One-shot entanglement-assisted quantum and classical communication

# One-shot entanglement-assisted quantum and classical communication

Nilanjana Datta and Min-Hsiu Hsieh Nilanjana Datta and Min-Hsiu Hsieh are with Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK. (E-mail: n.datta@statslab.cam.ac.uk and minhsiuh@gmail.com)
###### Abstract

We study entanglement-assisted quantum and classical communication over a single use of a quantum channel, which itself can correspond to a finite number of uses of a channel with arbitrarily correlated noise. We obtain characterizations of the corresponding one-shot capacities by establishing upper and lower bounds on them in terms of the difference of two smoothed entropic quantities. In the case of a memoryless channel, the upper and lower bounds converge to the known single-letter formulas for the corresponding capacities, in the limit of asymptotically many uses of it. Our results imply that the difference of two smoothed entropic quantities characterizing the one-shot entanglement-assisted capacities serves as a one-shot analogue of the mutual information, since it reduces to the mutual information, between the output of the channel and a system purifying its input, in the asymptotic, memoryless scenario.

## I Introduction

An important class of problems in quantum information theory concerns the evaluation of information transmission capacities of a quantum channel. The first major breakthrough in this area was made by Holevo [1, 2], and Schumacher and Westmoreland [3], who obtained an expression for the capacity of a quantum channel for transmission of classical information. They proved that this capacity is characterized by the so-called “Holevo quantity”. Expressions for various other capacities of a quantum channel were obtained subsequently, the most important of them perhaps being the capacity for transmission of quantum information. It was established by Lloyd [4], Shor [5], and Devetak [6]. Both of the above capacities require regularization over asymptotically many uses of the channel. The classical capacity formula is given in terms of a regularized Holevo quantity while the quantum capacity formula is given in terms of a regularized coherent information.

Even though these regularized expressions are elegant, they are not useful because the regularization prevents one from explicitly computing the capacity of any given channel. Moreover, since these capacities are in general not additive, surprising effects like superactivation can occur [7, 8]. Hence it is more desirable to obtain expressions for capacities which are given in terms of single-letter formulas, and therefore have the attractive feature of being exempt from regularization.

The quantum and classical capacities of a quantum channel can be increased if the sender and receiver share entangled states, which they may use in the communication protocol. From superdense coding we know that the classical capacity of a noiseless quantum channel is exactly doubled in the presence of a prior shared maximally entangled state (an ebit). For a noisy quantum channel too, access to an entanglement resource can lead to an enhancement of both its classical and quantum capacities. The maximum asymptotic rate of reliable transmission of classical (quantum) information through a quantum channel, in the presence of unlimited prior shared entanglement between the sender and the receiver, is referred to as the entanglement-assisted classical (quantum) capacity of the channel.

Quantum teleportation and superdense coding together imply the following simple relation between quantum and classical communication through a noiseless qubit channel: if the sender and receiver initially share an ebit of entanglement, then transmission of one qubit is equivalent to transmission of two classical bits111Without prior shared entanglement, one can only send one classical bit through a single use of a noiseless qubit channel. Moreover, a noiseless classical bit channel cannot be used to transmit a qubit.. This relation carries over to the asymptotic setting under the assumption of unlimited prior shared entanglement between sender and receiver [13]. One can then show that the entanglement-assisted quantum capacity of a quantum channel is equal to half of its entanglement-assisted classical capacity. In fact, the entanglement-assisted classical capacity was the first capacity for which a single-letter formula was obtained. This result is attributed to Bennett, Shor, Smolin and Thapliyal [9]. Its proof was later simplified by Holevo [10], and an alternative proof was given in [11]. A trade-off formula for the entanglement-assisted quantum capacity region was obtained by Devetak, Harrow, and Winter [12, 13].

