A-unital Operations and Quantum Conditional Entropy

Negative quantum conditional entropy states are key ingredients for information theoretic tasks such as superdense coding, state merging and one-way entanglement distillation. In this work, we ask: how does one detect if a channel is useful in preparing negative conditional entropy states? We answer this question by introducing the class of A-unital channels, which we show are the largest class of conditional entropy non-decreasing channels. We also prove that A-unital channels are precisely the completely free operations for the class of states with non-negative conditional entropy. Furthermore, we study the relationship between A-unital channels and other classes of channels pertinent to the resource theory of entanglement. We then prove similar results for ACVENN: a previously defined, relevant class of states and also relate the maximum and minimum conditional entropy of a state with its von Neumann entropy. The definition of A-unital channels naturally lends itself to a procedure for determining membership of channels in this class. Thus, our work is valuable for the detection of resourceful channels in the context of conditional entropy.


Introduction
The myriad information-theoretic applications of entanglement [1] have elevated it from being merely an interesting physical phenomenon to that of a resource. In the resourcetheoretic study of entanglement, the separable states (quantum states that can be written as convex combinations of pure product states) are called the free states, and those outside this set, the entangled states, are known as the resource states [2]. The resource theory also defines various classes of free operations, that is, quantum operations that cannot convert a free state to a resourceful one. Local Operations and Classical Communication (LOCC) is a physically defined class of free operations [1][2][3], characterized by the practical difficulty of communicating quantum information. A superset of LOCC is the class of separable channels [2,4], which has a more appealing mathematical structure than LOCC and is therefore used to prove properties of LOCC itself [4,5]. Entanglement-breaking channels form another important class of channels. These channels destroy all entanglement between a system and its environment [6].
Once these classes of channels have been defined, it is important to be able to decide whether or not a given channel belongs to any of these free classes. The quantum channel detection problem asks: given a channel and a well-defined class of channels, does the channel belong to the class [7][8][9][10]? The channel itself maybe provided as a blackbox, as in a laboratory setting, where it can be characterized by measuring its output on various input states, or as a full description (for instance, via its Kraus decomposition or Choi matrix). Unfortunately, for some classes defined with respect to entanglement, this problem is notoriously difficult. For example, it is currently unknown how to decide whether a given channel belongs to LOCC [2], and the problem of membership of a channel in the entanglement-breaking class is NP Hard [11]. There have been witness-based probabilistic approaches to deal with this problem, which are useful when some apriori information about a channel is known [7].
The interest in defining free operations for entanglement and studying mechanisms for detection of these operations is in part due to the resourcefulness of entanglement. While entanglement is certainly important in the characterization of quantum states for information-theoretic tasks, there are several phenomena for which merely possessing entanglement is not a sufficient condition to determine usefulness of a quantum state. Often, a state needs to have other properties beyond being entangled in order to be useful. One such property is the negativity of the quantum conditional entropy (also known as the conditional von Neumann entropy) of a quantum state [12]. The quantum conditional entropy (henceforth referred to as conditional entropy where there is no ambiguity) of a quantum state ρ AB is defined as where S(ρ) is the von Neumann entropy (henceforth referred to as entropy where there is no ambiguity) of the state ρ, given by S(ρ) = − Tr(ρ log ρ). All logarithms are taken to the base 2. Conditional entropy is used to define the information-theoretic quantities coherent information [13], which is just the negative of conditional entropy, and quantum discord [14][15][16], which is a measure of non-classical correlations in a quantum state. While all negative conditional entropy states are entangled, not all entangled states possess negative conditional entropy. Negative conditional entropy states are important for the informationtheoretic primitive quantum state merging [17,18], and provide quantum advantage in superdense coding [19][20][21]. Negativity of conditional entropy is a necessary feature for the one-way distillation of entanglement [22]. It is important in maximizing the rates of distributed private randomness distillation [23], as well as in the reduction of uncertainty in predicting the outcomes of two incompatible measurements [24]. Previously, the set of states that are not useful for any of the above tasks, that is, the set of states possessing non-negative conditional entropy (we refer to this set as F c (A|B)) was characterized as convex and compact [25,26]. 1 The set of states for which conditional entropy remains non-negative even under the application of global unitary channels (referred to as F ac (A|B) in this work) was also similarly characterized [27]. In this work, taking F c (A|B) to be a set of free states, we ask: what is the class of free operations for F c (A|B), and what are its properties? How does this class of operations relate to operations defined in the context of entanglement: separable channels and entanglement-breaking channels? How does one detect if a given channel belongs to the class of free operations for F c (A|B)? These questions are important in the context of preparation of quantum states for the aforementioned tasks, in order to determine if an operation one has access to is capable of generating resource. This paper is organized as follows: In Section 2, we introduce the notation, and recap the relevant preliminaries. In Section 3, we define a class of free operations NCVE(A|B), the largest class of conditional entropy non-decreasing channels. We prove that this class is exactly equal to O c cmax (A|B), the largest class of completely free channels for F c (A|B), and is elegantly characterized by a property we call A-unitality. We discuss properties of A-unital channels in Section 4. In Section 5, we discuss the relation between A-unital channels, separable channels and entanglement-breaking channels. While the set of separable states and F c (A|B) share a subset-superset relationship, we show that surprisingly, this relationship does not hold for A-unital channels and separable channels-there exist A-unital channels that are not separable, separable channels that are not A-unital, as well as channels in the intersection. In Section 6, we extend the class F ac (A|B), previously characterized only for the 2 ⊗ 2 case to the d ⊗ d case, and prove that unital channels are the largest class of completely free channels for F ac (A|B). Since both entropy and conditional entropy are heavily discussed in this manuscript, in Section 7, we provide upper and lower bounds for the conditional entropy of a state for a given entropy. We conclude in Section 8 with some concrete open questions that emerge from our work.

