A hypothesis testing approach for communication over entanglement ...

A hypothesis testing approach for communication over entanglement assisted compound quantum channel

arXiv:1706.08286v1 [quant-ph] 26 Jun 2017

Anurag Anshu∗

Rahul Jain†

Naqueeb Ahmad Warsi‡

Abstract We study the problem of communication over compound quantum channel in the presence of entanglement. Classically such channels are modeled as a collection of conditional probability distributions wherein neither the sender nor the receiver is aware of the channel being used for transmission, except for the fact that it belongs to this collection. We provide achievability and converse bounds for this problem in the one shot quantum setting in terms of quantum hypothesis testing divergence. Our achievability proof is similar in spirit to its classical counterpart. To arrive at our result, we use the technique of position based decoding along with a new approach for constructing a union of two projectors, which can be of independent interest.

1

Introduction

A typical assumption while communicating over a channel is that the communicating parties are aware of the channel characteristics. This means that the parties know the output distribution (or the quantum state) for a given input (or a quantum state). This leads to the well known model of point to point channel, which has been extensively studied in literature starting from the seminal work of Shannon [1]. It is not hard to imagine a real world setting which differs from this point of view. Let us consider the case where the communicating parties are not completely aware of the channel characteristics, potentially due to lack of sufficient statistical data or chaotic behavior of the channel. In such a case, the parties may have to work with the assumption that the channel is an element of a finite collection of channels. This setting is known as the compound channel. It has a straightforward quantum version where the channel is an element of a finite collection of quantum maps, that is, {N (1) , N (2) , . . . , N (s) }. In this work, we shall consider the entanglement assisted classical capacity of such compound quantum channels. By standard duality between teleportation and superdense coding, our results also apply to entanglement assisted quantum capacity. The classical version of this problem in the asymptotic i.i.d. (independent and identically distributed) setting was studied in [2, 3] (see also [4, Theorem 7.1]). The quantum capacities of compound quantum channels have been studied in several works, such as [5, 6, 7, 8, 9, 10, 11]. The work [11] studied the entanglement assisted capacities in one shot and asymptotic i.i.d. setting, where optimal results were obtained in the asymptotic i.i.d setting. The key tool used in [11] was that of the decoupling method and their bounds were obtained in terms of smooth min and max entropies. The classical version of this problem in the asymptotic i.i.d. (independent and identically distributed) setting was studied in [2, 3] (see also [4, Theorem 7.1]). The quantum capacities of compound quantum channels have been studied in several works, such as [5, 6, 7, 8, 9, 10, 11]. The work [11] studied the entanglement assisted capacities in one shot and asymptotic i.i.d. setting, where optimal results were obtained in the asymptotic i.i.d setting. The key tool used in [11] was that of the decoupling method and their bounds were obtained in terms of smooth min and max entropies. In this work, we give a one shot achievability result for this task in terms of the quantum hypothesis testing divergence. This divergence has been shown to characterize the near optimal amount of communication possible over the classical-quantum channel [12] and entanglement assisted quantum channel [13]. Broadly, our technique ∗

Centre for Quantum Technologies, National University of Singapore, Singapore. [email protected] Centre for Quantum Technologies, National University of Singapore and MajuLab, UMI 3654, Singapore. [email protected] ‡ Centre for Quantum Technologies, National University of Singapore and School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore and IIITD, Delhi. [email protected] †

1

follows the position-based decoding method introduced in [13], which allows for quantum hypothesis testing to be performed on several registers. But we require a new technical component along with that used in [13]. A key challenge that arises is to formulate a suitable quantum version of the union of two events in probability theory. In the classical achievability result for communication over compound channel, one uses a statement of the form Pr {E1 ∪ E2 } ≥ max{Pr {E1 } , Pr {E2 }}, for two events E1 and E2 (see for example, [4, Theorem 7.1]), which is a converse to the union bound that states Pr {E1 ∪ E2 } ≤ Pr {E1 } + Pr {E2 }. Quantum analogues of both these statements have been studied in previous works. Two well known examples of quantum version of the union bound are the Hayashi-Nagaoka inequality [14] and the results on sequential measurement [15, 16, 17]. The converse to the union bound has been studied in the quantum setting in [18] (where it is called “Quantum OR bound”) and in [19]. The result in [19, Corollary 11] is as follows: consider a collection of projectors {Π1 , Π2 , . . . Πs } and a quantum state ρ, with the property that either there exists an i such that Tr[Πi ρ] ≥ 1 − ε or Ei Tr[Πi ρ] ≤ δ. Then there exists a and in the second case, Tr[Πρ] ≤ 4δs. projector Π with the property that in first case, Tr[Πρ] ≥ (1−ε) 7 Above result cannot be used for our purpose since we require a one shot decoding strategy which makes an error of at most ε, for every ε ∈ (0, 1). To overcome this, we prove a new quantum analogue of the converse to the union bound in Theorem 2, which is more suited for our application. This is one of the main contributions of this paper and uses Jordan’s lemma (on the joint structure of two projectors) at its core. Informally, the statement of Theorem 2 is as follows. Consider a collection of projectors {Π1 , Π2 , . . . Πs } and quantum states {ρ1 , ρ2 , . . . ρs } such that for all i, Tr[Πi ρi ] ≥ 1 − ε. Then there exists a projector Π∗ such that it succeeds well on all ρi (that is, Tr[Π∗ ρi ] ≥ 1 − O(ε)) log(s) and for all quantum states σ, Tr[Π∗ σ] ≤ sO(log ε ) Ei Tr[σΠi ]. Using Theorem 2 and the aforementioned position based decoding, we give our achievability result in Theorem 3. Our achievability result is in terms of a one shot quantity derived from quantum hypothesis testing divergence. More precisely, let DεH (ρkσ) denote the smooth quantum hypothesis testing divergence. We consider the following variant of the smooth quantum hypothesis testing divergence, for a bipartite quantum state ρAB : IεH (A : B)ρAB := min DεH (ρAB kσA ⊗ ρB ) , σA

