Private and Quantum Capacities of More Capable and Less Noisy Quantum Channels

Two new classes of quantum channels, which we call more capable and less noisy, are introduced. The more capable class consists of channels such that the quantum capacities of the complementary channels to the environments are zero. The less noisy class consists of channels such that the private capacities of the complementary channels to the environment are zero. For the more capable class, it is clarified that the private capacity and quantum capacity coincide. For the less noisy class, it is clarified that the private capacity and quantum capacity can be single letter characterized.


I. INTRODUCTION
One of the most important problem in quantum information theory is to determine the quantum capacity of a noisy quantum channel. The capacity is defined as the transmission rate optimized over all possible quantum error correcting codes such that decoding errors vanish in the limit of asymptotically many uses of the channel.
Mathematically, a quantum channel can be described by the trace preserving completely positive (TPCP) map from the input system to the output system. By using the Stinespring dilation of the TPCP map, we can naturally define a complementary channel to an environment system, and we can regard the noisy quantum channel as a wire-tap channel [1,2] from the sender to the legitimate receiver and the eavesdropper who can observe the environment system of the channel (eg. see [3]). Then we can define the private capacity of the noisy quantum channel as the transmission rate optimized over all possible wire-tap codes such that decoding errors and information leakage vanish in the limit of asymptotically many uses of the channel.
The private capacity and quantum capacity of noisy quantum channels were established in [4][5][6][7]. However unlike the capacity formula of a classical noisy channel or the private capacity formula of a classical wire-tap channel, the private capacity and quantum capacity formulae are not single letter characterized, i.e., the formulae involve the limit with respect to the number of channel uses, and they are not computable. Indeed, some numerical evidences clarified that the expressions in the capacity formulae are not additive [8][9][10][11], and the single letter characterization is not possible in general at least by using the same expressions.
A quantum channel is called degradable if there exists another degrading channel such that the conjunction of the channel to the legitimate receiver and the degrading channel coincide with the complementary channel to the eavesdropper. In such a case, the single letter charac- * Electronic address: shun-wata@is.tokushima-u.ac.jp terizations of the private capacity and quantum capacity were established [3,12].
A quantum channel is called conjugate degradable if there exists another degrading channel such that the conjunction of the channel to the legitimate receiver and the degrading channel coincide with the complementary channel to the eavesdropper up to complex conjugation. In such a case, the single letter characterizations were also established [13].
To date, all quantum channel whose capacities are single letter characterized are degradable or conjugate degradable 1 , and it is important to clarify a broader class of quantum channels such that the single letter characterizations are possible.
Aside from the possibility of the single letter characterizations, there is also another interesting problem. In the quantum information theory, the private information transmission and the quantum information transmission are closely related [4,[14][15][16], and the possibility of the latter implies the possibility of the former. However, the private information transmission and the quantum information transmission are not exactly equivalent. Indeed, although the private capacity and quantum capacity coincide for degradable quantum channels [17], the former can be strictly larger than the latter in general. Especially the private capacity can be positive even if the quantum capacity is zero [18]. Thus it is important to clarify a condition on quantum channels such that the private capacity and quantum capacity coincide or not.
To shed light on the above mentioned two problems, we introduce two new classes of quantum channels, which we call more capable and less noisy. The more capable class consists of channels such that the quantum capacities of the complementary channels are zero. The less noisy class consists of channels such that the private capacities of the complementary channels are zero. Later, these definitions turn out to be natural analogies of the partial orderings, more capable and less noisy, between classical channels [19].
The inclusive relation of the degradable, the conjugate degradable, the less noisy, and the more capable classes are summarized in Fig. 1. In this paper, we show that the private capacity and quantum capacity coincide for channels in the more capable class. Furthermore, we also show that the private capacity and quantum capacity can be single letter characterized for channels in the less noisy class. These results provide partial solutions to the above mentioned two problems.
The rest of the paper is organized as follows. In Section II, we review some known results on the private capacity and quantum capacity of quantum channels. In Section III, the more capable and less noisy classes are introduced, and we state our main results. In Section IV, we summarize certain properties implied by more capable and less noisy, and show proofs of our main results. We finalize the paper with conclusion in Section IV.