The different capacities of a quantum channel were originally evaluated in the so-called asymptotic, memoryless scenario, that is, in the limit of asymptotically many uses of the channel, under the assumption that the channel was memoryless (i.e., there is no correlation in the noise acting on successive inputs to the channel). In reality however, the assumption of channels being memoryless, and the consideration of an asymptotic scenario is not necessarily justified. A more fundamental and practical theory of information transmission through quantum channels is obtained instead in the so-called one-shot scenario (see e.g. [14] and references therein) in which channels are available for a finite number of uses, there is a correlation between their successive actions, and information transmission can only be achieved with finite accuracy. The optimal rate at which information can be transmitted through a single use of a quantum channel (up to a given accuracy) is called its one-shot capacity. Note that a single use of the channel can itself correspond to a finite number of uses of a channel with arbitrarily correlated noise. The one-shot capacity of a classical-quantum channel was studied in [15, 16, 17], whereas the the one-shot capacity of a quantum channel for transmission of quantum information was evaluated in [14, 18].

The fact that the one-shot scenario is more general than the asymptotic, memoryless one is further evident from the fact that the asymptotic capacities of a memoryless channel can directly be obtained from the corresponding one-shot capacities. Moreover, one-shot capacities also yield the asymptotic capacities of channels with memory (see e.g. [18]). In [19] the asymptotic entanglement-assisted classical capacity of a particular class of quantum channels with long-term memory, given by convex combinations of memoryless channels, was evaluated. The classical capacity of this channel was obtained in [20]. A host of results on asymptotic capacities of channels with memory can be attributed to Bjelakovic et al (see [21, 22] and references therein).

In this paper we study the one-shot entanglement-assisted quantum and classical capacities of a quantum channel. The requirement of finite accuracy is implemented by imposing the constraint that the error in achieving perfect information transmission is at most , for a given . We completely characterize these capacities by deriving upper and lower bounds for them in terms of the same smoothed entropic quantities.

Our lower and upper bounds on the one-shot entanglement-assisted quantum and classical capacities converge to the known single-letter formulas for the corresponding capacities in the asymptotic, memoryless scenario. Our results imply that the difference of two smoothed entropic quantities characterizing these one-shot capacities serves as a one-shot analogue of the mutual information, since it reduces to the mutual information between the output of a channel and a system purifying its input, in the asymptotic, memoryless setting.

The paper is organized as follows. We begin with some notations and definitions of various one-shot entropic quantities in Section II. In Section III, we first introduce the one-shot entanglement-assisted quantum communication protocol and then give upper and lower bounds on the corresponding capacity in Theorem 8. The proof of the upper bound is given in the same section, whereas the proof of the lower bound is given in Appendix B. The case of one-shot entanglement-assisted classical communication is considered in Section IV, and the bounds on the corresponding capacity is given in Theorem 13, the proof of which is given in the same section. In Section V, we show how our results in the one-shot setting can be used to recover the known single-letter formulas in the asymptotic, memoryless scenario. Finally, we conclude in Section VI.

## Ii Notations and Definitions

Let denote the algebra of linear operators acting on a finite-dimensional Hilbert space , and let be the set of positive operators of unit trace (states):

 D(H)={ρ∈B(H):Trρ=1}.

Furthermore, let

 D≤(H):={ρ∈B(H):Trρ≤1}.

Throughout this paper, we restrict our considerations to finite-dimensional Hilbert spaces and denote the dimension of a Hilbert space by .

For any given pure state , we denote the projector simply as . For an operator , let denote its restriction to the subsystem . For given orthonormal bases and in isomorphic Hilbert spaces of dimension , we define a maximally entangled state (MES) of Schmidt rank to be

 |Φ⟩AB=1√dd∑i=1|iA⟩⊗|iB⟩. (1)

Let denote the identity operator in , and let denote the completely mixed state in .

In the following we denote a completely positive trace-preserving (CPTP) map simply as , and denote the identity map as . Similarly, we denote an isometry simply as .