where the minimization is over all quantum states σA . This quantity appeared earlier in the work [20] in context of converse bounds for entanglement assisted classical communication over noisy quantum channels. Using the techniques developed in [20], we also give a converse bound for our task (which was also observed in the asymptotic i.i.d. setting in [11]) in terms of IεH (A : B)ρ . The converse appears as Theorem 4. We note that the achievability and converse bounds differ by an additive factor of O(log s) log logε s . The additive loss of O(log sε ) in the amount of communication can also be found in the one shot classical case (for example, in the one shot version of the argument given in [4, Theorem 7.1]). An important question is to establish the asymptotic i.i.d. properties of our bound. For this, we relate the quantity IεH (A : B)ρAB to the quantity DεH (ρAB kρA ⊗ ρB ) in Theorem 5, where we use the converse result in [20] and the achievability result in [13]. The achievability result in the asymptotic i.i.d. setting can then be obtained by appealing to the asymptotic i.i.d. behavior of DεH (ρAB kρA ⊗ ρB ) [21, 22]. A matching converse in the asymptotic i.i.d. setting has been given in [11], again using the ideas developed in [20]. An interesting variant of the compound channel is that of a compound channel with informed sender. In this setting, the sender knows which channel is acting from the given collection, but the receiver has no such information. This was considered in the classical asymptotic i.i.d. setting in [23] and in the quantum one shot and asymptotic i.i.d. settings in [11]. We give our achievability result in terms of a variant of the aforementioned one shot version of quantum mutual information, which appears in Section 5.

2

Preliminaries

For a natural numbers n, m with n ≤ m, let [n : m] represent the set {n, n + 1, . . . m}. For N > 0, log N is with respect to the base 2 and ln N is with respect to base e. Consider a finite dimensional Hilbert space H endowed with an inner product h·, ·i (in this√ paper, we only consider X † X and ℓ2 norm is finite dimensional Hilbert-spaces). The ℓ norm of an operator X on H is kXk := Tr 1 1 √ kXk2 := TrXX † . A quantum state (or a density matrix) is a positive semi-definite matrix on H with trace equal to 1. It is called pure if and only if its rank is 1. A sub-normalized quantum state is a positive semi-definite matrix on H with trace less than or equal to 1. Let |ψi be a unit vector on H, that is hψ, ψi = 1. With some abuse of notation, we

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use ψ to represent the quantum state and also the density matrix |ψihψ|, associated with |ψi. Given a quantum state ρ on H, support of ρ, called supp(ρ) is the subspace of H spanned by all eigen-vectors of ρ with non-zero eigenvalues. A quantum register A is associated with some Hilbert space HA . A quantum state ρ on register A is represented as ρA . If two registers A, B are associated with the same Hilbert space, we shall represent the relation by A ≡ B. Composition of two registers A and B, denoted AB, is associated with Hilbert space HA ⊗ HB . For two quantum states ρ and σ, ρ ⊗ σ represents the tensor product (Kronecker product) of ρ and σ. The identity operator on HA (and associated register A) is denoted IA . For normal operators A and B we will use the notation B  A (B ≺ A) if (A − B) is a positive semi-definite operator (positive definite operator). P Let ρAB be a quantum state. We define ρB := TrA ρAB := i (hi| ⊗ IB )ρAB (|ii ⊗ IB ), where {|ii}i is an orthonormal basis for the Hilbert space HA . The quantum state ρB is referred to as the marginal quantum state of ρAB . Unless otherwise stated, a missing register from subscript in a quantum state will represent partial trace over that register. Given a ρA , a purification of ρA is a pure quantum state ρAB such that TrB ρAB = ρA . Purification of a quantum state is not unique. A unitary operator UA : HA → HA is such that UA† UA = UA UA† = IA . An isometry V : HA → HB is such that † V † V = IP A and V V = IB . A POVM on the register A is a collection of positive semi-definite operators {Mi }i∈I such that i Mi = IA .

