5δ, (175) = Tr Gm n πd − ρd (183) then we find that n m,δ ⊗mo = Tr Gm πd − ρd ≤ δ max g(j), (184) j∈[d] n,δ −nδ/2 ⊗n {F (U C , L )} ≥ 1 − ε − 2 , (176) EUKn (V ) e Kn n e 1 where [d] ≡ {1, . . . , d} and the inequality follows from and we see that the RHS can be made arbitrarily close to applying a bound from [74] (also called “typical average one by taking n large enough. We can then conclude that lemma” in [75]). Now we can apply the above inequality there exists a unitary U , such that the codespace de- Kn to find that fined by UKn Cn achieves the same entanglement fidelity ˆ h i⊗n given above, implying that the rate H(M(πV ))−M(πV )) Tr G πm,δ is achievable for entanglement transmission over M. mn d Now we apply the methods of Holevo [37] and further ⊗m ≤ Tr{Gmρd } + δ max g(j) (185) arguments of Klesse [68] to see how to achieve the rate j∈[d] given in the statement of the theorem for the channel N = Tr{Gρd} + δ max g(j) (186) while meeting the desired energy constraint. We follow j∈[d] 0 the reasoning in [37] very closely. Consider that G is a = P + εd + δ max g(j). (187) non-constant operator. Thus, the image of the convex set j∈[d] of all density operators under the map ρ → Tr{Gρ} is an For all d large enough, we can then find δ such that the interval. Suppose first that P is not equal to the mini- 0 last line above is ≤ P/(1 + δ ) for δ, δ ∈ (0, δ ]. mum eigenvalue of G. Then there exists a real number 1 1 0 The quantum coding scheme we use is that of Klesse P 0 and a density operator ρ in D(H ) such that A [68] discussed previously, now setting M = N ⊗m and the 0 ⊗m Tr{Gρ} ≤ P < P. (177) subspace V to be the frequency-typical subspace of ρd , m,δ ∞ so that ΠV = Πd . Letting πCn denote the maximally Let ρ = P λ |jihj| be a spectral decomposition of ρ, ⊗n j=1 j mixed projector onto the codespace Cn ⊂ V , we find and define that [68, Section 5.3] d ⊗n X ˜ † h m,δi ρd ≡ λj|jihj|, where (178) ⊗n {U π U } = π ⊗n = π . (188) EUKn (V ) Kn Cn Kn V d j=1 −1  d  So this and the reasoning directly above imply that ˜ X λj ≡ λj  λj . (179) † ⊗n {Tr{G U π U }} ≤ P/(1 + δ ), (189) j=1 EUKn (V ) mn Kn Cn Kn 1

for δ, δ1 ≤ δ0. Furthermore, from (176), for arbitrary Then kρ − ρdk1 → 0 as d → ∞. Let g(j) ≡ hj|G|ji, so that ε ∈ (0, 1) and suﬃciently large n, we ﬁnd that

d ⊗mn U (V ⊗n){1 − Fe(UK Cn, N )} ≤ ε, (190) X ˜ 0 E Kn n Tr{Gρd} = λjg(j) = P + εd, (180) j=1 as long as the rate where εd → 0 as d → ∞. Consider the density operator ⊗m m,δ ˆ ⊗m m,δ 0 ⊗m m,δ R = [H(N (πd )) − H(N (πd ))]/m − δ (191) ρd , and let Πd denote its strongly typical projector, deﬁned as the projection onto the strongly typical sub- for δ0 > 0. At this point, we would like to argue the space existence of a code that has arbitrarily small error and meets the energy constraint. Let E0 denote the event m m ˜ √ span{|j i : N(j|j )/m − λj ≤ δ}, (181) ⊗mn 1−Fe(UKn Cn, N ) ≤ ε and let E1 denote the event 16

Tr{G U π U † } ≤ P . We can apply the union We have thus proven that the rate H(N (ρ)) − H(Nˆ (ρ)) mn Kn Cn Kn bound and Markov’s inequality to find that is achievable for entanglement transmission with average energy constraint for all ρ satisfying Tr{Gρ} < P . Pr {E0 ∩ E1} We can extend this argument to operators ρ such that ⊗n UKn (V ) Tr{Gρ} = P by approximating them with operators ρξ = c c = Pr {E0 ∪ E1} (192) (1 − ξ)ρ + ξ|eihe|, where |ei is chosen such that he|G|ei < U (V ⊗n) Kn P . Suppose now that P is the minimum eigenvalue of G. ⊗mn √ ≤ Pr {1 − Fe(UKn Cn, N ) ≥ ε} In this case, the condition Tr{Gρ} ≤ P reduces to the U (V ⊗n) Kn support of ρ being contained in the spectral projection n † o + Pr Tr{GmnUKn πCn UK } ≥ P (193) of G corresponding to this minimum eigenvalue. The U (V ⊗n) n Kn condition in Definition3 implies that the eigenvalues of G 1 ⊗mn have finite multiplicity, and so the support of ρ is a fixed ≤ √ ⊗n {1 − F (U C , N )} EUKn (V ) e Kn n ε finite-dimensional subspace. Thus we can take ρd = ρ, 1 † and we can repeat the above argument with the equality ⊗n + EUK (V ){Tr{GmnUKn πCn U }} (194) P n Kn Tr{Gρ} = P holding at each step. √ As a consequence, we can conclude that ≤ ε + 1/(1 + δ1). (195) ˆ Since we can choose n large enough to have ε arbitrar- sup H(N (ρ)) − H(N (ρ)) (201) Tr{Gρ}≤P ily small, there exists such an n such that the last line is strictly less than one. This then implies the exis- is achievable as well. Finally, we can repeat the whole ar- ⊗mn √ tence of a code Cn such that Fe(Cn, N ) ≥ 1 − ε (k) ⊗k (k) gument for all ρ ∈ D(HA ) satisfying Tr{Gkρ } ≤ P , ⊗k and Tr{GmnπCn } ≤ P (i.e., it has arbitrarily good en- take the channel as N , and conclude that the following tanglement fidelity and meets the average energy con- rate is achievable: straint). Furthermore, the rate achievable using this code ⊗m m,δ ˆ ⊗m m,δ 1 ⊗k (k) ⊗k (k) is equal to [H(N (πd )) − H(N (πd ))]/m. We sup H(N (ρ )) − H(Nˆ (ρ )). (202) k (k) have shown that this rate is achievable for all δ > 0 and Tr{Gkρ }≤P all integer m ≥ 1. By applying the limiting argument from [74] (see also [76]), we thus have that the following Taking the limit as k → ∞ gives the statement of the is an achievable rate as well: theorem.