The trace distance between two operators and is given by

 ||A−B||1:=Tr[{A≥B}(A−B)]−Tr[{A

where denotes the projector onto the subspace where the operator is non-negative, and . The fidelity of two states and is defined as

 F(ρ,σ):=Tr√√ρσ√ρ=∣∣∣∣√ρ√σ∣∣∣∣1. (2)

Note that the definition of fidelity can be naturally extended to subnormalized states. The trace distance between two states and is related to the fidelity as follows (see e. g. [23]):

 1−F(ρ,σ)≤12||ρ−σ||1≤√1−F2(ρ,σ). (3)

The entanglement fidelity of a state , with purification , with respect to a CPTP map is defined as

 Fe(ρQ,A):=⟨ΨQR|(idR⊗A)(|Ψ⟩⟨Ψ|RQ)|ΨRQ⟩. (4)

We will also make use of the following fidelity criteria. The minimum fidelity of a map is defined as

 Fmin(T):=min|ϕ⟩∈H⟨ϕ|T(|ϕ⟩⟨ϕ|)|ϕ⟩. (5)

The average fidelity of a map is defined as

 Fav(T):=∫dϕ⟨ϕ|T(|ϕ⟩⟨ϕ|)|ϕ⟩. (6)

The results in this paper involve various entropic quantities. The von Neumann entropy of a state is given by . Throughout this paper we take the logarithm to base . For any state the quantum mutual information is defined as

 I(A:B)ρ:=H(A)ρ+H(B)ρ−H(AB)ρ. (7)

The following generalized relative entropy quantity, referred to as the max-relative entropy, was introduced in [24]:

###### Definition 1

The max-relative entropy of two operators and is defined as

 Dmax(ρ||σ):=logmin{λ:ρ≤λσ}. (8)

We also use the following min- and max- entropies defined in [25, 26, 27]:

###### Definition 2

Let . The min-entropy of conditioned on is defined as

 Hmin(A|B)ρ=maxσB∈D(HB)[−Dmax(ρAB||IA⊗σB)].
###### Definition 3

For any , we define the -ball around as follows

 Bε(ρ)={¯¯¯ρ∈D≤(H):F2(¯¯¯ρ,ρ)≥1−ε2}.
###### Definition 4

Let and . The -smoothed min-entropy of conditioned on is defined as

 Hεmin(A|B)ρ=max¯¯ρAB∈Bε(ρAB)Hmin(A|B)¯¯ρ.

The max-entropy is defined in terms of the min-entropy via the following duality relation [25, 26, 28]:

###### Definition 5

Let and let be an arbitrary purification of . Then for any

 Hεmax(A|C)ρ:=−Hεmin(A|B)ρ. (9)

In particular, if is a pure state, then

 Hεmin(A|B)ρ=−Hεmax(A)ρ. (10)

For any state , the smoothed max-entropy can be equivalently expressed as [25, 28]

 Hεmax(A|B)ρ:=min¯¯ρAB∈Bε(ρAB)Hmax(A|B)¯¯ρ, (11)

where

 Hmax(A|B)¯¯ρ=maxσB∈D(HB)2logF(¯¯¯ρAB,IA⊗σB). (12)

Moreover, for any ,

 Hmax(A)ρ:=2logTr√ρA. (13)

Various properties of the entropies defined above, which we employ in our proofs, are given in Appendix A.

## Iii One shot Entanglement-Assisted Quantum Capacity of a Quantum Channel

Entanglement-assisted quantum information transmission through a quantum channel is also referred to as the “father” protocol [12, 13]. The goal of this section is to analyse the one-shot version of this protocol. In order to do so, we first study the protocol of one-shot entanglement assisted entanglement transmission through a quantum channel, which is detailed below. We obtain bounds on its capacity in terms of smoothed entropic quantities, and then prove how these bounds readily yield bounds on the capacity of the one-shot “father” protocol.