Definition 1. Let ρ, σ be quantum states. The smooth quantum hypothesis testing divergence is defined as DεH (ρkσ) :=

max

0MI Tr[Mρ]≥1−ε

− log Tr [M σ] .

Definition 2. Let ρAB be a quantum state. The following quantity is a variant of the smooth quantum hypothesis testing divergence: IεH (A : B)ρAB := min σA

= min σA

DεH (ρAB kσA ⊗ ρB ) max

0MI Tr[MρAB ]≥1−ε

− log Tr [M (σA ⊗ ρB )] .

We note that the minimization in above quantity is over the first register in the argument. Fact 1 (Minimax theorem, [24]). Let X , Y be convex compact sets and f : X × Y → R be a continuous function that satisfies the following properties: f (·, y) : X → R is convex for fixed y, and f (x, ·) : Y → R is concave for fixed x. Then it holds that min max f (x, y) = max min f (x, y). x∈X y∈Y

y∈Y x∈X

The following lemma follows from Definition 2 and Fact 1. Lemma 1. Let ρAB be a quantum state. There exists an operator M ∗ satisfying Tr[M ∗ ρAB ] ≥ 1 − ε, such that for all quantum states σA , −Iε (A:B)ρ AB . Tr[M ∗ (σA ⊗ ρB )] ≤ 2 H Proof. From Definition 2, we conclude that 2

−IεH (A:B)ρ

AB

= max σA

a

=

min

0MI Tr[MρAB ]≥1−ε

min

Tr [M (σA ⊗ ρB )]

max Tr [M (σA ⊗ ρB )]

σA 0MI Tr[MρAB ]≥1−ε

b

= max Tr [M ∗ (σA ⊗ ρB )] , σA

where a follows from the minimax theorem (Fact 1) and the facts that Tr [M (σA ⊗ ρB )] is linear in M for a fixed σA (and vice-versa), σA belongs to a convex compact set and M belongs to a convex compact set and b follows by defining M ∗ to the operator that achieves the infimum in second equality. The lemma concludes with the observation that M ∗ also satisfies Tr[M ∗ ρAB ] ≥ 1 − ε.

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Lemma 2 (Jordan’s lemma, [25]). For any two projectors Π(1) and Π(2) , there exists a set of orthogonal projectors {Πα }kα=1 (each of dimension either one or two), for some natural number k, such that P 1. α∈[1:k] Πα = I. 2. Πα Π(i) = Π(i) Πα , for all i ∈ {1, 2}, α ∈ [1 : k].

3. Πα Π(i) Πα is a one dimensional projector, for i ∈ {1, 2}, α ∈ [1 : k]. Fact 2 (Hayashi-Nagaoka inequality, [14]). Let 0 ≺ S ≺ I, T be positive semi-definite operators. Then 1

1

I − (S + T )− 2 S(S + T )− 2  2(I − S) + 4T. Theorem 1 (Neumark’s theorem). For any POVM {Mi }i∈I acting on a system S, there exists a unitary USP and an orthonormal basis {|iiP }i∈I such that for all quantum states ρS , we have h i † Tr USP (IS ⊗ |iihi|P ) USP (ρS ⊗ |0ih0|P ) = Tr [Mi ρS ] .

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A union of projectors

In this section, we prove a quantum version of the following classical statement. Let E1 , E2 ⊆ X be two sets and p be a probability distribution over X . Then there exists a set E ∗ ⊆ X (more precisely, E ∗ = E1 ∪ E2 ) such that Prp {E ∗ } ≥ max{Prp {E1 } , Prp {E2 }}. The following lemma is a quantum version of this statement.

Lemma 3. Let {Π(1) , Π(2) } be two projectors. For every δ > 0, there exists a projector Π⋆ such that for all quantum states ρ, Tr [Π⋆ ρ] ≥ max{Tr[Π(1) ρ], Tr[Π(2) ρ]} − δ;  2  Π⋆  2 Π(1) + Π(2) . δ

k

(1) (2)

Proof. Let {Πα }α=1 be the set of orthogonal projectors (each either one or two dimensional) obtained by Jordan’s Pk lemma (Lemma 2) applied on Π(1) , Π(2) such that α=1 Πα = I. Furthermore, for α ∈ [1 : k] and i ∈ {1, 2} let Π(i) (α) := Πα Π(i) Πα .

Observe that Π(i) =

P

α

Π(i) (α). Also, let o n h i Far := α : Tr Π(1) (α)Π(2) (α) < 1 − δ 2 ,

and let the set Near be the compliment of the set Far. For every α ∈ [1 : k], let Π⋆ (α) be defined as follows: ( Πα if α ∈ Far; ⋆ Π (α) = (1) Π (α) otherwise. We now show that Π⋆ :=

P

α∈[1:k]

(3)

Π⋆ (α) satisfies the properties mentioned in Equations (1) and (2).