II. PRELIMINARIES
Let N B be a quantum channel from an input system H A to an output system H B . By using the Stinespring dilation (eg. see [3]), there exist an environment system H E and an isometry U BE from H A to the joint system for every input ρ, where Tr E is the partial trace with respect to the environment system. By using this representation, we can naturally define another channel which is usually called the complementary channel of N B . Although the Stinespring dilation is not unique, the following arguments do not depend on the choice of the dilation because two dilations can be converted to each other by applying a local unitary to the environment systems.
Throughout the paper, we basically follow the notations from [3,20]. The von Neumann entropy of a density matrix ρ is defined by H(ρ) = −Trρ log ρ, and the quantum relative entropy between ρ and σ is defined by D(ρ σ) = Trρ(log ρ − log σ). For input state ρ A to the channel N B , the coherent information is de- When the input state is clear from the context, we omit the subscript and denote I c (A B). The quantum mutual information of ρ XB on the joint system is defined by then the quantum mutual information can be written as When the legitimate receiver can observe the output of N B and the eavesdropper can observe the output of N E , the private capacity is characterized by [4,5] C p (N B ) = lim where where {ρ u A } are states (not necessarily pure states) on H A indexed by u ∈ U, and P U is a probability distribution on a finite set U.
On the other hand, when the sender want to transmit quantum information to the receiver through the channel N B , the quantum capacity is characterized by [4,6] where Both Eqs. (1) and (2) cannot be single letter characterized, i.e., C p (N B ) > C (1) p (N ) and Q(N ) > Q (1) (N ) in general [8][9][10][11]. Furthermore, the private capacity can be strictly larger than the quantum capacity, i.e., C p (N B ) > Q(N B ) for some channels [18].
The channel N B is said to be degradable if there exists a TPCP map D such that D • N B = N E . This is a quantum analogue of degraded broadcast channel [21]. When N B is degradable, then it is known [3,12] that the single letter formulae hold, i.e., C p (N B ) = C (1) Let C denote entry-wise complex conjugation with respect to a fixed basis of H E . Then, the channel N B is said to be conjugate degradable if there exists a TPCP map D such that D • N B = C • N E . When N B is conjugate degradable, it is known that Q(N B ) = Q (1) (N B ) [13]. Later, it will turn out that C p (N B ) = Q(N B ) = Q (1) (N B ) also holds.

III. MAIN STATEMENTS
In this section, we introduce two new classes of quantum channels, and show our main results.

Definition 1
The quantum channel N B is said to be more capable than the environment, or just more capable, if the quantum capacity of the complementary channel to the environment is zero, i.e., Q(N E ) = 0. (3)

Definition 2
The quantum channel N B is said to be less noisy than the environment, or just less noisy, if the private capacity of the complementary channel to the environment is zero, i.e., Since C p (N E ) ≥ Q(N E ), less noisy implies more capable. By using the eigenvalue decomposition of ρ A n , we can rewrite the coherent information as where we set |X | = dim H A . Thus, by noting Eq. (2), the quantum channel N B being more capable can be also described as holds for every n ≥ 1. Furthermore, by noting Eq. (1), the quantum channel N B being less noisy can be also described as holds for every n ≥ 1. Eqs. (6) and (7) resemble the definitions of more capable and less noisy for classical channels [19], and it is justified to call quantum channels satisfying Eqs. (3) or (4) more capable or less noisy.
In [19], alternative description of less noisy, less divergence contracting, was introduced, and we can also extend such a description to the quantum channel. The quantum channel N B is said to be less divergence contracting if holds for every n ≥ 1. Later, we will show that the quantum channel is less noisy if and only if less divergence contracting (Proposition 4). This alternative description plays a crucial role when we prove Theorem 2.
The followings are our main results.
Theorem 1 Suppose that the quantum channel N B is more capable. Then, we have Theorem 2 Suppose that the quantum channel N B is less noisy. Then, we have When N B is conjugate degradable, we can show that C p (N E ) = 0 as follows. Suppose that the sender sends a state ρ i that corresponds to the message i, and the eavesdropper 2 uses a POVM {M i }. Then, for the entrywise complex conjugate POVM {M i }, we have , where the last equality follows becauseM T i = M i and (C ⊗n • N ⊗n E (ρ i )) T = N ⊗n E (ρ i ). Thus, the legitimate receiver can get exactly the same information as the eavesdropper and the private information transmission to the eavesdropper is impossible. From this argument, conjugate degradable implies less noisy.
When the quantum capacity of the channel is 0 but it can be used to share bound entanglement, then the channel is called a binding entanlement channel [22]. Especially when the channel produces a positive partial transpose (PPT) bound entanglement, the channel is called PPT binding entanglement channel. If a complementary channel is a binding entanglement channel, then the main channel obviously belongs to the more capable class. Since the complementary channel of the conjugate degradable channel can only produce PPT bipartite state [13], if there exists a conjectured negative partial transpose (NPT) binding entanglement channel, the complementary of such a channel belongs to the more capable class but not to the conjugate degradable class.
It is known that there exists a channel such that the quantum capacity is zero (PPT binding entanglement channel) but the private capacity is strictly positive [18]. Let the complementary channel N E be such a channel. Then the channel N B is more capable but not less noisy 3 . Thus, the more capable class is strictly broader than the less noisy class. However, it is not yet clear whether the less noisy class is strictly broader than the degradable or conjugate degradable classes.