The one-shot -error entanglement-assisted entanglement transmission protocol is as follows (see Fig. 1). The goal is for Alice to transmit half of a maximally entangled state, , that she shares with a reference , to Bob through a quantum channel , with the help of a maximally entangled state which she initially shares with him, such that finally the maximally entangled state is shared between Bob and the reference . We denote the latter as to signify that Alice’s system has been transferred to Bob. Note that denotes the number of ebits of entanglement consumed in the protocol and denotes the number of qubits transmitted from Alice to Bob. We require that Alice achieves her goal up to an accuracy , for some fixed .

Let Alice and Bob initially share a maximally entangled state . Without loss of generality, a one-shot -error entanglement-assisted entanglement transmission code of rate can then be defined by a pair of encoding and decoding operations as follows:

1. Alice performs some encoding (CPTP map) . Let us denote the encoded state as

 ξB1RA′=(idB1R⊗EA0A1→A′)(ΦA0R⊗ΦA1B1).

and denote the channel output state as

 |Ω⟩B1RBEE1=(IB1R⊗UA′→BEN)|ξ⟩B1RA′E1 (14)

where is a Stinespring extension of the quantum channel and is some purification of the encoded state .

2. After receiving the channel output , Bob performs a decoding operation on the systems in his possession. Denote the output state of Bob’s decoding operation by

 ˆΩB0B′REE1:=(idREE1⊗DB1B→B0B′)(ΩB1RBEE1). (15)

For a quantum channel , and any fixed , a real number is said to be an -achievable rate if there exists a pair of encoding and decoding maps such that,

 Fe(τA0,˜D∘N∘˜E)=⟨ΦRA0|ˆΩRB0|ΦRA0⟩≥1−ε, (16)

where , , and is the CPTP map defined through the relation222This relation uniquely defines the map because it specifies its Choi-Jamiolkowski state.

 (idR⊗˜E)(ΦA0R):=(idRB1⊗E)(ΦA0R⊗ΦA1B1).

Note that

 ˆΩRB0:=(idR⊗˜D∘N∘˜E)(ΦRA0).
###### Definition 6 (Entanglement transmission fidelity)

For any Hilbert space , the entanglement transmission fidelity of a quantum channel , in the presence of an assisting maximally entangled state, is defined as follows.

 Fe(N,HA0):=max˜E,˜DFe(τA0,˜D∘N∘˜E), (17)

where denotes a completely mixed state in , and , are CPTP maps.

###### Definition 7

Given a quantum channel and a real number , the one-shot -error entanglement-assisted entanglement transmission capacity of is defined as follows:

 E(1)ea,ε(N):=max{log|A0|:Fe(N,HA0)≥1−ε}.

Our main result of this section is the following theorem, which gives upper and lower bounds on in terms of smoothed min- and max- entropies.

###### Theorem 8

For any fixed , , and being a positive number such that , the one-shot -error entanglement-assisted entanglement transmission capacity of a noisy quantum channel , in the case in which the assisting resource is a maximally entangled state, satisfies the following bounds:

 maxϕA′∈D(HA′)12[Hε′min(A)ψ−Hε′max(A|B)ψ]+2logε′≤E(1)ea,ε(N)≤maxϕA′∈D(HA′)12[Hεmin(A)ψ−H2ε+2√κmax(A|B)ψ]+log√2ε, (18)

where the maximisation is over all possible inputs to the channel. In the above, denotes the following state

 |ψ⟩ABE:=(IA⊗UA′→BEN)|ϕ⟩AA′ (19)

where is a Stinespring isometry realizing the channel, and denotes a purification of the input to the channel.

Proof of the lower bound in (18). The proof is given in Appendix B.

Proof of the upper bound in (18). As stated in the beginning of Sec. III, any one shot -error entanglement-assisted entanglement transmission protocol (see Fig. 1) of a quantum channel consists of a pair of encoding-decoding maps such that the condition (16) holds. However, this condition along with Theorem 4 of [30] imply that there exists a partial isometry such that

 Fe(τA0,A∘V)≥1−2ε, (20)

where , with being the channel and being the decoding map followed by a partial trace over . The condition (20) in turn implies that

 F(ˆΩRB0,ΦRA0)≥1−2ε, (21)

where

 ˆΩRB0:=(idR⊗˜D∘N∘V)(ΦRA0).