  Proof of Equation (1): Fix a quantum state ρ and let ρ(α) := Πα ρΠα . Let i ∈ {1, 2} be such that Tr Π(i) ρ =

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    max{Tr Π(1) ρ , Tr Π(2) ρ }. Consider, Tr [Π⋆ ρ] =

X

α∈Far a

= ≥ =

α∈Near

h i Tr Π(1) (α)ρ(α)

Tr [Πα ρ(α)] +

X

h i h i X Tr Π(i) (α)ρ(α) + Tr Π(1) (α)ρ(α)

α∈Far

X

X

α∈Near

α∈[1:k] c

Tr [Π⋆ (α)ρ(α)]

X

α∈Far b

X

Tr [Π⋆ (α)ρ(α)] +

α∈Near

 h i  X Tr Π(i) (α)ρ(α) + Tr[ρ(α)]Tr Π(1) (α) − Π(i) (α) α∈Near

h i X

≥ Tr Π(i) ρ − Tr[ρ(α)] Π(1) (α) − Π(i) (α) α∈Near

ρ(α) Tr[ρ(α)]



∞

h i X d ≥ Tr Π(i) ρ − δ Tr[ρ(α)] α∈Near

(i)

≥ Tr[Π ρ] − δ,

(i) where a followsP from the definition of Π⋆(α) mentioned    in Equation (3); b follows from the identity Πα  Π (α); c follows since α∈[1:k] Tr Π(i) (α)ρ(α) = Tr Π(i) ρ and d follows from the property of the set Near.

Proof of Equation (2): To prove the property mentioned in Equation (2) we assume the following claim:

Claim 1. For α ∈ [1 : k],

 2  (1) (2) Π (α) + Π (α) . δ2 The proof of this claim is given towards the end of this proof. Notice the following set of inequalities: X Π⋆ = Π⋆ (α) Π⋆ (α) 

α

This completes the proof of the lemma.

 X 2   Π(1) (α) + Π(2) (α) 2 δ α   2 = 2 Π(1) + Π(2) . δ

Proof of Claim 1: Claim trivially follows for α ∈ Near. We now consider the case when α ∈ Far. Towards this notice that Πi (α) is a one dimensional projector as guaranteed by Jordan’s lemma, for each i ∈ {1, 2}. Further, let Π(2) (α) be defined as follows: |Π(2) (α)i = γ|Π(1) (α)i + β|Π1 (α)i⊥ ; (4)

where |Π(1) (α)i⊥ is the unit vector orthogonal to |Π(1) (α)i in the subspace corresponding to Πα . From the definition √ 2 of the set Far we conclude that |γ| < 1 − δ . Now consider the operator Π(1) (α)+Π(2) (α) which can be represented as follows:   1 + |γ|2 γ⋆β , (5) γβ ⋆ 1 − |γ|2 where in the above we have used the fact that |β|2 = 1 − |γ|2 . The characteristic equation of the matrix in Equation (5) satisfies the following: (1 − λ)2 − |γ|4 − |γ|2 (1 − |γ|2 ) = 0, (6) where λ is an eigenvalue. From Equation (6) we have that λ ≥ 1 − |γ| > proves the claim.

δ2 2 .

Thus,

δ2 2 Πα

Lemma 3 allows us to the prove the following theorem, which is our main result.

5

 Π(1) (α) + Π(2) (α). This

Theorem 2. Let ε, δ > 0. For i ∈ [1 : s], let ρ(i) be a quantum state and Π(i) be a projection operator such that i h Tr Π(i) ρ(i) ≥ 1 − ε. Then there exists a projection operator Π⋆ such that i h Tr Π⋆ ρ(i) ≥ 1 − ε − (δ log(2s));  log(2s)   2 (1) (2) (s) . Π + Π + . . . + Π Π⋆  δ2

Proof. Without loss of generality, we assume that s = 2t for some integer t. The proof for general s proceeds in similar fashion. We group the projectors Π(i) into pairs {Π(1) , Π(2) }, {Π(3) , Π(4) }, . . . , {Π(s−1) , Π(s) }. Applying Lemma 3 to each of the pair with given δ, we obtain a collection of projectors {Π(1,2) , Π(3,4) , . . . Π(s−1,s) } such that for every odd i < s, we have   h i h i  2 (i,i+1) (i) (i,i+1) (i+1) (i,i+1) (i) (i+1) Tr Π ρ ≥ 1 − ε − δ, Tr Π ρ ≥ 1 − ε − δ, Π  Π + Π . δ2 Now, we further group the projectors {Π(1,2) , Π(3,4) , . . . Π(s−1,s) } into consecutive pairs and by applying Lemma 3 with given δ, obtain projectors {Π(1,2,3,4) , Π(5,6,7,8) , . . . Π(s−3,s−2,s−1,s) } such that for every i satisfying i mod 4 = 1, we have i h Tr Π(i,i+1,i+2,i+3) ρ(j) ≥ 1 − ε − 2δ ∀j ∈ [i, i + 3]; Π(i,i+1,i+2,i+3) 



2 δ2

    2 2  (i) (i+1) (i+2) (i+3) Π + Π + Π + Π Π(i,i+1) + Π(i+2,i+3)  . δ2

Continuing in this way till log(s) steps (for a general s, the number of steps will be at most log(2s)), we obtain the desired projector Π∗ . This completes the proof.