1 ⊗m m,δ ˆ ⊗m m,δ lim lim [H(N (πd )) − H(N (πd ))] δ→0 m→∞ m VII. ENERGY-CONSTRAINED QUANTUM = H(N (ρ )) − H(Nˆ (ρ )), (196) AND PRIVATE CAPACITY OF DEGRADABLE d d CHANNELS 0 where Tr{Gρd} ≤ P + εd ≤ P . Given that both It is unknown how to compute the quantum and pri- H(N (ρd)) and H(Nˆ (ρd)) are finite, we can apply (32)– (35) and rewrite vate capacities of general channels, but if they are degradable, the task simplifies considerably. That is, it is known H(N (ρ )) − H(Nˆ (ρ )) = I (ρ , N ). (197) from [51] and [77], respectively, that both the uncon- d d c d strained quantum and private capacities of a degradable Finally, we take the limit d → ∞ and find that channel N are given by the following formula:

Q(N ) = P (N ) = sup Ic(ρ, N ). (203) lim inf Ic(ρd, N ) ≥ Ic(ρ, N ), (198) d→∞ ρ where we have used the representation Here we prove the following theorem, which holds for the energy-constrained quantum and private capacities Ic(ρd, N ) = I(ρd, N ) − H(ρd), (199) of a degradable channel N : applied that the mutual information is lower semicon- Theorem 3 Let G be a Gibbs observable and P ∈ [0, ∞). tinuous [45, Proposition 1], the entropy H is continuous Let a quantum channel N be degradable and satisfy for all states σ such that Tr{Gσ} < P (following from Condition1. Then the energy-constrained capacities a variation of [39, Lemma 11.8]), and the fact that a Q(N , G, P ), E(N , G, P ), P (N , G, P ), and K(N , G, P ) ρ Pd ˜1/2 are finite, equal, and given by the following formula: purification |ψdi ≡ j=1 λj |ji ⊗ |ji has the conver- gence k|ψρihψρ| − |ψρihψρ|k → 0 as d → ∞. Now since d d 1 sup H(N (ρ)) − H(Nˆ (ρ)), (204) H(N (ρ)) and H(Nˆ (ρ)) are each finite, we can rewrite ρ:Tr{Gρ}≤P

Ic(ρ, N ) = H(N (ρ)) − H(Nˆ (ρ)). (200) where Nˆ denotes a complementary channel of N . 17

Proof. That the quantity in (204) is ﬁnite follows di- Applying the partial trace and the assumption in (70), it rectly from the assumption in Condition1 and Lemma1. follows that From Theorem1, we have that M 1 X m Tr{Gρ } = Tr{G ρ n } ≤ P. (213) Q(N , G, P ) = E(N , G, P ) A M n A ≤ P (N , G, P ) = K(N , G, P ). (205) m=1 Let σB denote the average single-channel output state: Theorem2 implies that the rate in (204) is achievable. n So this gives that 1 X σ ≡ N (ρ ) = Tr n {σ n }, (214) B A n B \Bi B sup H(N (ρ)) − H(Nˆ (ρ)) i=1 ρ:Tr{Gρ}≤P and let σ denote the average single-channel environment ≤ Q(N , G, P ) = E(N , G, P ). (206) E state: To establish the theorem, it thus suﬃces to prove the n 1 X following converse inequality σ ≡ Nˆ (ρ ) = Tr n {σ n }. (215) E A n E \Ei E i=1 K(N , G, P ) ≤ sup H(N (ρ)) − H(Nˆ (ρ)). (207) ρ:Tr{Gρ}≤P It follows from non-negativity, subadditivity of entropy, concavity of entropy, (213), and the assumption that G To do so, we make use of several ideas from [4, 51, 62, 77]. is a Gibbs observable that Consider an (n, M, G, P, ε) code for secret key transmission with an average energy constraint, as described in M ! 1 X m SectionIIID. Using such a code, we take a uniform distri- 0 ≤ H ρ n M A bution over the codewords, and the state resulting from m=1 an isometric extension of the channel is as follows: n M ! X 1 X m ≤ H Tr n {ρ n } M M A \Ai A 1 X N ⊗n m i=1 m=1 σ ≡ |mihm| ⊗ [U ] (ρ n ). (208) MBˆ nEn M Mˆ A m=1 ≤ nH(ρA) < ∞. (216) Now consider that each codeword in such a code has a Similar reasoning but applying Condition1 implies that spectral decomposition as follows: n X ∞ 0 ≤ H(Bn) ≤ H(B ) ≤ nH(B) < ∞. (217) m X l,m l,m σ i σ σ ρAn ≡ pL|Mˆ (l|m)|ψ ihψ |An , (209) i=1 l=1 Similar reasoning but applying Lemma1 implies that for a probability distribution pL|Mˆ and some orthonormal l,m n basis {|ψ iAn }l for HAn . Then the state σMBˆ nEn has n X the following extension: 0 ≤ H(E )σ ≤ H(Ei)σ ≤ nH(E)σ < ∞. (218) i=1

1 M ∞ ˆ X X Furthermore, the entropy H(M)σ = log2 M because the σLMBˆ nEn ≡ pL|Mˆ (l|m)|lihl|L ⊗ |mihm|Mˆ M reduced state σM is maximally mixed with dimension m=1 l=1 N ⊗n l,m l,m equal to M. ⊗ [U ] (|ψ ihψ |An ). (210) Our analysis makes use of several other entropic quantities, each of which we need to argue is ﬁnitely bounded We can also deﬁne the state after the decoding measure- from above and below and thus can be added or sub- ment acts as tracted at will in our analysis. The quantities involved M ∞ are as follows, along with bounds for them [47, 59, 61]: 1 X X σ ≡ p (l|m)|lihl| ⊗|mihm| LMMˆ 0En M L|Mˆ L Mˆ ˆ n m,m0=1 l=1 0 ≤ I(M; B )σ ≤ min{log2 M, nH(B)σ}, (219) m0 N ⊗n l,m l,m 0 0 0 ≤ I(Mˆ ; En) ≤ min{log M, nH(E) }, (220) ⊗ TrBn {ΛBn [U ] (|ψ ihψ |An )} ⊗ |m ihm |M 0 . σ 2 σ (211) ˆ n 0 ≤ H(M|E )σ ≤ log2 M, (221)

Let ρA denote the average single-channel input state, as well as deﬁned as n n M n 0 ≤ I(MLˆ ; B )σ,I(L; B |Mˆ )σ, 1 X X m n ρA ≡ TrA \Ai {ρAn }. (212) n ˆ Mn H(B |LM)σ ≤ nH(B)σ, (222) m=1 i=1 18

n and I(Mˆ ; E )σ:

ˆ n ˆ n n n I(M; B )σ − I(M; E )σ 0 ≤ I(MLˆ ; E )σ,I(L; E |Mˆ )σ, ˆ n n ˆ n = I(ML; B )σ − I(L; B |M)σ H(E |LMˆ )σ ≤ nH(E)σ. (223) h n n i − I(MLˆ ; E )σ − I(L; E |Mˆ )σ (235) We now proceed with the converse proof: n n = I(MLˆ ; B )σ − I(MLˆ ; E )σ h n ˆ n ˆ i ˆ − I(L; B |M)σ − I(L; E |M)σ (236) log2 M = H(M)σ (224) 0 0 ˆ n ˆ n = I(Mˆ ; M )σ + H(Mˆ |M )σ (225) ≤ I(ML; B )σ − I(ML; E )σ (237) ˆ 0 n n ˆ ≤ I(M; M )σ + h2(ε) + ε log2(M − 1) (226) = H(B )σ − H(B |LM)σ ˆ n h n n i ≤ I(M; B )σ + h2(ε) + ε log2 M. (227) − H(E )σ − H(E |LMˆ )σ (238) = H(Bn) − H(Bn|LMˆ ) The first equality follows because the entropy of a uni- σ σ h n n i form distribution is equal to the logarithm of its cardi- − H(E )σ − H(B |LMˆ )σ (239) nality. The second equality is an identity. The first in- n n equality follows from applying Fano’s inequality in (48) = H(B )σ − H(E )σ. (240) to the condition in (71). The second inequality follows from applying the Holevo bound [78, 79]. The direct sum The first equality follows from the chain rule for mutual property of the trace distance and the security condition information. The second equality follows from a rear- in (72) imply that rangement. The first inequality follows from the assumption of degradability of the channel, which implies that Bob’s mutual information is never smaller than Eve’s: 1 n n I(L; B |Mˆ )σ ≥ I(L; E |Mˆ )σ. The third equality follows σ ˆ n − π ˆ ⊗ ωEn 2 ME M 1 from definitions. The fourth equality follows because the M marginal entropies of a pure state are equal, i.e., 1 X 1 ˆ ⊗n m = N (ρAn ) − ωEn ≤ ε, (228) M 2 1 n m=1 H(B |LMˆ )σ

1 X N ⊗n l,m l,m = p (l|m)H(Tr n {[U ] (|ψ ihψ | n )}) which, by the AFW inequality in (53) for classical– M L|Mˆ E A quantum states, means that l,m 1 X N ⊗n l,m l,m = p (l|m)H(Tr n {[U ] (|ψ ihψ | n )}) L|Mˆ B A ˆ n ˆ n M H(M|E )π⊗ω − H(M|E )σ ≤ ε log2(M) + g(ε). l,m n (229) = H(E |LMˆ )σ. (241) But Continuing, we have that n n H(Mˆ |E )π⊗ω − H(Mˆ |E )σ (240) = H(B1)σ − H(E1)σ + H(B2 ··· Bn)σ n = H(Mˆ )π − H(Mˆ |E )σ (230) − H(E1 ··· En)σ ˆ ˆ n = H(M)σ − H(M|E )σ (231) − [I(B1; B2 ··· Bn)σ − I(E1; E2 ··· En)σ] n (242) = I(Mˆ ; E )σ, (232) ≤ H(B1)σ − H(E1)σ so then + H(B2 ··· Bn)σ − H(E1 ··· En)σ (243) n X ˆ n ≤ H(Bi)σ − H(Ei)σ (244) I(M; E )σ ≤ ε log2(M) + g(ε). (233) i=1 Returning to (227) and inserting (233), we find that ≤ n H(B)U(ρ) − H(E)U(ρ) (245) " # ≤ n sup H(N (ρ)) − H(Nˆ (ρ)) . (246) ˆ n ˆ n ρ:Tr{Gρ}≤P log2 M ≤ I(M; B )σ − I(M; E )σ + 2ε log M + h (ε) + g(ε). (234) 2 2 The first equality follows by exploiting the definition of mutual information. The first inequality follows n We now focus on bounding the term I(Mˆ ; B )σ − from the assumption of degradability, which implies that 19

I(B1; B2 ··· Bn)σ ≥ I(E1; E2 ··· En)σ. The second in- Theorem 4 Let G be a Gibbs observable and P ∈ equality follows by iterating the argument. The third in- [0, ∞). Let a quantum channel N satisfy Condition1. equality follows from the concavity of the coherent infor- Then the energy-constrained capacities P (N , G, P ) and mation for degradable channels (Proposition1), with ρA K(N , G, P ) are finite, equal, and bounded from above by defined as in (212) and satisfying (213). Thus, the final the regularized energy-constrained private information: inequality follows because we can optimize the coherent information with respect all density operators satisfying 1 ⊗k P (N , G, P ) = K(N , G, P ) ≤ lim Cp(N , Gk,P ). the energy constraint. k→∞ k Putting everything together and assuming that ε < (251) 1/2, we find the following bound for all (n, M, G, P, ε) Proof. Theorem1 implies that private communication codes: P (N , G, P ) = K(N , G, P ). (252) 1 1 (1 − 2ε) log M − [h2(ε) + g(ε)] n 2 n To establish the theorem stated above, it thus suffices to ≤ sup H(N (ρ)) − H(Nˆ (ρ)). (247) prove the following converse inequality ρ:Tr{Gρ}≤P 1 ⊗k K(N , G, P ) ≤ lim Cp(N , Gk,P ). (253) Now taking the limit as n → ∞ and then as ε → 0, we k→∞ k can conclude the inequality in (207). This concludes the proof. To do so, we follow all of the steps of Theorem3 un- ⊗n til (234). Now let µ0 ∈ M(H ) denote the dis- crete measure induced by the (n, M, G, P, ε) secret-key VIII. REGULARIZED CONVERSES FOR transmission code. For this measure, the condition ENERGY-CONSTRAINED QUANTUM AND Tr{Gnρ(µ0)} ≤ P holds by definition, being the same PRIVATE CAPACITY OF GENERAL CHANNELS as (70). Thus, picking up from (234), we obtain the following: In this section, we establish regularized converses for ˆ n ˆ n the energy-constrained quantum and private capacities of I(M; B )σ − I(M; E )σ general channels. We start with private capacity, but be- = χ(N (µ0)) − χ(Nˆ (µ0)) (254) fore doing so, we should give some further background ⊗n (available in [39, 70, 80]) and recall the definition of ≤ Cp(N , Gn,P ), (255) the energy-constrained private information of a channel with the inequality holding for the simple reason that we [80]. A generalized (continuous) ensemble corresponds can never achieve a smaller value by optimizing over all to a Borel probability measure on the set of quantum generalized ensembles satisfying the energy constraint. states. Let M(H) denote the set of all Borel probability We then conclude that measures on D(H) having the topology of weak conver- gence. The average state ρ(µ) of a generalized ensemble 1 1 ⊗n µ ∈ M(H) is the barycenter of the measure µ defined by (1 − 2ε) log M ≤ Cp(N , Gn,P ) n 2 n the following Bochner integral: 1 + [h (ε) + g(ε)] . (256) Z n 2 ρ(µ) ≡ µ(dρ) ρ. (248) D(H) Now taking the limit as n → ∞ and then as ε → 0, we can conclude the inequality in (253). (The notation µ(dρ) indicates that µ is a measure over all mixed states.) We let N (µ) denote the generalized en- We now turn to the quantum capacity: semble resulting from applying the channel to the states in the generalized ensemble specified by µ. The Holevo Theorem 5 Let G be a Gibbs observable and P ∈ quantity for a generalized ensemble is defined as [0, ∞). Let a quantum channel N satisfy Condition1. Z Then the energy-constrained capacities Q(N , G, P ) and χ(µ) ≡ µ(dρ) D(ρkρ(µ)). (249) E(N , G, P ) are finite and equal to the regularized energy- D(H) constrained coherent information: The energy-constrained private information of a channel 1 ⊗k N is then defined as [80] Q(N , G, P ) = E(N , G, P ) = lim Ic(N , Gk,P ). k→∞ k (257) Cp(N , G, P ) ≡ sup χ(N (µ)) − χ(Nˆ (µ)), µ∈M(H):Tr{Gρ(µ)}≤P (250) Proof. Theorem2 establishes the following lower bound: ˆ where N denotes a complementary channel of N . We 1 ⊗k Q(N , G, P ) ≥ lim Ic(N , Gk,P ), (258) can now state our first result for general channels: k→∞ k 20 and Theorem1 the following equality: δ ∈ (0, 1), we have that