IV. PROOF OF THEOREMS
A. Properties of C In this section, we summarize the properties of C (1) p (N B ) and Q (1) (N B ) when Eqs. (6) or (7) hold for n = 1. For n ≥ 2, we can also show similar properties of C (1) p (N ⊗n B ) and Q (1) (N ⊗n B ) when Eqs. (6) or (7) hold for each n by considering n times extension of N B . The following properties can be regarded as quantum extensions of the properties shown for classical channels in the literatures [2,19,23,24] Proposition 1 Suppose that Eq. (6) holds for n = 1. Then we have be an eigenvalue decomposition. LetX be the random variable on U × X such that where Since P U and {ρ u A } are arbitrary, we have The opposite inequality is obvious from the definitions of C Proposition 2 Suppose that Eq. (6) does not hold for n = 1, and the density operator ρ * A maximizing I c (A B) is full rank. Then, we have Especially when dim H A = 2 and C (1) p (N B ) > 0, the sufficient and required condition on is that Eq. (6) holds for n = 1.
Proof. Since Eq. (6) does not hold for n = 1, there existŝ ρ A such that Since ρ * A is full rank, there exists 0 < λ < 1 such that ρ * A − λρ A is positive semidefinite. We construct P U and {ρ u A } as follows. Let be eigenvalue decompositions. Let U = {0} ∪ X . Then, we set P U (0) = λ, P U (u) = β u for u ∈ X , ρ 0 A =ρ A , and ρ u A = |φ u φ u | for u ∈ X . LetX be the random variable on U × X such that where Eq. (10) follows from I(X; B|U = 0) − I(X; E|U = 0) = I c (A B)ρ < 0 and Eq. (11) follows from Next, we show the latter statement of the proposition. The sufficient condition follows from Proposition 1. Suppose that holds. Since C Then, since dim H A = 2, ρ * A must be full rank. Thus, the required condition follows from the former statement of the proposition.

Proof. Let
be an eigenvalue decomposition.
Then, let U = {1, . . . , m}, P U (u) = p i ,X be the random variable on U × X such that It should be noted that the coherent information is known to be concave if the quantum channel N B is degradable [3, Eq. Proposition 4 The following two conditions are equivalent.
Next, we show that (ii) implies (i). For any P U and {ρ u A }, we have Since Eq. It is a straight forward consequence of Proposition 1. Since N B is more capable, Eq. (6) holds for every n ≥ 1. Thus, we have C (1) p (N ⊗n ) = Q (1) (N ⊗n B ) for every n ≥ 1, and the statement of the theorem follows from Eqs. (1) and (2).

C. Proof of Theorem 2
Since less noisy implies more capable, by Theorem 1, it suffice to show Q(N B ) = Q (1) (N B ). For this purpose, we will show that holds for every n ≥ 1. For any input state ρ A k A ℓ on H ⊗(k+ℓ) A , let ρ A k and ρ A ℓ be the partial traces. Then, we have Since Eq. (7) holds for n = (k + ℓ), by (n times extension of) Proposition 4, Eq. (8) also holds for n = (k+ℓ), which implies Thus, Eq. (14) can be proved by induction.

V. CONCLUSION
In this paper, we introduced two new classes of quantum channels, which we call more capable and less noisy. For the more capable class, we showed that the private capacity and quantum capacity coincide. For the less noisy class, we showed that the private capacity and quantum capacity can be single letter characterized.
Our results shed light on further understanding of the private and quantum capacities of quantum channels. However, the conditions such that a certain channel belongs to the more capable class or the less noisy class are hard to verify in general, and we do not yet know whether there exists a channel that belongs to the less noisy class but not to the degradable or conjugate degradable classes, which is an important future research agenda.