By using Uhlmann’s theorem [29], and the second inequality in (3), we infer from (21) that

 ∥ˆΩB0B′RE−ΦRB0⊗σB′E∥1≤2√2√4ε, (22)

for some state , where denotes the output state of Bob’s decoding operation, which in this case is defined by

 ˆΩB0B′RE:=(idRE⊗DB1B→B0B′)(ΩB1RBE), (23)

with

 |Ω⟩B1RBE=(IB1R⊗UA′→BEN)|ξ⟩RA′B1 (24)

where denotes the state resulting from the isometric encoding on .

By the monotonicity of the trace distance under the partial trace, we have

 ∥ˆΩRE−τR⊗σE∥1≤2√2√4ε:=κ. (25)

From (23) it follows that , since the decoding map does not act on the systems and . This fact, together with (25) implies that and .

Let us set in the state defined in (24), and let and . Then

 −Hε+2ε′+ε′′max(A|B)Ω (26) = Hε+2ε′+ε′′min(RB1|E)Ω ≥ Hε′′min(R|E)Ω+Hε′min(B1|RE)Ω−log2ε2 ≥ Hmin(R)τ+Hε′min(B1|RE)Ω−log2ε2 ≥ Hmin(R)τ+Hε′min(B1|REB)Ω−log2ε2 = Hmin(R)τ−Hε′max(B1)Ω−log2ε2 ≥ log|A0|−log|A1|−log2ε2.

The first equality holds because of the duality relation (9) between the conditional smoothed min- and max- entropies. The first inequality follows from the chain rule for smoothed min-entropies (Lemma 19). The second inequality follows from Lemma 20 of Appendix A and , whereas the third inequality follows from Lemma 21 of Appendix A. The second equality follows from the duality relation (9) and the fact that is a pure state. The last inequality holds because

 log|A0| = log|R|=Hmin(R)τ log|A1| = log|B1|≥Hε′max(B1)Ω.

Moreover, for any ,

 Hεmin(A)Ω≥Hmin(RB1)τ⊗τ=log|A0|+log|A1| (27)

since . Combining (26) and (27) and choosing yields

 log|A0|≤12[Hεmin(A)Ω−H2ε+2√κmax(A|B)Ω]+log√2ε.

This completes the proof of the upper bound in (18) since we can choose to be the pure state corresponding to the channel output (see (24)) when the optimal isometric encoding is applied.

### Iii-a One-shot entanglement-assisted quantum (EAQ) capacity of a quantum channel

###### Definition 9 (Minimum output fidelity)

For any Hilbert space , we define the minimum output fidelity of a quantum channel , in the presence of an assisting maximally entangled state, as follows:

 Fmin(N,H):=max˜E,˜Dmin|ϕ⟩∈HF2(|ϕ⟩,˜D∘N∘˜E(|ϕ⟩⟨ϕ|)), (28)

where , are CPTP maps.

###### Definition 10

For any fixed , the one-shot entanglement-assisted quantum (EAQ) capacity of a quantum channel is defined as follows:

 Q(1)ea,ε(N):=max{log|H|:Fmin(N,H)≥1−ε}. (29)

The following theorem allows us to relate the one-shot entanglement-assisted entanglement transmission capacity to the one-shot entanglement-assisted quantum (EAQ) capacity .

###### Theorem 11

For any fixed , for a quantum channel , and an assisting entanglement resource in the form of a maximally entangled state,

 E(1)ea,ε(N)−1≤Q(1)ea,2ε(N)≤E(1)ea,4ε(N). (30)
###### Proof:

The proof is given in Appendix C. \qed

## Iv One shot Entanglement-Assisted Classical Capacity of a Quantum Channel

We consider entanglement-assisted classical (EAC) communication through a single use of a noisy quantum channel, in the case in which the assisting resource is given by a maximally entangled state. The scenario is depicted in Fig. 2. The sender (Alice) and the receiver (Bob) initially share a maximally entangled state , where Alice possesses while Bob has , and . The goal is for Alice to transmit classical messages labelled by the elements of the set to Bob, through a single use of the quantum channel , with the help of the prior shared entanglement.

