On Partially Trace Distance Preserving Maps and Reversible Quantum Channels

We give a characterization of trace-preserving and positive linear maps preserving trace distance partially, that is, preservers of trace distance of quantum states or pure states rather than all matrices. Applying such results, we give a characterization of quantum channels leaving Helstrom’s measure of distinguishability of quantum states or pure states invariant and show that such quantum channels are fully reversible, which are unitary transformations.


Introduction
Linear preserver problems concern the characterization of linear maps on matrix spaces that leave certain functions, subsets, relations, and so forth, invariant, and have been an active research area in matrix theory.The earliest paper on linear preserver problems dates back to the year of 1897, and a great deal of effort has been devoted to the study of this type of questions (see [1][2][3][4] and their references).In 1975, Choi in [4] gave a characterization of completely positive linear maps on matrix algebras.In the theory of quantum information, a quantum channel just be a completely positive and trace preserving linear map.So Choi gave a mathematical characterization of quantum channels.Such a result is applied in quantum information extensively.In recent years, more and more researchers on linear preserver problems pay their attention to the theory of quantum information (see [5][6][7][8][9] and their references).In this paper, we will give a characterization of trace-preserving and positive linear maps preserving trace distance partially, that is, preservers of trace distance of quantum states or pure states rather than all matrices (see Theorems 6 and 10).Applying such results, we give a characterization of quantum channels leaving Helstrom's measure of distinguishability of quantum states or pure states invariant and show that such quantum channels are fully reversible, which are unitary transformations (see Theorems 7 and 11).
In the mathematical framework of quantum information, quantum states are positive operators with trace 1 on complex Hilbert space , and we denote by S() the set of all quantum states on , which is a convex subset of the space of trace-class operators T().Pure states are rank one projections.If dim =  < ∞, then T() is identical with B(), that is, the  ×  complex matrix algebra.In the case of dim =  < ∞, a quantum channel Φ : B() → B() has the following form: where    ∈ B() and ∑  =1  *    =  with the identity .For a quantum channel Φ, the aim of quantum error correction is to find another quantum channel Ψ such that Here, we call that Φ is reversible for the state ; if Φ is reversible for all states  ∈ C, then Φ is reversible on C. C is a subset of S() and is called error correction code.
It is easy to check that if Φ is reversible for , , then Φ is reversible for the arbitrary convex combination of , .So, if C is an error correction code, we can assume that C is a convex set.The topic of quantum error correction or reversibility of quantum channels naturally arises in the analysis of questions in quantum information and plays an important role in the theory of quantum information (see [10][11][12][13] and their references).In the theory of reversibility of quantum channels, Helstrom's measure of distinguishability of quantum states plays an important role and is defined as follows.
Definition (see Theorem 7).We call such a channel the fully reversible channel as follows.
Definition 3. A quantum channel Φ : B() → B() is fully reversible if there exists a quantum channel Ψ such that Ψ ∘ Φ() =  for all quantum states .
In Definition 2, taking  = 1/2, one can define the map preserving trace distance of quantum states as follows.
The map preserving Helstrom's measure of distinguishability must preserve trace distance.Robin Blume-Kohout et al. in [10] showed that with the assumption ( †), the channel preserving Helstrom's measure of distinguishability is reversible for an error correction code C, but the channel preserving trace distance of quantum states is not.In this paper, we will show that the channel preserving trace distance of all quantum states is also fully reversible (see Theorem 7).Indeed a fully reversible channel Φ is a unitary transformation; that is, there exists a unitary operator  such that Φ() =  * for all quantum states .Also many authors pay their attention to characterizing preservers of trace distance (see [14,15] and their references).Let P 1 () be the set of all pure states; furthermore, we introduce the following general partial preservers of trace distance of pure states.
The map preserving trace distance of quantum states must preserve trace distance of pure states.We also give a characterization of trace preserving and positive linear maps preserving trace distance of pure states and show that the channel preserving trace distance of all pure states is also fully reversible (see Theorems 10 and 11).