4

Communication over compound quantum channel

Definition 3. Let |θiEA EB be the shared entanglement between Alice n and Bob. o Let M be the message register. An s

(i)

(R, ε)-entanglement assisted code for a compound quantum channel NA→B

i=1

consists of

• An encoding operation E : M EA → A for Alice .

• A decoding operation D : BEB → M ′ for Bob, with M ′ ≡ M being the output register such that for all m and for all i ∈ [1 : s], Pr {M ′ 6= m|M = m} ≤ ε.

Achievability result n os (i) Theorem 3. Let NA→B be a compound quantum channel and let ε ∈ (0, 1). Let A′ ≡ A be a purifying i=1 register. Then, for any R satisfying     ε ε + log , (7) min IεH (B : A′ )N (i) (|ψihψ| ′ ) + (2 log 2s) log R ≤ max AA A→B 6 log(2s) s |ψihψ|AA′ i∈[1:s] n os (i) there exists an (R, 8ε)-entanglement assisted code for the compound quantum channel NA→B . i=1

Proof. Fix |ψihψ|AA′ and R as given in Equation (7). Introduce the registers A1 , A2 , . . . A2R , such that Ai ≡ A and A′1 , A′2 , . . . A′2R such that A′i ≡ A′ . Alice and Bob share the quantum state |ψihψ|A1 A′1 ⊗ |ψihψ|A2 A′2 , . . . |ψihψ|A2R A′ R , 2

6

where Alice holds the registers A1 , A2 , · · · , A2R and Bob holds the registers A′1 , A′2 , · · · , A′2R . For i ∈ [1 : s], let (i) 0  MBA′  I be such that for all j ∈ [1 : s], we have i  h (j) (i) IεH (B : A′ )N (i) (|ψihψ| ′ ) ≤ − log Tr MBA′ NA→B (ψA ) ⊗ ψA′ . A→B

AA

(i) operator MBA′

The existence of such an is guaranteed by Lemma 1. Further, as guaranteed by the Neumark’s theorem (i) (Theorem 1), let ΠBA′ P be such that ∀i, j ∈ [1 : s], h  i h  i (i) (j) (i) (j) Tr MBA′ NA→B (ψA ) ⊗ ψA′ = Tr ΠBA′ P NA→B (ψA ) ⊗ ψA′ ⊗ |0ih0|P , and

h  i h  i (i) (i) (i) (i) Tr MBA′ NA→B (|ψihψ|AA′ ) = Tr ΠBA′ P NA→B (|ψihψ|AA′ ) ⊗ |0ih0|P .

Let Π⋆BA′ P be the operator obtained by setting ε (i) ← δ, NA→B (|ψihψ|AA′ ) ⊗ |0ih0|P ← ρ(i) , 3 log(2s)

(i)

ΠBA′ P ← Π(i)

in Theorem 2. Our protocol is as follows: Encoding: Alice on receiving the message m ∈ [1 : 2R ] sends the register Am over the channel. Assuming that the (i) channel NA→B was used for transmission, the quantum state in Bob’s possession is the following:
(i) (i) ΘBA′ ···A′ := ψA′1 ⊗ · · · ⊗ NAm →B |ψihψ|Am A′m · · · ⊗ ψA′ R , 1

2

2R

(i) ΘBA′ j

for some i ∈ [1 : s]. Further, notice that the quantum state between the register A′j and the channel output B is the following ( (i) NA→B (|ψihψ|AA′ ) if j = m; (i) ΘBA′ = (i) j NA→B (ψA ) ⊗ ψA′ otherwise.

Decoding: The decoding technique is derived from the work [13]. For each m ∈ [1 : 2R ] consider the following operator Λ(m) := IA′1 ⊗ IA′2 ⊗ · · · Π⋆BA′m P ⊗ · · · ⊗ IA′ R , 2

where Π⋆BA′m P is as discussed above. The decoding POVM element corresponding to m is:  − 12  − 21 X X Λ(m′ ) . Ω(m) :=  Λ(m′ ) Λ(m)  m′ ∈[1:2R ]

m′ ∈[1:2R ]

P

It is easy to observe that m Ω(m)  I. Bob on receiving the channel output appends an ancilla |0ih0|P to his registers and then measuresP his registers using the POVM defined above. He outputs ‘0’ for the outcome corresponding to the POVM element I − m Ω(m). Probability of error: Let M be the message which was transmitted by Alice using the strategy above and let M ′ be the decoded message by Bob using the above mentioned decoding POVM. Further, let us assume that the channel (i) NA→B be used for this transmission. By the symmetry of the encoding and decoding strategy, it is enough to show that Pr {M ′ 6= 1 | M = 1} ≤ 8ε, under the event that M = 1 is the transmitted message. i h (i) Pr {M ′ 6= 1|M = 1} = Tr (I − Ω(1)) ΘBA′ A′ ···A′ ⊗ |0ih0|P 1 1 2R i i h h X a (i) (i) Tr Λ(m′ )ΘBA′ A′ ···A′ ⊗ |0ih0|P ≤ 2 · Tr (I − Λ(1)) ΘBA′ A′ ···A′ ⊗ |0ih0|P + 4 1

1

2R

1

m′ 6=1

i h  h X b (i) Tr Π⋆BA′ = 2 · Tr I − Π⋆BA′1 P ΘBA′ ⊗ |0ih0|P + 4 1

c

≤ 2(2ε) + 4s × d



6 log(2s) ε

m′ 6=1

2 log(2s)

≤ 4ε + 4ε.