c nh ˆ i o Q(N , G, P ) = E(N , G, P ). (259) Pr EL ≤ P ∩ FL ≤ 2ε/δ L n o = Pr EL > P ∪ FˆL > 2ε/δ (272) We now establish the upper bound L n o ≤ Pr {EL > P } + Pr FˆL > 2ε/δ (273) L L 1 ⊗k E(N , G, P ) ≤ lim Ic(N , Gk,P ). (260) {E } {Fˆ } k→∞ k ≤ EL L + EL L (274) P 2ε/δ Fix δ ∈ (0, 1). Consider an (n, M, G, P (1 − δ), ε) code P (1 − δ) ε ≤ + (275) for entanglement transmission with an average energy P 2ε/δ constraint, as described in SectionIIIB. Let = 1 − δ + δ/2 = 1 − δ/2. (276)

n ⊗n n ωRS ≡ (idR ⊗[D ◦ N ◦ E ])(ΦRS), (261) Thus, PrL{EL ≤ P ∩ FˆL ≤ 2ε/δ} > δ/2 > 0, and we can ⊗n n conclude that there exists at least one realization l of L κRBn ≡ (idR ⊗[N ◦ E ])(ΦRS), (262) for which the conditions El ≤ P and Fˆl ≤ 2ε/δ hold. For this value, we have by (25) that where the symbols on the right-hand side are described in SectionIIIB. Note that M = dim(HR), by deﬁnition. 1 l p P l l ωRS − ΦRS ≤ 2ε/δ. (277) Let l p(l)|φ ihφ |RAn be a spectral decomposition of the 2 1 n state (idR ⊗ E )(ΦRS), and deﬁne Now consider that l n ⊗n l l ω ≡ (id ⊗[D ◦ N ])(|φ ihφ | n ), (263) RS R RA log dim(HR) = I(RiS)Φ (278) l ⊗n l l κ n ≡ (idR ⊗N )(|φ ihφ |RAn ), (264) p RB ≤ I(RiS)ωl + 2 2ε/δ log dim(HR) + g(p2ε/δ). (279) so that The equality follows from a direct calculation and the X l X l ωRS = p(l)ωRS, κRBn = p(l)κRBn . (265) inequality from (277) and the continuity bound in (51). l l Continuing, we have that

n I(RiS) l ≤ I(RiB ) l (280) By the condition in (63), we have that ω κ ⊗n l ˆ ⊗n l = H(N (φAn )) − H(N (φAn )) (281) ⊗n ε ≥ 1 − hΦ|RSωRS|ΦiRS (266) ≤ Ic(N , Gn,P ). (282) X l = p(l) 1 − hΦ|RSωRS|ΦiRS (267) The first inequality follows from data processing of coher- l ent information recalled in (37). The equality follows by X ≡ p(l)Fˆl. (268) rewriting the coherent information, given that the var- l ious entropies involved are finite. The final inequality ⊗n follows because the definition of Ic(N , Gn,P ) involves an optimization with respect to all input states ρ(n) satis- Also, the energy constraint in (61) implies that (n) l fying Tr{Gnρ } ≤ P and φAn is one such state. Putting everything together, we find that n P (1 − δ) ≥ Tr{GnE (πS)} (269) X l p 1 1 ⊗n = p(l) Tr{GnφAn } (270) (1 − 2 2ε/δ) log dim(HR) ≤ Ic(N , Gn,P ) l n n 1 X + g(p2ε/δ). (283) ≡ p(l)El. (271) n l Now taking the limit as n → ∞ and then as ε → 0, we We would like to conclude that there exists at least one conclude that l value of l for which the state φ n realizes a good en- RA 1 ⊗k tanglement generation code, in the sense of [4], while at E(N , G, P (1 − δ)) ≤ lim Ic(N , Gk,P ). (284) k→∞ k the same time meeting the energy constraint. Let L be a random variable with probability distribution p(l). By However, we have proved that the above inequality holds the union bound and Markov’s inequality, for constant for all δ ∈ (0, 1), and so we can take a supremum over 21

δ ∈ (0, 1) and arrive at the conclusion that By the assumption of the theorem, this means that ˆ sup E(N , G, P (1 − δ)) = E(N , G, P ) (285) D(N (ρ)kθβ1 ) ≥ D(N (ρ)kθβ2 ), (295) δ∈(0,1)

where β1 and β2 are such that Tr{Gθβ1 } = P1 and 1 ⊗k ≤ lim Ic(N , Gk,P ), Tr{Gθβ2 } = P2. After a rewriting using deﬁnitions and k→∞ k the fact that all terms below are ﬁnite, the inequality (286) above becomes which is the inequality in (260). This concludes the proof. ˆ Tr{N (ρ) log θβ2 } − Tr{N (ρ) log θβ1 } ≥ H(N (ρ)) − H(Nˆ (ρ)). (296)

−β1G −β2G IX. THERMAL STATE AS THE OPTIMIZER Set Z1 ≡ Tr{e } and Z2 ≡ Tr{e }. We can then rewrite the upper bound as In this section, we prove that the function ˆ Tr{N (ρ) log θβ2 } − Tr{N (ρ) log θβ1 }