Preliminaries
In this section, we review relevant definitions and concepts from literature. A quantum state ρ AB is called separable if it can be written as a convex combination of pure product states. That is, Separable states are the free states in the resource theory of entanglement. We denote the set of separable states in D(AB) by F s (A|B). The set of states ρ AB ∈ D(AB) for which S A|B (ρ AB ) ≥ 0, that is, the set of states possessing non-negative conditional entropy (Eq. 1) is denoted by F c (A|B).
Free and completely free channels for a given set of free states were defined in [2]. We define these here for F c (A|B). Separable channels [4] form an important class of channels in the entanglement resource theory. We define these channels here: is called separable if and only if its action on a state ρ ∈ D(AB) can be expressed as for some integer n ≥ 1 and linear maps K i A : The class of all separable channels mapping D(AB) to D(CD) is denoted by SEP(A|B → C|D). When the input and output spaces are the same, the class is denoted by SEP(A|B) ≡ SEP(A|B → A|B).
It turns out that SEP(A|B) is exactly equal to O s cmax (A|B) [2]. That is, the class of separable channels form the largest class of completely free channels for F s (A|B).
Entanglement-breaking channels constitute another relevant class of channels [13]: Unital channels play an important role in the context of entropy, as the class of unital channels mapping D(A) to itself are exactly the ones that do not decrease the entropy of any state ρ ∈ D(A) [13]. We define them as

The class of all unital channels mapping D(A) to D(A) is denoted by UNI(A).
Having reviewed the definitions, we now characterize the class of completely free operations for F c (A|B).