7

(i) P ΘBA′ m′

m′

1

2R

⊗ |0ih0|P

  × exp ln 2 · R − min IεH (B : A′ )N (j) j∈[1:s]

i

A→B (|ψihψ|AA′ )



where a follows from Hayashi-Nagaoka operator inequality (Fact 2); b follows from the definition of Λ(m), c follows from the properties of Π⋆BA′ P (see Theorem 2) and d follows from our choice of R.

Converse bound We introduce the following definition for our converse bound. Definition 4. Let |θiEA EB be the shared entanglement between Alice and Bob. Let M be the message register with uniform distribution over the messages. An (R, ε, ψA )-entanglement assisted code for a quantum channel NA→B consists of • An encoding operation E : M EA → A for Alice, such that the quantum state at the input register A, averaged over the message in register M , is ψA . • A decoding operation D : BEB → M ′ for Bob, with M ′ ≡ M being the output register such that for all i ∈ [1 : s], Pr {M ′ 6= M } ≤ ε. n os (i) Theorem 4. Let NA→B be a compound quantum channel and let ε ∈ (0, 1). For any (R, ε)-entanglement i=1 assisted code for this compound quantum channel, it holds that   min IεH (B : A′ )N (i) (|ψihψ| ′ ) . R ≤ max |ψihψ|AA′

i∈[1:s]

A→B

AA

Proof. We consider the case where the message of Alice is drawn from a uniform distribution. This implies the result in the theorem, which is for the worst case over the messages. Fix an i ∈ [1 : s] and the corresponding quantum (i) channel NA→B . As shown in [20, Equation 76], for any (R, ε, ψA )-entanglement assisted code for the quantum (i) channel NA→B , we have R ≤ IεH (B : A′ )N (i) (|ψihψ| ′ ) , A→B

where |ψihψ|

AA

is a purification of ψA . In the case of compound quantum channel, the value of i is unknown to osthe n (i) communicating parties. Hence, for any (R, ε)- entanglement assisted code for the compound channel NA→B i=1 with average quantum state ψA at the input, we have AA′

R ≤ min IεH (B : A′ )N (i)

A→B (|ψihψ|AA′ )

i∈[1:s]

.

Since the choice of the quantum state ψA is arbitrary, maximizing above expression over all possible pure quantum states |ψihψ|AA′ establishes the result. Finally, we relate IεH (B : A′ )ρBA′ to the quantity DεH (ρBA′ kρB ⊗ ρA′ ) in the following theorem. This helps us to provide optimal bounds in the asymptotic i.i.d. case. Theorem 5. Let ρBA′ be a quantum state and ε ∈ (0, 1). For every δ > 0, it holds that DεH (ρBA′ kρB ⊗ ρA′ ) − 2 log

1 2ε+2δ 2ε+2δ ≤ IH (B : A′ )ρBA′ ≤ DH (ρBA′ kρB ⊗ ρA′ ) . δ

Proof. Consider a purification |ρihρ|ABA′ of the quantum state ρBA′ . Let NAB→B be the channel that traces out register A. From the achievability result [13, Theorem 1], there exists an (R, 2ε + 2δ, ρAB )-entanglement assisted code for the channel NAB→B such that R = DεH (NAB→B (ρABA′ )kNAB→B (ρAB ) ⊗ ρA′ ) − 2 log

1 1 = DεH (ρBA′ kρB ⊗ ρA′ ) − 2 log . δ δ

On the other hand, as shown in [20, Equation 76], for any (R, 2ε + 2δ, ρAB )-entanglement assisted code for the channel NAB→B , it holds that 2ε+2δ 2ε+2δ R ≤ IH (B : A′ )NAB→B (ρABA′ ) = IH (B : A′ )ρBA′ . 2ε+2δ 2ε+2δ This establishes the lower bound on IH (B : A′ )ρBA′ . The upper bound on IH (B : A′ )ρBA′ follows from the definition. This completes the proof.