ˆ −β2G sup H(N (ρ)) − H(N (ρ)) (287) = Tr{Nˆ (ρ) log e /Z2 } Tr{Gρ}=P −β1G − Tr{N (ρ) log e /Z1 } (297) is optimized by a thermal state input if the channel N is ˆ degradable and satisfies certain other properties. In what = log [Z1/Z2] − β2 Tr{GN (ρ)} + β1 Tr{GN (ρ)} (298) follows, for a Gibbs observable G, we define the thermal ≤ log [Z1/Z2] − β2P2 + β1P1. (299) state θ of inverse temperature β > 0 as β Thus, we have established a uniform upper bound on the e−βG coherent information of states subject to the constraints θβ ≡ . (288) given in the theorem: Tr{e−βG} H(N (ρ))−H(Nˆ (ρ)) ≤ log [Z /Z ]−β P +β P . (300) Theorem 6 Let G be a Gibbs observable and P ∈ [0, ∞). 1 2 2 2 1 1 Let N : T (HA) → T (HB) be a degradable quantum chan- This bound is saturated when we choose the input ρ = nel satisfying Condition1. Let θβ denote the thermal θβ, where β is such that Tr{Gθβ} = P , because state of G, as in (288), satisfying Tr{Gθ } = P for some β ˆ β > 0. Suppose that N and a complementary channel log [Z1/Z2] − β2P2 + β1P1 = H(N (θβ)) − H(N (θβ)). Nˆ : T (HA) → T (HE) are Gibbs preserving, in the sense (301) that there exist β1, β2 > 0 such that This concludes the proof. Remark 2 Note that we can also conclude that P ≥ P N (θβ) = θβ , Nˆ (θβ) = θβ . (289) 1 2 1 2 for channels satisfying the hypotheses of the above the- Set orem because the channel is degradable, implying that H(θ ) ≥ H(θ ), and the entropy of a thermal state ˆ β1 β2 P1 ≡ Tr{GN (θβ)},P2 ≡ Tr{GN (θβ)}. (290) is a strictly increasing function of the energy (and thus invertible) [65, Proposition 10]. Suppose further that N and Nˆ are such that, for all input states ρ such that Tr{Gρ} = P , the output energies Remark 3 The assumptions in Theorem6 might seem satisfy somewhat artificial, but the next section demonstrates several natural examples of channels that satisfy the as- ˆ Tr{GN (ρ)} ≤ P1, Tr{GN (ρ)} ≥ P2. (291) sumptions. Then the function sup H(N (ρ)) − H(Nˆ (ρ)), (292) X. APPLICATION TO GAUSSIAN QUANTUM Tr{Gρ}=P CHANNELS is optimized by the thermal state θβ. We can now apply all of the results from previous sec- Proof. Let D : T (H ) → T (H ) be a degrading chan- tions to the particular case of quantum bosonic Gaussian B E channels [20, 21]. These channels model natural physical nel such that D ◦ N = Nˆ . Consider a state ρ such that processes such as photon loss, photon amplification, ther- Tr{Gρ} = P . The monotonicity of quantum relative en- malizing noise, or random kicks in phase space. They sat- tropy with respect to quantum channels (see (29)) implies isfy Condition1 when the Gibbs observable for m modes that is taken to be m D(N (ρ)kN (θβ)) ≥ D((D ◦ N )(ρ)k(D ◦ N )(θβ)) (293) ˆ X † Em ≡ ωjaˆjaˆj, (302) = D(Nˆ (ρ)kNˆ (θβ)). (294) j=1 22 where ωj > 0 is the frequency of the jth mode anda ˆj It is also well known that thermal states can be written is the photon annihilation operator for the jth mode, as a Gaussian mixture of displacement operators acting † so thata ˆjaˆj is the photon number operator for the jth on the vacuum state: mode. Z 2m ⊗m † We start with a brief review of Gaussian states and θβ = d ξ p(ξ) D(ξ)[|0ih0|] D (ξ), (312) channels (see [20, 21, 81] for more comprehensive reviews, but note that here we mostly follow the conventions of where p(ξ) is a zero-mean, circularly symmetric Gaussian [20]). Let distribution. From this, it also follows that randomly displacing a thermal state in such a way leads to another ˆ R ≡ [ˆq1,..., qˆm, pˆ1,..., pˆm] ≡ [ˆx1,..., xˆ2m] (303) thermal state of higher temperature: denote a row vector of position- and momentum- Z 2m † quadrature operators, satisfying the canonical commu- θβ = d ξ q(ξ) D(ξ)θβ0 D (ξ), (313) tation relations: 0 h i 0 1 where β ≥ β and q(ξ) is a particular circularly symmet- Rˆ , Rˆ = iΩ , where Ω ≡ ⊗ I , (304) j k j,k −1 0 m ric Gaussian distribution. A 2m × 2m matrix S is symplectic if it preserves the T and Im denotes the m × m identity matrix. We take symplectic form: SΩS = Ω. According to Williamson’s theorem [82], there is a diagonalization of the covariance the annihilation√ operator for the jth mode asa ˆj = (ˆqj + 2m matrix V ρ of the form, ipˆj)/ 2. For z a column vector in R , we define the uni- † tary displacement operator D(z) = D (−z) ≡ exp(iRzˆ ). T V ρ = Sρ (Dρ ⊕ Dρ)(Sρ) , (314) Displacement operators satisfy the following relation: ρ ρ i where S is a symplectic matrix and D ≡ D(z)D(z0) = D(z + z0) exp − zT Ωz0 . (305) diag(ν , . . . , ν ) is a diagonal matrix of symplectic eigen- 2 1 m values such that νi ≥ 1 for all i ∈ {1, . . . , m}. Computing Every state ρ ∈ D(H) has a corresponding Wigner char- this decomposition is equivalent to diagonalizing the ma- acteristic function, defined as trix iV ρΩ[83, Appendix A]. The entropy H(ρ) of a quantum Gaussian state ρ is a χρ(z) ≡ Tr{D(z)ρ}, (306) direct function of the symplectic eigenvalues of its covari- ρ and from which we can obtain the state ρ as ance matrix V [20]: m Z 2m d z † X ρ ρ = m χρ(z) D (z). (307) H(ρ) = g((νj − 1)/2) ≡ g(V ), (315) (2π) j=1 A quantum state ρ is Gaussian if its Wigner characteristic where g(·) is defined in (52) and we have indicated a function has a Gaussian form as shorthand for this entropy as g(V ρ). 1 T ρ ρ T The Hilbert–Schmidt adjoint of a Gaussian quantum χρ(ξ) = exp − z V z + i [µ ] z , (308) 4 channel NX,Y from m modes to m modes has the following effect on a displacement operator D(z)[20]: where µρ is the 2m×1 mean vector of ρ, whose entries are defined by µρ ≡ hRˆ i and V ρ is the 2m × 2m covariance 1 j j ρ D(z) 7−→ D(Xz) exp − zT Y z + izT d , (316) matrix of ρ, whose entries are defined as 4 ρ ˆ ρ ˆ ρ Vj,k ≡ h{Rj − µj , Rk − µk}iρ. (309) where X is a real 2m × 2m matrix, Y is a real 2m × 2m positive semi-definite matrix, and d ∈ 2m, such that The following condition holds for a valid covariance ma- R they satisfy trix: V ≥ iΩ, which is a manifestation of the uncertainty principle. Y − iΩ + iXT ΩX ≥ 0. (317) A thermal Gaussian state θβ of m modes with respect ρ to Eˆm from (302) and having inverse temperature β > 0 The effect of the channel on the mean vector µ and the thus has the following form: covariance matrix V ρ is thus as follows:

ˆ ˆ ρ T ρ −βEm −βEm µ 7−→ X µ + d, (318) θβ = e / Tr{e }, (310) V ρ 7−→ XT V ρX + Y. (319) and has a mean vector equal to zero and a diagonal 2m× 2m covariance matrix. One can calculate that the photon All Gaussian channels are covariant with respect to dis- number in this state is equal to placement operators. That is, the following relation holds