Without loss of generality, any EAC communication protocol can be assumed to have the following form: Alice encodes her classical messages into states of the system in her possession. Let the encoding (CPTP) map corresponding to her classical message be denoted by , for each . Alice then sends the system through the noisy quantum channel . After Bob receives the channel output , he performs a POVM on the system in his possession, which yields the classical register containing his inference of the message sent by Alice.

###### Definition 12 (One-shot ε-error EAC capacity)

Given a quantum channel and a real number , the one-shot -error entanglement-assisted classical capacity of is defined as follows:

 C(1)ea,ε(N):=max{log|K|:∀k∈K,Pr[k≠^k]≤ε} (31)

where the maximization is over all possible encoding operations and POVMs.

The following theorem gives bounds on the one-shot -error EAC capacity of a quantum channel.

###### Theorem 13

For any fixed , the one-shot -error entanglement-assisted classical capacity of a noisy quantum channel , in the case in which the assisting resource is a maximally entangled state, satisfies the following bounds

 maxϕA′[Hε′′min(A)ψ−Hε′′max(A|B)ψ]+4logε′′−2≤ C(1)ea,ε(N)≤maxϕA′[H4εmin(A)ψ−H8ε+2√κ′max(A|B)ψ]+log12√2ε, (32)

where the maximization is over all possible input states, to the channel, and is such that . In the above, is the pure state defined in (19).

### Iv-a Acheivability

The lower bound in (32) is obtained by employing the one-shot version of the entanglement-assisted quantum communication (or “father”) protocol.

From Theorem 8 and Theorem 11, it follows that the one-shot -error entanglement-assisted quantum communication protocol for a quantum channel that consumes ebits of entanglement and transmits qubits can be expressed in terms of the following one-shot resource inequality:

 ⟨N⟩+e(1)ε[qq]≥εq(1)ε[q→q]. (33)

Here represents one qubit of quantum communication from Alice (the sender) to Bob (the receiver); represents an ebit shared between Alice and Bob, and the notation is used to emphasize that the error in achieving the goal of the protocol is at most . In the above, and , given below, respectively, follow from (82) and (83):

 e(1)ε =12[Hε′min(A)ψ+Hε′max(A|B)ψ] (34) q(1)ε =12[Hε′min(A)ψ−Hε′max(A|B)ψ]+2logε′−1, (35)

where is such that , and is defined in (19).

The resource inequality (33) readily yields a resource inequality for one-shot EAC communication through a noisy quantum channel, which in turn can be used to obtain a lower bound on the one-shot EAC capacity. This can be seen as follows. Combining (33) with the resource inequality for superdense coding:

 [qq]+[q→q]≥2[c→c],

yields the following resource inequality for one-shot EAC communication through the noisy channel :

 ⟨N⟩+q(1)ε[qq]+e(1)ε[qq] ≥εq(1)ε[q→q]+q(1)ε[qq] ≥√ε2q(1)ε[c→c]. ⇒⟨N⟩+(q(1)ε+e(1)ε)[qq] ≥√ε2q(1)ε[c→c]. (36)

Replacing by in (36) directly yields the following lower bound on the -error one-shot EAC capacity333The necessity of replacing by arises from the different fidelity criteria used in defining the one-shott entanglement assisted quantum and classical capacities (see (28) and (31)):

 C(1)ea,ε(N) ≥2q(1)ε2 =[Hε′′min(A)ψ−Hε′′max(A|B)ψ]+4logε′′−2, (37)

where is as defined in Theorem 13. Note that (37) reduces to the lower bound in (32) when the optimal input state is used.

### Iv-B Proof of the Converse

We prove the upper bound in (32) by showing that if it did not hold then one would obtain a contradiction to the upper bound (in (18)) on the one-shot entanglement-assisted quantum capacity of a channel.