8

5

The case of informed sender

n os (i) In this section we discuss the case where the sender is aware about which channel in the set NA→B is being i=1 used for transmission. Towards this we make the following definitions: Definition 5. Let ρAB , σB be quantum states. Define: ε ˜Iε (A : B) H ρAB ,σB := min DH (ρAB kτA ⊗ σB ) . τA

Definition 6. Let ρAB be a quantum state. Following is another variant of the smooth quantum hypothesis testing divergence: ε ˜ε ˆIεH (A : B) ρAB := min IH (A : B)ρAB ,σB = min DH (ρAB kτA ⊗ σB ) . σB ,τA

σB

The following lemma is an analogue of Lemma 1 and follows from Definition 5 and Fact 1. Lemma 4. Let ρAB , σB be quantum states. There exists an operator M ∗ satisfying Tr[M ∗ ρAB ] ≥ 1 − ε, such that for all quantum states τA −˜Iε (A:B)ρ AB ,σB . Tr[M ∗ (τA ⊗ σB )] ≤ 2 H Proof. From Definition 5, we conclude that 2

−IεH (A:B)ρ

AB ,σB

=

max τA

a

= b

=

min

0MI Tr[MρAB ]≥1−ε

min

Tr [M (τA ⊗ σB )]

max Tr [M (τA ⊗ σB )]

τA 0MI Tr[MρAB ]≥1−ε

max Tr [M ∗ (τA ⊗ σB )] , τA

where a follows from the minimax theorem (Fact 1) and the facts that Tr [M (τA ⊗ σB )] is linear in M for a fixed τA (and vice versa), τA belongs to a convex compact set and M belongs to a convex compact set and b follows by defining M ∗ to the operator that achieves the infimum in second equality. The lemma concludes with the observation that Tr[M ∗ ρAB ] ≥ 1 − ε. Let M denote the message register. Definition 7. Let |θiEA EB be the shared entanglement between Alice and n Bob. o s (i) An (R, ε)-entanglement assisted code for a compound quantum channel NA→B in the case of informed sender i=1 consists of • An encoding operation Ei : M EA → A for Alice (notice the dependence of the encoding function on the index i). • A decoding operation D : BEB → M ′ for Bob, with M ′ ≡ M being the output register such that for all m and for all i ∈ [1 : s], Pr {M ′ 6= m|M = m} ≤ ε. We prove the following achievability result: os n (i) Theorem 6. Let NA→B be a compound quantum channel and let ε ∈ (0, 1). Let A′ ≡ A be a purifying i=1 register. Then for any R satisfying     ε ε + log 2 , (8) max ˆIεH (B : A′ )N (i) (|ψihψ| ′ ) + (log 2s) log R ≤ min AA A→B 6 log(2s) s i∈[1:s] |ψihψ|AA′ n os (i) there exists an (R, 8ε)-entanglement assisted code for compound channel NA→B in the case of informed sender. i=1

9

Proof. Fix R as given in Equation (8). Introduce the registers A1 , A2 , . . . As2R , such that Ai ≡ A and A′1 , A′2 , . . . A′s2R such that A′i ≡ A′ . Further, for every message m ∈ [1 : 2R ], Alice and Bob share a band of s entangled quantum states of the following form: (1)

(2)

|ψihψ|As(m−1)+1 A′

s(m−1)+1

(s)

⊗ |ψihψ|As(m−1)+2 A′

s(m−1)+2

, . . . |ψihψ|Asm A′sm ,

where Alice holds the registers As(m−1)+1 , As(m−1)+2 , · · · , Asm , and Bob holds the registers A′s(m−1)+1 , A′s(m−1)+2 , · · · , A′sm , (i)

and for every i ∈ [1 : s], |ψihψ|AA′ is such that it achieves the maximum in max ˆIεH (B : A′ )N (i)

A→B (|ψihψ|AA′ )

|ψihψ|AA′

Define the quantum state σA′ :=

1 s

max ˆIεH (B : A′ )N (i)

P

(i)

ψA′ . We observe that for all i ∈ [1 : s],

i∈[1:s]

A→B (|ψihψ|AA′ )

|ψihψ|AA′

.

= ˆIεH (B : A′ )N (i)

(i) A→B (|ψihψ|AA′ )

≤ ˜IεH (B : A′ )N (i)

(i) A→B (|ψihψ|AA′ ),σA′

.

(i)

For i ∈ [1 : s], let 0  MBA′  I be such that for all j ∈ [1 : s], we have i  h (j) (j) (i) ˜IεH (B : A′ ) (i) ′ . ) ⊗ σ (ψ N ≤ − log Tr M (i) ′ A A A→B BA N (|ψihψ| ),σ ′ A→B

AA′

A

(i)

The existence of such an MBA′ is guaranteed by Lemma 4. Further, as guaranteed by the Neumark’s theorem (Theo(i) rem 1), ∀i ∈ [1 : s], let ΠBA′ P be such that ∀j ∈ [1 : s] i  i h  h (j) (j) (i) (j) (j) (i) Tr MBA′ NA→B (ψA ) ⊗ σA′ = Tr ΠBA′ P NA→B (ψA ) ⊗ σA′ ⊗ |0ih0|P ,

and

i  i h  h (i) (i) (i) (i) (i) (i) Tr MBA′ NA→B (|ψihψ|AA′ ) = Tr ΠBA′ P NA→B (|ψihψ|AA′ ) ⊗ |0ih0|P .