Completely Free Operations for Conditional Entropy
Since the negativity of conditional entropy implies usefulness for various tasks, a natural candidate for a class of operations that are free for F c (A|B) is the class of channels that does not decrease the conditional entropy of any state. Conditional entropy non-decreasing channels are defined as follows:  It turns out that this class of channels is exactly equal to O c cmax (A|B) which was introduced in Def. 2. We prove this equivalence by showing that both NCVE(A|B) and O c cmax (A|B) are in fact equal to a third class of operations, defined as follows: for some ρ A ∈ D(A), and the set of B-unital channels on D(AB) is denoted as UNI(B|A).
Not only does the class of A-unital channels serve to show that both NCVE(A|B) and O c cmax (A|B) are equal, but its definition also makes this class amenable to detection, as discussed in Section 4.3. We now prove the equivalence of these three classes. Proof. The structure of the proof is depicted in Fig. 1. Let N : D(AB) → D(AB) be a quantum channel. In Lemma 1, we show that N ∈ UNI(A|B) =⇒ N ∈ NCVE(A|B), labelled as (a). In Lemma 2, we show that N ∈ NCVE(A|B) =⇒ N ∈ UNI(A|B), which is labelled as (b). In Lemma 3, we show that N ∈ UNI(A|B) =⇒ (id A B ⊗ N ) ∈ UNI(A A|B B). We then use Lemma 1 and Lemma 3 to prove Lemma 4, which is N ∈ UNI(A|B) =⇒ N ∈ O c cmax (A|B), labelled as (c) in the figure. Finally, we prove Lemma 5, which is N ∈ O c cmax (A|B) =⇒ N ∈ NCVE(A|B), labelled as (d) in the figure. These lemmas establish the equivalence between the three classes.
To prove the Lemma, we require two results related to quantum relative entropy. The first result is a characterization of the conditional entropy in terms of relative entropy [13,28]: Next, we require the monotonicity of relative entropy, which is From Eq. 5, for all ρ ∈ D(AB) and σ B ∈ D(B), we have In particular, we must have where From Eq. 6, we have Thus, for all ρ ∈ D(AB), we have S A|B (N (ρ)) ≥ S A|B (ρ), which implies that N ∈ NCVE(A|B).

Lemma 2. N ∈ NCVE(A|B) =⇒ N ∈ UNI(A|B).
Proof. For any ρ AB ∈ D(AB), the maximum value of S A|B (ρ AB ) is log d A and is attained if and only if ρ [13]. We use the above fact to prove the lemma by contradiction. Let N ∈ NCVE(A|B). Assume that N / ∈ UNI(A|B). Then, there exists some σ B ∈ D(B) such that However, this is a contradiction since N ∈ NCVE(A|B). Therefore, the assumption that N / ∈ UNI(A|B) must be false, and we have N ∈ NCVE(A|B) =⇒ N ∈ UNI(A|B). Proof. Let N AB ∈ UNI(A|B). Then, for all ρ B ∈ D(B), we have N (I A ⊗ ρ B ) = I A ⊗ ρ B for some ρ B ∈ D(B). It is always possible to find a basis for any D(X), in which each basis element is a density matrix, that is, each basis element also belongs to D(X) [13].
Let {β i } i be such a basis for D(B ) and {β j } j be such a basis for D(B). Thus, {β i ⊗ β j } ij is a basis of density matrices for D(B B). We have, Now, for all σ ∈ D(B B), we have Thus, id A B ⊗ N AB ∈ UNI(A A|B B). Proof. In order to prove the lemma, we prove the contrapositive statement, that is N AB / ∈ NCVE(A|B) =⇒ N AB / ∈ O c cmax (A|B). Let N AB : D(AB) → D(AB) be a quantum channel that does not belong to NCVE(A|B). Thus there exists ρ ∈ D(AB) such that S A|B (N (ρ)) < S A|B (ρ). Consider Hilbert spaces H A and H Note that because −S A|B (ρ) < −S A|B (N (ρ)) and both these values lie in , the state σ always exists.

Now, consider the state σ A B ⊗ ρ AB ∈ D(A B AB). Using Eq. 13, we have
We now apply the operation id A B ⊗ N AB on the state σ A B ⊗ρ AB . Once again, using Eq. 13, we have (A A|B B). Therefore, the channel id A B ⊗ N AB takes a state from inside F c (A A|B B) to outside, and using Def. 2, Thus, Lemmas 1-5 prove all the implications depicted in Fig. 1, and Theorem 1 follows.