X 1 † T † T . (311) NX,Y (D(z)ρD (z)) = D(X z)NX,Y (ρ)D (X z). eβωj − 1 j (320) 23

Just as every quantum channel can be implemented as It thus remains to prove that H(NX,Y (θβ)) − a unitary transformation on a larger space followed by a ˆ H(NXE ,YE (θβ)) is increasing with decreasing β. This fol- partial trace, so can Gaussian channels be implemented lows from the covariance property in (320), the concavity as a Gaussian unitary on a larger space with some extra of coherent information in the input for degradable chan- modes prepared in the vacuum state, followed by a partial nels (Proposition1), and the fact that thermal states can trace [20]. Given a Gaussian channel NX,Y with Z such be realized by random Gaussian displacements of thermal T that Y = ZZ we can find two other matrices XE and states with lower temperature. Consider that ZE such that there is a symplectic matrix H(NX,Y (θβ0 )) − H(NˆX ,Y (θβ0 )) T E E X Z Z S = T , (321) 2m h i 0 ˆ 0 XE ZE = d ξ q(ξ) H(NX,Y (θβ )) − H(NXE ,YE (θβ )) which corresponds to the Gaussian unitary transforma- (327) Z tion on a larger space. The complementary channel 2m h † ˆ = d ξ q(ξ) H(D(Xξ)NX,Y (θβ0 )D (Xξ)) NXE ,YE from input to the environment then effects the following transformation on mean vectors and covariance † i ˆ 0 matrices: − H(D(XEξ)NXE ,YE (θβ )D (XEξ)) (328) Z ρ T ρ 2m h † µ 7−→ XE µ , (322) = d ξ q(ξ) H(NX,Y (D(ξ)θβ0 D (ξ))) V ρ 7−→ XT V ρX + Y , (323) E E E † i ˆ 0 − H(NXE ,YE (D(ξ)θβ D (ξ))) (329) T where YE ≡ ZEZE . 0 0 ≤ H(N (θ )) − H(Nˆ (θ )). (330) A quantum Gaussian channel for which X = X ⊕ X , X,Y β XE ,YE β 0 0 0 0 Y = Y ⊕ Y , and d = d ⊕ d is known as a phase- The first equality follows by placing a probability distri- insensitive Gaussian channel, because it does not have bution in front, and the second follows from the unitary a bias to either quadrature when applying noise to the invariance of quantum entropy. The third equality fol- input state. lows from the covariance property of quantum Gaussian The main result of this section is the following theo- channels, given in (320). The inequality follows because rem, which gives an explicit expression for the energy- the coherent information of degradable channels is con- constrained capacities of all phase-insensitive degradable cave in the input state (Proposition1) and from (313). Gaussian channels that satisfy the conditions of Theo- rem6 for all β > 0:

Theorem 7 Let NX,Y be a phase-insensitive degradable Gaussian channel, having a dilation of the form in (321). A. Special cases: Single-mode pure-loss and quantum-limited amplifier channels Suppose that NX,Y satisfies the conditions of Theorem6 for all β > 0. Then its energy-constrained capacities We can now discuss some special cases of the above Q(NX,Y , Eˆm,P ), E(NX,Y , Eˆm,P ), P (NX,Y , Eˆm,P ), result, some of which have already been known in the and K(N , Eˆ ,P ) are equal and given by the following X,Y m literature. Suppose that the channel is a single-mode formula: pure-loss channel Lη, where η ∈ [1/2, 1] characterizes T θβ T θβ the average fraction of photons that make it through the g(X V X + Y ) − g(XE V XE + YE), (324) channel from sender to receiver [84]. In this case, the √ where θβ is a thermal state of mean photon number P . channel has X = ηI2 and Y = (1 − η)I2. We take the Gibbs observable to be the photon-number operator Proof. Since the channel is degradable, satisfies Condi- aˆ†aˆ and the energy constraint to be N ∈ [0, ∞). Such ˆ S tion1, and Em is a Gibbs observable, Theorem3 applies a channel is degradable [85] and was conjectured [33] to and these capacities are given by the following formula: have energy-constrained quantum and private capacities ˆ equal to sup H(NX,Y (ρ)) − H(NXE ,YE (ρ)). (325) ρ:Tr{Eˆmρ}≤P g(ηNS) − g((1 − η)NS). (331) By assumption, the channel satisfies the conditions of Theorem6 as well for all β > 0, so that the following This conjecture was proven for the quantum capacity in [28, Theorem 8], and the present paper establishes the function is optimized by a thermal state θβ of mean photon number P : statement for private capacity. This was argued by exploiting particular properties of the g function (estab- sup H(N (ρ)) − H(Nˆ (ρ)) lished in great detail in [86]) to show that the thermal X,Y XE ,YE state input is optimal for any fixed energy constraint. ρ:Tr{Eˆmρ}=P Here we can see this latter result as a consequence of the ˆ = H(NX,Y (θβ)) − H(NXE ,YE (θβ)). (326) more general statements in Theorems6 and7, which are 24 based on the monotonicity of relative entropy and other scenarios considered in prior works for other capaci- properties of this channel, such as covariance and degrad- ties [32, 90, 91]. Let the Gibbs observable be Eˆm, as ability. Taking the limit NS → ∞, the formula in (331) given in (302), and suppose that the energy constraint converges to is P ∈ [0, ∞). Suppose that the channel is an m-mode channel consisting of m parallel pure-loss channels Lη, log2(η/[1 − η]), (332) each with the same transmissivity η ∈ [1/2, 1]. Then for Eˆ and such an m-mode channel, the conditions of Theo- which is consistent with the formula stated in [27]. m rems6 and7 are satisfied, so that the energy-constrained Suppose that the channel is a single-mode quantum- quantum and private capacities are given by limited amplifier channel√ Aκ of gain κ ≥ 1. In this case, the channel has X = κI2 and Y = (κ − 1)I2. Again we m take the energy operator and constraint as above. This X g(ηNj(β)) − g((1 − η)Nj(β)), (335) channel is degradable [85] and was recently proven [35] j=1 to have energy-constrained quantum and private capacity equal to where

g(κNS + κ − 1) − g([κ − 1] [NS + 1]). (333) βωs Ns(β) ≡ 1/(e − 1), (336)

The result was established by exploiting particular prop- Pm erties of the g function in addition to other arguments. and β is chosen such that P = j=1 Nj(β), so that the However, we can again see this result as a consequence energy constraint is satisfied. A similar statement applies of the more general statements given in Theorems6 and to m parallel quantum-limited amplifier channels each 7. Taking the limit N → ∞, the formula converges to having the same gain κ ≥ 1. In this case, the conditions S of Theorems6 and7 are satisfied, so that the energy- constrained quantum and private capacities are given by log2(κ/ [κ − 1]), (334)

m which is consistent with the formula stated in [27] and X recently proven in [29, 30]. g(κNj(β) + κ − 1) − g([κ − 1] [Nj(β) + 1]), (337) j=1 Remark 4 Ref. [27] has been widely accepted to have provided a complete proof of the unconstrained quantum where Nj(β) is as deﬁned above and β is chosen to satisfy Pm capacity formulas given in (332) and (334). The impor- P = j=1 Nj(β). tant developments of [27] were to identify that it suﬃces Theorems6 and7 can be applied indirectly to a more to optimize coherent information of these channels with general scenario. Let m = k+l, where k and l are positive respect to a single channel use and Gaussian input states. integers. Suppose that the channel consists of k pure-loss