Let Π⋆BA′ P be the operator obtained by setting ε ← δ, 3 log(2s)

(i)

(i)

(i)

ΠBA′ P ← Π(i)

NA→B (|ψihψ|AA′ ) ⊗ |0ih0|P ← ρ(i) ,

in Theorem 2. Our protocol is as follows: (i)

Encoding: Alice on receiving the message m ∈ [1 : 2R ] and getting informed that the channel NA→B will be used for transmission sends chooses the register As(m−1)+i from the band corresponding to the message m and transmits it (m,i) over the channel. Let ΘBA′ ···A′ be the quantum state on Bob’s registers after Alice’s transmission over the channel. 1

s2R

(m,i)

Notice that the quantum state ΘBA′ between the register A′k and the channel output B is the following k

(m,i) ΘBA′ k

=

(

(i)

(i)

NA→B (|ψihψ|AA′ ) (i) (i) (k mod s) NA→B (ψA ) ⊗ ψA′

if k = s(m − 1) + i; otherwise,

where k mod s is interpreted as s instead of 0, when k is a multiple of s. Decoding: For each k ∈ [1 : s2R ] consider the following operator Λ(k) := IA′1 ⊗ IA′2 ⊗ · · · Π⋆BA′ P ⊗ · · · ⊗ IA′ k

10

s2R

,

where Π⋆BA′ P is as discussed above. The decoding POVM element corresponding to m is: k



Ω(m) := 

X

k′ ∈[1:s2R ]

− 21 

Λ(k ′ )



X

k∈[s(m−1)+1:sm]



Λ(k) 

X

k′ ∈[1:s2R ]

− 12

Λ(k ′ )

.

P It is easy to observe that m Ω(m)  I. Bob on receiving the channel output appends an ancilla |0ih0|P to his registers and then measuresP his registers using the POVM defined above. He outputs ‘0’ for the outcome corresponding to the POVM element I − m Ω(m). Probability of error: Let M be the message which was transmitted by Alice using the strategy above and let M ′ be the decoded message by Bob using the above mentioned decoding POVM. Further, let us assume that the channel (i) NA→B is used for this transmission. By the symmetry of the encoding and decoding strategy, it is enough to show that Pr {M ′ 6= 1 | M = 1} ≤ 8ε, under the event that M = 1 is the transmitted message. h i (1,i) Pr {M ′ 6= 1|M = 1} = Tr (I − Ω(1)) ΘBA′ A′ ···A′ ⊗ |0ih0|P 1 1 s2R i h a (1,i) ≤ 2 · Tr (I − Λ(i)) ΘBA′ A′ ···A′ ⊗ |0ih0|P 1 1 R h s2 i X X (1,i) Tr Λ(k)ΘBA′ A′ ···A′ ⊗ |0ih0|P +4 1

m′ 6=1 k∈[m′ (s−1)+1:sm′ ]

b

= 2 · Tr

1

s2R

i  h X (1,i) I − Π⋆BA′i P ΘBA′ ⊗ |0ih0|P + 4 i

X

m′ 6=1 k∈[m′ (s−1)+1:sm′ ]

i h (1,i) Tr Π⋆BA′ P ΘBA′ ⊗ |0ih0|P k

k

i h i  h X X (j) (i) (i) (1,i) Tr Π⋆BA′ P NA→B (ψA ) ⊗ ψA′ ⊗ |0ih0|P = 2 · Tr I − Π⋆BA′i P ΘBA′ ⊗ |0ih0|P + 4 i

m′ 6=1 j∈[1:s]

h  i (1,i) = 2 · Tr I − Π⋆BA′i P ΘBA′ ⊗ |0ih0|P i     X X 1 (i) (i) (j) + 4s Tr Π⋆BA′ P NA→B (ψA ) ⊗  ψA′  ⊗ |0ih0|P  s m′ 6=1 j∈[1:s] i i h  h X (i) (i) (1,i) Tr Π⋆BA′ P NA→B (ψA ) ⊗ σA′ ⊗ |0ih0|P = 2 · Tr I − Π⋆BA′i P ΘBA′ ⊗ |0ih0|P + 4s i

c

2

≤ 2(2ε) + 4s × 2

≤ 2(2ε) + 4s × d



6 log(2s) ε



6 log(2s) ε

m′ 6=1

log(2s) log(2s)

  × exp ln 2 · R − min ˜IεH (B : A′ )N (j) j∈[1:s]

(j) A→B (|ψihψ|AA′ ),σA′

  × exp ln 2 · R − min ˆIεH (B : A′ )N (j) j∈[1:s]

(j) A→B (|ψihψ|AA′ )



≤ 4ε + 4ε. P where a follows from Hayashi-Nagaoka operator inequality (Fact 2) and the identity Λ(i)  k∈[1:s] Λ(k); b follows from the definition of Λ(m), c follows from the properties of Π⋆BA′ P (see Theorem 2) and d follows from our choice of R.

Acknowledgment This work is supported by the Singapore Ministry of Education and the National Research Foundation, through the Tier 3 Grant “Random numbers from quantum processes” MOE2012-T3-1-009 and NRF RF Award NRF-NRFF201313.

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