Partially Trace Distance Preservers and Fully Reversible Channels
In this section, we are first devoted to characterizing a class of positive and trace-preserving linear maps preserving trace distance of quantum states.
Theorem 6.Let  be a finite dimensional complex Hilbert space with dim  = , Φ : B() → B() being a positive and trace-preserving linear map; then the following statements are equivalent: (I) Φ preserves trace distance of quantum states; that is, ‖Φ() − Φ()‖ 1 = ‖ − ‖ 1 for all ,  ∈ S(); (II) there exists a unitary operator  on  such that Φ() =  * for all states  ∈ S() or Φ() =    * for all states  ∈ S(), where   is the transpose of  with respect to an orthonormal basis.
Applying Theorem 6, we will have the following main result.
Theorem 7. Let  be a finite dimensional complex Hilbert space with dim  = , Φ : B() → B() being a quantum channel, that is, a completely positive and trace-preserving linear map; then the following statements are equivalent: (III) Φ preserves trace distance of quantum states; that is, ‖Φ() − Φ()‖ 1 = ‖ − ‖ 1 for all ,  ∈ S(); (IV) Φ is a unitary transformation; that is, there exists a unitary operator  on  such that Φ() =  * for all input states .
Remark 8. Theorem 7 shows that quantum channels leaving Helstrom's measure of distinguishability or trace distance of all quantum states invariant is fully reversible and vice versa.Also we prove that a quantum channel is fully reversible if and only if it is a unitary transformation.Indeed, such a result also can be induced by some other characterization of preservers in existence.Here, let us give a short proof; if a quantum channel is fully reversible, by [11], then the channel preserves the relative entropy of all quantum states.Such a map has been characterized in [16], where Molnár showed that the map preserving relative entropy of all quantum states has the following form:   →  * , where  is a unitary or antiunitary operator.Applying the result in [16], it follows from linearity and complete positivity of quantum channels that fully reversible quantum channels are unitary transformations.
Before the proof of Theorems 6 and 7, we first give some primary observations.It is mentioned that a quantum state  is a pure state if and only if rank  = 1 if and only if  2 = ; that is, pure states are rank one projections.Lemma 9.For a quantum state  ∈ S() with dim  =  < ∞,  is a pure state if and only if there exist  − 1 pairwise orthogonal states  1 , . . .,  −1 such that   = 0 for all 1 ≤  ≤  − 1.

Proof of Theorem 6. (II)⇒(I) is clear; we only need to check that (I)⇒(II).
The proof is divided into the following claims.
Since Φ preserves orthogonality of rank one projections, so the case (I) does not occur.Therefore, this claim holds true.The proof is completed.
Next we will give the proof of Theorem 7.
Next we show that (III) ⇒ (IV).Since the channel Φ satisfies the assumptions in Theorem 6, so there exists a unitary operator  on  such that (1) Φ() =  * for all input states  or (2) Φ() =    * for all input states , where   is the transpose of  with respect to an orthonormal basis.One can note that the transpose map is positive but is not completely positive (see [9]), so for a quantum channel Φ, case (2) does not occur.So (IV) holds true.The proof is completed.

Characterizing Channels Reversible for Pure States
In the section, we first give a characterization of positive and trace-preserving linear maps preserving trace distance of pure states.
Theorem 10.Let  be a finite dimensional complex Hilbert space with dim  = , Φ : B() → B() being a positive and trace-preserving linear map; then the following statements are equivalent: (I) Φ preserves trace distance of pure states; that is, ‖Φ() − Φ()‖ 1 = ‖ − ‖ 1 for all ,  ∈  1 (); (II) there exists a unitary operator  on  such that Φ() =  * for all states  ∈ S() or Φ() =    * for all states  ∈ S(), where   is the transpose of  with respect to an orthonormal basis.
Applying Theorem 10, we will have the following refined result.
Theorem 11.Let  be a finite dimensional complex Hilbert space with dim  = , Φ : B() → B() being a quantum channel, that is, a completely positive and trace-preserving linear map; then the following statements are equivalent: (IV) Φ preserves trace distance of pure states; that is, ‖Φ() − Φ()‖ 1 = ‖ − ‖ 1 for all ,  ∈ P 1 (); (V) Φ is a unitary transformation; that is, there exists a unitary operator  on  such that Φ() =  * for all input states .
Next we show that Φ() is a pure state for any pure state .For any pure state , by Lemma 9, then there exist  − 1 pairwise orthogonal nonzero pure states  1 , . . .,  −1 such that    = 0 for all 1 ≤  ≤  − 1.Since Φ preserves orthogonality of pure states and is injective on pure states, it follows that Φ( 1 ), . . ., Φ( −1 ) are states, pairwise orthogonal, and Φ(  )Φ() = 0 for all 1 ≤  ≤  − 1.So by Lemma 9 again, Φ() is a rank one projection.Now we have that Φ is linear and maps pure states to pure states; similar to the proof of Theorem 6, one can complete the proof.
Next we will give the proof of Theorem 11.
Next we show that (IV)⇒(V).Since the channel Φ satisfies the assumptions in Theorem 10, so there exists a unitary operator  on  such that (1) Φ() =  * for all input states  or (2) Φ() =    * for all input states , where   is the transpose of  with respect to an orthonormal basis.Since the transpose map is positive, but is not completely positive (see [9]), so for a quantum channel Φ, the case (2) does not occur.So (IV) holds true.The proof is completed.
Remark 12.However, it is also mentioned that if the error correction code C is the proper convex subset of S(), that is, S() \ C ̸ = 0, the channel preserving trace distance of quantum states in C may not be reversible on C (see [10,Example 7]).Also a natural question is to give a characterization of the channel on a proper subset of S() preserving trace distance, and such a result may help us understand more deeply the theory of reversibility of quantum channels.