The issue is that [27] relied on an “optimization proce- channels Lηi , each of transmissivity ηi ∈ [1/2, 1], and l dure carried out in” [26] in order to establish the infinite- quantum-limited amplifier channels Aκj , each of gain κj energy quantum capacity formula given there (see just be- for j ∈ {1, . . . , l}. In this scenario, Theorems6 and7 fore [27, Eq. (12)]). However, a careful inspection of [26, apply to the individual channels, so that we know that Section V-B] reveals that no explicit optimization proce- a thermal state is the optimal input to each of them for dure is given there. The contentious point is that it is a fixed input energy. The task is then to determine how necessary to show that, among all Gaussian states, the to allocate the energy such that the resulting capacity thermal state is the input state optimizing the coherent is optimal. Let P denote the total energy budget, and k l information of the quantum-limited attenuator and am- suppose that a particular allocation {{Ni}i=1, {Mj}j=1} plifier channels. This point is not argued or in any way is made such that justified in [26, Section V-B] or in any subsequent work k m or review on the topic [39, 87–89]. As a consequence, we X X have been left to conclude that the proof from [27] features P = ωiNi + ωjMj. (338) a gap which was subsequently closed in [28, Section III-G- i=1 j=1 1] and [35]. The result in [29, 30] gives a completely different approach for establishing the unconstrained quan- Then Theorems6 and7 apply to the scenario when the tum and private capacities of the quantum-limited ampli- allocation is fixed and imply that the resulting quantum fier channel, which preceded the development in [35]. and private capacities are equal and given by

k X B. Special cases: Multi-mode pure-loss and g(ηiNi) − g((1 − ηi)Ni) quantum-limited ampliﬁer channels i=1 l X Our results from Theorems6 and7 allow for mak- + g(κMj + κ − 1) − g([κ − 1] [Mj + 1]). (339) ing more general statements, applicable to broadband j=1 25

However, we can then optimize this expression with re- photon numbers more severely than the typical photon spect to the energy allocation, leading to the following number constraint. constrained optimization problem: For the case of the pure-loss and quantum-limited amplifier channels, we can give concrete bounds for energy- k constrained quantum and private capacity, usingn ˆ2 as X max g(ηiNi) − g((1 − ηi)Ni) the energy observable, by employing an idea put for- {{N }k ,{N }l } i i=1 j j=1 i=1 ward recently in [40, Remark 21], as well as other ar- l guments. Suppose that the Gibbs observable is nown ˆ2 X + g(κMj + κ − 1) − g([κ − 1] [Mj + 1]), (340) (the square of the photon number operator). We first j=1 discuss how to obtain an upper bound on the capacities. Due to these channels being degradable, Theorem3 such that applies, and it suffices to consider optimizing the single-

k m copy energy-constrained coherent information in (204), X X 2 P = ω N + ω M . (341) subject to the constraint Tr{nˆ ρ} ≤ P on the input state i i j j ρ. By concavity of the square-root function, and due to i=1 j=1 2 P∞ 2 the fact that Tr{nˆ ρ} = n=0 p(n)n for some proba- This problem can be approached using Lagrange multi- bility distribution p(n), it follows that every√ state satis- 2 plier methods, and in some cases handled analytically, fying Tr√{nˆ ρ} ≤ P also satisfies Tr{nρˆ } ≤ P . Setting while others need to be handled numerically. Many dif- NS = P , we then find that the formulas in (331) and ferent scenarios were considered already in [32], to which (333) with this value of NS give an upper bound on the we point the interested reader. However, we should note capacities. that [32] was developed when the formulas above were To find a lower bound on the capacities, we can op- only conjectured to be equal to the capacity and not timize the single-copy energy-constrained coherent infor- proven to be so. mation in (204) with respect to all Gaussian state inputs. The coherent information can also be rewritten in this case as a particular conditional entropy (see [27, 51]) XI. DISCUSSION OF NON-GAUSSIAN that is a function of the input state ρ. Now we ap- CHANNELS AND OTHER ENERGY ply an argument from [40, Remark 21]. The pure-loss CONSTRAINTS and quantum-limited amplifier channels and their complementary channels are phase-covariant, meaning that We stress again here that the framework for energy- a unitary phase operator einφˆ acting on the input state constrained quantum and private capacity given in this commutes with the channels and acts as a unitary phase paper applies to more general situations beyond bosonic operator on the output. Since for any state ρ, the value of Gaussian channels with photon number constraints, just Tr{nˆ2ρ} is unchanged by applying a random phase to ρ, as the frameworks from [36–38, 70] do for other kinds of but the conditional entropy does not decrease under this communication capacities. All we require for our theo- operation and the phase-randomized state becomes num- rems to apply is that the energy observable be a Gibbs ber diagonal, it suffices to perform the optimization over observable (Definition3) and the channel satisfy the fi- all states that are both Gaussian (by assumption) and nite output-entropy condition (Condition1). number diagonal. For a single-mode state, the only such There are some interesting cases to consider. For ex- possibility is a thermal state. Finally, since the func- ample, it could be the case that the initial state of the tions in (331) and (333) are equal to the coherent in- environment in a thermal channel has not reached its formations of these channels when sending in a thermal equilibrium state and is in a non-Gaussian state different state of mean photon number NS, and these functions from a thermal state. This kind of channel is related to are monotone increasing with respect to NS, it suffices those presented and analyzed recently in [92]. If the ini- to pick a thermal state θ(NS) of mean photon number NS tial environment state is an approximate thermal state that meets the energy constraint P with equality. Since 2 (has trace distance close to a thermal state of a certain Tr{nˆ θ(NS)} = NS(2NS + 1), by solving√ the equation 1 photon number), then the tools of the present paper, as NS(2NS + 1) = P , we find that NS = 4 ( 1 + 8P − 1) well as those detailed in the recent work [40], could be and then lower bounds on the energy-constrained quan- used to estimate the quantum and private capacity of tum and private capacities of these channels are given by this non-equilibrium thermal channel. (331) and (333) with this value of NS. Even in the bosonic setting, one could also consider In summary, the energy-constrained quantum and pri- other energy observables besides photon number observ- vate capacities of the pure-loss channel, with Gibbs ob- 2 ables. For example, one could consider the square or servable set ton ˆ , are bounded from√ above by the func- higher powers of the photon number observables, which tion in (331) evaluated at NS = P and from√ below by 1 might be relevant in situations in which the transmit- the formula in (331) evaluated at NS = 4 ( 1 + 8P − 1). ter is highly sensitive to higher photon numbers. Using One obtains related bounds for the energy-constrained the square of the photon number would penalize higher quantum and private capacities of the quantum-limited 26 amplifier channel, with Gibbs observable set ton ˆ2, by channels, to which the general framework could apply. It evaluating the formula in (333) at the same values of NS. could be interesting to explore generalizations of the re- We note that similar arguments can be employed for sults and settings from [94–98] regarding fermionic Gaus- any power of the photon number operatorn ˆ, and one sian channels. A more particular question we would like would find bounds for the energy-constrained capacities to see answered is whether concavity of coherent informa- in a similar way. tion of degradable channels could hold in settings beyond Going beyond the bounds given above, it is an intrigu- that considered in Proposition1. We suspect that an ap- ing open question to identify the actual capacities with proximation argument along the lines of that given in the these modified Gibbs observables. In this scenario, it proof of [45, Proposition 1] should make this possible. We is not clear that the extremality of Gaussian states [93] also think it should be possible to establish an equality applies, because the constraint is not on the covariance in Theorem4, but we leave this for future endeavors. matrix, but rather on the expectation of a four-point cor- relator. ACKNOWLEDGMENTS