Properties of A-Unital channels 4.1 Serial and Parallel Concatenation
Since we have UNI(A|B) = O c cmax (A|B), it follows that serial concatenation of two Aunital channels is A-unital, and parallel concatenation of an A -unital channel and an A-unital channel is A A-unital [2]. However, this is also readily seen from the definition of UNI(A|B). Let M AB , N AB ∈ UNI(A|B). Then, we have Thus, Then, from the closedness under serial concatenation above, it follows that

Theorem 2. A convex combination of A-unital channels is an A-unital channel.
Proof. Let N 1 , N 2 ∈ UNI(A|B). Then we have,

Detection
The characterization of both conditional entropy non-decreasing channels and completely free channels for F c (A|B) as A-unital channels makes apparent how these channels can be detected. For any D(B), it is always possible to find d 2 B density matrices {β i } i from D(B) that form a basis for for D(B) [13]. For some unknown channel N given as a blackbox, a necessary and sufficient condition for A-unitality is states β i from the basis. This follows using the same proof technique as in Lemma 3.

Relationship between A-Unital Channels and Other Classes
We now turn our attention to the relationship between A-unital channels and other classes. If we detect that a channel is A-unital (or not A-unital), can anything be said about the membership of the channel in SEP(A|B)? It turns out that this is not the case, as there exist A-unital channels that are not separable, separable channels that are not Aunital, as well as channels in the intersection. We discuss this in the following subsections. We also discuss the relationship between entanglement breaking channels and A-unital channels. These relationships are depicted in Fig. 2. Throughout this section, we assume

A-Unital Non-Separable Channels
We prove the existence of channels that belong to UNI(A|B) but not SEP(A|B) by constructing such a channel. Let {|i } i be an orthonormal basis for H A , and {|j } j be an orthonormal basis for H B .  . We have

• The Preparation of I
Finally, we define the Swap and Prepare Channel:

Theorem 3. There exist A-unital non-separable channels.
Proof. In Lemma 6, we show that the channel P AB ∈ UNI(A|B), and in Lemma 7, we show that P AB / ∈ SEP(A|B).
Proof. From the discussion following Def. 3 The state χ is a product state across the A A|B B cut, and hence χ ∈ F s (A A|B B). We now apply id A B ⊗ P AB to the state χ.
Observe that χ is entangled across the A A|B B cut. Thus, χ / ∈ F s (A A|B B). From the definition of O s cmax (A|B), this implies that P AB / ∈ O s cmax (A|B), hence P AB / ∈ SEP(A|B). The action of P AB is depicted in Fig. 3.
From Lemmas 6 and 7, we have that P AB ∈ UNI(A|B) but P AB / ∈ SEP(A|B), and thus, Theorem 3 follows.  (2). Application of Tr A is depicted in subfigure (3), and finally, P r A prepares the state I A d and this is depicted in subfigure (4). Thus, P AB is A-unital, but not separable, as the channel generates entanglement across the A A|B B cut.

Non-A-Unital Separable Channels
Examples of channels that are non-A-unital but separable are replacement channels [2] R which trace out a given state and prepare a product state , R is not an A-unital channel. However, since both tracing out as well as preparation of a product state are separable channels, R is a separable channel. A specific example of a channel which prepares the state |00 00| AB is where {|i } i and {|j } j are orthonormal bases for H A and H B respectively.

A-Unital Separable Channels
A trivial example of a channel that is both A-unital and separable is the identity channel. A more non-trivial example is any channel of the form A is a unitary channel. C AB is separable by definition. It is also A-unital as shown below. Let σ ∈ D(B). We have