Let us assume that

 C(1)ea,ε(N)>Δ(2ε,N), (38)

where

 Δ(2ε,N):=maxϕA′[H4εmin(A)ψ−H8ε+2√κ′max(A|B)ψ]+log12√2ε, (39)

with , being given by (19), and the maximization being over all possible input states to the channel444The factor of in front of arises from the different fidelity criteria used in defining the one-shott entanglement assisted quantum and classical capacities (see (28) and (31)), and the relation (3) between the fidelity and the trace distance.. This is equivalent to the assumption that more than bits of classical information can be communicated through a single use of with an error , in the presence of an entanglement resource in the form of a maximally entangled state.

Now since unlimited entanglement is available for the protocol, we infer that by quantum teleportation more than qubits can be transmitted over a single use of with an error . Then from the definition (29) of the one-shot -error entanglement assisted quantum capacity of the channel , it follows that . However, this contradicts the upper bound to as obtained from (30) and (18).

## V Entanglement-assisted classical and quantum capacities for multiple uses of a memoryless channel

### V-a Entanglement-assisted classical capacity for multiple uses of a memoryless channel

###### Definition 14

We define the entanglement-assisted classical capacity in the asymptotic memoryless scenario as follows:

 C∞ea(N):=limε→0liminfn→∞1nC(1)ea,ε(N⊗n)

where denotes the one-shot -error EAC capacity for independent uses of the channel .

Next we show how the known achievable rate for EAC communication in the asymptotic, memoryless scenario can be recovered from Theorem 13. We also prove that this rate is indeed optimal [9, 10].

###### Theorem 15

[9, 10] The entanglement-assisted classical capacity in the asymptotic memoryless scenario is given by the following:

 C∞ea(N)=maxϕA′∈D(HA′)I(A:B)ψ (40)

where the maximization is over all possible input states to the channel , is defined in (19), and denotes the mutual information of the state .

###### Proof:

First we prove that

 C∞ea(N)≥maxϕA′∈D(HA′)I(A:B)ψ. (41)

From the lower bound in Theorem 13, we have

 C∞ea(N)≥limε→0liminfn→∞1n(maxϕA′n∈D(H⊗nA′)[Hε′′min(An)ψn−Hε′′max(An|Bn)ψn]+4logε′′−2) (42)

where is defined as

 ψn≡ψAnBn:=(idAn⊗N⊗n)(ϕAnA′n), (43)

where denotes a purification of the input state , and is as defined in Theorem 13. By restricting the maximization in the above inequality to the set of input states of the form

 C∞ea(N) ≥ limε→0liminfn→∞1n(maxϕA′∈D(HA′)[Hε′′min(An)ψ⊗n (44) −Hε′′max(An|Bn)ψ⊗n]+4logε′′−2)

where is defined through (19). Let be the state such that

 I(A:B)ˆψ=maxϕA′∈D(HA′)I(A:B)ψ.

Further restricting to the state , we can obtain the following from (44)

 C∞ea(N) ≥ limε→0liminfn→∞1n[Hε′′min(An)ˆψ⊗n (45) −Hε′′max(An|Bn)ˆψ⊗n+4logε′].

Then from the superadditivity of the limit inferior and the fact that the limits on the right-hand side of the above equation exist [26], we obtain

 C∞ea(N)≥limε→0limn→∞1nHε′′min(An)ˆψ⊗n−limε→0limn→∞1nHε′′max(An|Bn)ˆψ⊗n. (46)

Finally, by using Lemma 22, we obtain the desired bound (41):

 C∞ea(N) ≥H(A)ˆψ−H(A|B)ˆψ =I(A:B)ˆψ.

Next we prove that

 C∞ea(N)≤maxϕA′∈D(HA′)I(A:B)ψ. (47)

From the upper bound in Theorem 13, we have

 C∞ea(N)≤limε→0liminfn→∞1n