XII. CONCLUSION We are grateful to Saikat Guha, Alexander Holevo, Anna Kuznetsova, and Maksim Shirokov for discussions This paper has provided a general theory of energy- related to this paper. We thank the anonymous referees constrained quantum and private communication over for several comments that helped to improve the paper. quantum channels. We defined several communication MMW acknowledges the NSF under Award No. CCF- tasks (SectionIII), and then established ways of con- 1350397, as well as the Office of Naval Research. HQ verting a code for one task to that of another task (Sec- is supported by the Air Force Office of Scientific Re- tionIV). These code conversions have implications for ca- search, the Army Research Office, and the National Sci- pacities, establishing non-trivial relations between them ence Foundation. (SectionV). We showed that the regularized, energy- constrained coherent information is achievable for entanglement transmission with an average energy constraint, Appendix A: Minimum fidelity and minimum under the assumption that the energy observable is of entanglement fidelity the Gibbs form (Definition3) and the channel satisfies the finite-output entropy condition (Condition1). We The following proposition states that a quantum code then proved that the various quantum and private ca- with good minimum fidelity implies that it has good min- pacities of degradable channels are equal and character- imum entanglement fidelity with negligible loss in param- ized by the single-letter, energy-constrained coherent in- eters. This was first established in [66] and reviewed in formation (SectionVII). We finally applied our results [67]. Here we follow the proof available in [69], which to Gaussian channels and recovered some results already therein established a relation between trace distance and known in the literature in addition to establishing new diamond distance between an arbitrary channel and the ones. identity channel. We have left open the question of proving that the regularized, energy-constrained private information, defined Proposition 5 Let C : T (H) → T (H) be a quantum in (250), is an achievable rate for private communica- channel with finite-dimensional input and output. Let 0 tion. We think that this should certainly be possible. H be a Hilbert space isomorphic to H. If One particular method for doing so would be to extend min hφ|C(|φihφ|)|φi ≥ 1 − ε, (A1) the results of [11] such that they apply to coding with en- |φi∈H ergy constraints and over infinite-dimensional channels. Other approaches, like that along the lines of [36, 37] then for public classical communication, in conjunction with √ the method from [4,5], could also be employed. The min hψ|(idH0 ⊗C)(|ψihψ|)|ψi ≥ 1 − 2 ε, (A2) 0 first approach mentioned above could potentially lead to |ψi∈H ⊗H a simpler proof of Theorem2 (regarding quantum com- where the optimizations are with respect to state vectors. munication instead of private communication), but the details remain to be worked out. Proof. The inequality in (A1) implies that the following Going forward from here, a great challenge is to es- inequality holds for all state vectors |φi ∈ H: tablish a general theory of energy-constrained private and quantum communication with a limited number of hφ| [|φihφ| − C(|φihφ|)] |φi ≤ ε. (A3) channel uses. Recent progress in these scenarios with- out energy constraints [30, 31] suggests that this might By the inequalities in (25), this implies that be amenable to analysis. Another question is to iden- √ tify and explore other physical systems, beyond bosonic k|φihφ| − C(|φihφ|)k1 ≤ 2 ε, (A4) 27 for all state vectors |φi ∈ H. We will show that the ∞-norm. The third inequality follows because the ∞- norm of a traceless Hermitian operator is bounded from ⊥ ⊥ ⊥ √ hφ| |φihφ | − C(|φihφ |) |φ i ≤ 2 ε, (A5) above by half of its trace norm [99, Lemma 4]. The final inequality follows from applying (A4). Let |ψi ∈ H0 ⊗ H be an arbitrary state vector. All for every orthonormal pair |φi, |φ⊥i of state vectors such state vectors have a Schmidt decomposition of the in H. Set following form: |φi + ik|φ⊥i X p |wki ≡ √ (A6) |ψi = p(x)|ζxi ⊗ |ϕxi, (A12) 2 x for k ∈ {0, 1, 2, 3}. Then it follows that where {p(x)}x is a probability distribution and {|ζxi}x and {|ϕxi}x are orthonormal sets, respectively. Then 3 consider that 1 X |φihφ⊥| = ik|w ihw |. (A7) 2 k k k=0 1 − hψ|(idH0 ⊗C)(|ψihψ|)|ψi = hψ|(idH0 ⊗ idH − idH0 ⊗C)(|ψihψ|)|ψi Consider now that = hψ|(idH0 ⊗ [idH −C])(|ψihψ|)|ψi X ⊥ ⊥ ⊥ = p(x)p(y)hϕ | [|ϕ ihϕ | − C(|ϕ ihϕ |)] |ϕ i. hφ| |φihφ | − C(|φihφ |) |φ i x x y x y y x,y ⊥ ⊥ ≤ |φihφ | − C(|φihφ |) ∞ (A8) (A13) 3 1 X ≤ k|w ihw | − C(|w ihw |)k (A9) Now applying the triangle inequality and (A5), we find 2 k k k k ∞ k=0 that 1 3 X 1 − hψ|(id 0 ⊗C)(|ψihψ|)|ψi ≤ k|wkihwk| − C(|wkihwk|)k (A10) H 4 1 k=0 √ X ≤ 2 ε. (A11) = p(x)p(y)hϕx| [|ϕxihϕy| − C(|ϕxihϕy|)] |ϕyi x,y X The first inequality follows from the characterization of ≤ p(x)p(y) |hϕx| [|ϕxihϕy| − C(|ϕxihϕy|)] |ϕyi| the operator norm as kAk = sup |hφ|A|ψi|, where x,y ∞ |φi,|ψi √ the optimization is with respect to state vectors |ϕi and ≤ 2 ε. (A14) |ψi. The second inequality follows from substituting (A7) and applying the triangle inequality and homogeneity of This concludes the proof.

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