A-Unital and Entanglement-Breaking Channels
Both UNI(A|B) and SEP(A|B) are concerned with correlations across the A|B split. Thus, we relate both these classes with channels of the form id A ⊗Φ B where Φ B ∈ EB(B), that is Φ B is an entanglement-breaking channel (Def. 4), as they destroy all entanglement across the A|B split. These channels are A-unital, as (id  Figure 2. On the contrary, channels of the form Φ A ⊗ id B where Φ A ∈ EB(A) are not necessarily A-unital, although they are separable. As an example, consider the following classical-quantum channel M A : D(A) → D(A), that acts as All classical-quantum channels are entanglement-breaking [13], however, the channel . A-unital channels are closely related to unital channels. Indeed, as given in Def. 5, while A-unital channels map states of the form In the next section, we explore this connection further. We show that while A-unital channels are the largest class of completely free channels for F c (A|B), unital channels form the largest class of completely free channels for a previously defined, related set F ac (A|B), the set of all states whose conditional entropy remains non-negative under global unitary operations. The relationship between A-unital, B-unital and unital channels is depicted in Fig. 4.
6 Unital Operations and Absolute Non-Negative Conditional Entropy

Absolutely Non-Negative Conditional Entropy States
Previously, the absolute version of F c (A|B) (analogous to absolutely separable states [29,30]) was defined and characterized for the case where d A = d B = 2 [27]. Here, we extend the definition by characterizing it for the case where d A = d B = d. The set F ac (A|B) is well-characterized in terms of the von Neumann entropy, as shown in the following theorem.  Proof. Let ρ AB ∈ D(AB) and S(ρ AB ) ≥ log d. As the maximum von Neumann entropy of the subsystem B is upperbounded by log d, To prove the "only if" part of the statement, consider the following set of vectors {|ψ nm } nm which is a basis for the vector space H A ⊗ H B [31] Notice that for each |ψ nm , we have (|ψ nm ψ nm |) B = I d . Now, let σ AB ∈ F ac (A|B). Let the set of vectors {|γ nm } nm be the eigenbasis of σ AB . Consider the unitary channel U e ≡ nm |ψ nm γ nm |. The state U e σ AB U † e is diagonal in the |ψ nm basis, hence, U e σ AB U † e B = I d , and S (U e σ AB U † e ) B = log d. By definition, we have S A|B (U σ AB U † ) ≥ 0 for all U , in particular for U e . Therefore,

Completely Free Operations for Absolute Non-Negative Conditional Entropy States
Here, we define the class of completely free operations for F ac (A|B), respecting the d A = d B = d condition in the definition of F ac (A|B). We now prove that for the above definition of O ac cmax (A|B), unitality is a necessary and sufficient condition for complete freeness. to itself. This unitality implies we have S (N (ρ)) ≥ S (ρ) ∀ ρ ∈ D(A AB B) [13]. Hence, ∀ σ ∈ F ac (A A|B B), we have S (N (σ)) ≥ S (σ) ≥ log dd , using Theorem 4. Thus, N (σ) ∈ F ac (A A|B B). This implies that N is a free operation, which in turn implies that N AB ∈ O ac cmax (A|B). Let M AB be a non-unital channel in D(AB). Hence, M AB Thus, S(σ AB ) < log d 2 , since the maximum entropy for ρ ∈ D(AB) = log d 2 and is achieved only at I AB d 2 . Consider the state χ = γ A B ⊗ I AB d 2 where γ is a pure state and d = d. Hence S(χ) = log d 2 which implies that χ ∈ F ac (A A|B B). We have χ = (id A B ⊗ M AB ) (χ) = γ A B ⊗ σ AB . Therefore, S(χ ) < log d 2 and χ / ∈ F ac (A A|B B). This implies that id A B ⊗ M AB is not a free operation, and therefore, M AB / ∈ O ac cmax (A|B).

Upper and Lower Bounds for Conditional Entropy given Entropy
The characterization of F ac (A|B) via entropy suggests that the entropy of a state can determine the values of conditional entropy that the state can possess. In the following section, we discuss upper and lower bounds for conditional entropy, given the entropy of a state ρ AB , where d A = d B = d.
S(R(p x )) = x (Eq. 31) implies that S A|B (R(p x )) = x − log d ∈ E x (Eq. 32). Thus, as x − log d is a lower bound of E x , and x − log d ∈ E x , it is the tight lower bound.