In quantum technologies, quantum channels are essential for transmitting quantum states. Quantum channels usually introduce noise, thereby reducing the information in a state. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems and are also special cases of Hermitian-preserving trace-preserving (HPTP) linear maps. Concatenating a quantum channel with another quantum channel makes it noisier and degrades its information and resource preservability. In this work, we demonstrate a physically meaningful way to compare a pair of quantum channels using Hermitian-preserving trace-preserving linear maps. Precisely, given a pair of quantum channels $Λ_1, Λ_2$ and an arbitrary unknown input state, we show that the output state of the quantum channel $Λ_2$ from the pair can be uniquely identified from the output statistics of the other channel $Λ_1$ from the pair, using some POVM, iff $Λ_2$ can be obtained from $Λ_1$ by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, $Λ_2$ may not always be obtained from $Λ_1$ through post-processing. The relation between these channels is a preorder, and we study its characterization in this work. In other words, we provide an operational interpretation of this preorder induced by HPTP linear maps in a communication scenario. Furthermore, we characterize the difficulty of implementing $Λ_2$ given that $Λ_1$ has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example. In short, we provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.In quantum technologies, quantum channels are essential for transmitting quantum states. Quantum channels usually introduce noise, thereby reducing the information in a state. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems and are also special cases of Hermitian-preserving trace-preserving (HPTP) linear maps. Concatenating a quantum channel with another quantum channel makes it noisier and degrades its information and resource preservability. In this work, we demonstrate a physically meaningful way to compare a pair of quantum channels using Hermitian-preserving trace-preserving linear maps. Precisely, given a pair of quantum channels $Λ_1, Λ_2$ and an arbitrary unknown input state, we show that the output state of the quantum channel $Λ_2$ from the pair can be uniquely identified from the output statistics of the other channel $Λ_1$ from the pair, using some POVM, iff $Λ_2$ can be obtained from $Λ_1$ by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, $Λ_2$ may not always be obtained from $Λ_1$ through post-processing. The relation between these channels is a preorder, and we study its characterization in this work. In other words, we provide an operational interpretation of this preorder induced by HPTP linear maps in a communication scenario. Furthermore, we characterize the difficulty of implementing $Λ_2$ given that $Λ_1$ has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example. In short, we provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.

quant-pharXiv:2512.07822Dec 9, 20250 citations

The Information of Quantum Channels

Mark M. Wilde

Cambridge University Press eBooksNov 2, 20160 citations

Reversibility of a quantum channel: General conditions and their applications to Bosonic linear channels

Максим Евгеньевич Широков

The method of complementary channel for analysis of reversibility (sufficiency) of a quantum channel with respect to families of input states (pure states for the most part) are considered and applied to Bosonic linear (quasi-free) channels, in particular, to Bosonic Gaussian channels. The obtained reversibility conditions for Bosonic linear channels have clear physical interpretation and their sufficiency is also shown by explicit construction of reversing channels. The method of complementary channel gives possibility to prove necessity of these conditions and to describe all reversed families of pure states in the Schrodinger representation. Some applications in quantum information theory are considered. Conditions for existence of discrete classical-quantum subchannels and of completely depolarizing subchannels of a Bosonic linear channel are presented.The method of complementary channel for analysis of reversibility (sufficiency) of a quantum channel with respect to families of input states (pure states for the most part) are considered and applied to Bosonic linear (quasi-free) channels, in particular, to Bosonic Gaussian channels. The obtained reversibility conditions for Bosonic linear channels have clear physical interpretation and their sufficiency is also shown by explicit construction of reversing channels. The method of complementary channel gives possibility to prove necessity of these conditions and to describe all reversed families of pure states in the Schrodinger representation. Some applications in quantum information theory are considered. Conditions for existence of discrete classical-quantum subchannels and of completely depolarizing subchannels of a Bosonic linear channel are presented.

arXiv:1212.2354Journal of Mathematical PhysicsNov 1, 201310 citations

Coherence of quantum channels

Chandan Datta, Sk Sazim, Arun Kumar Pati +1 more

arXiv:1710.05015Annals of PhysicsAug 23, 201828 citations

Structure of irreducibly covariant quantum channels for finite groups

Marek Mozrzymas, Michał Studziński, Nilanjana Datta

We obtain an explicit characterization of linear maps, in particular, quantum channels, which are covariant with respect to an irreducible representation (U) of a finite group (G), whenever U⊗Uc is simply reducible (with Uc being the contragradient representation). Using the theory of group representations, we obtain the spectral decomposition of any such linear map. The eigenvalues and orthogonal projections arising in this decomposition are expressed entirely in terms of representation characteristics of the group G. This in turn yields necessary and sufficient conditions on the eigenvalues of any such linear map for it to be a quantum channel. We also obtain a wide class of quantum channels which are irreducibly covariant by construction. For two-dimensional irrreducible representations of the symmetric group S(3), and the quaternion group Q, we also characterize quantum channels which are both irreducibly covariant and entanglement breaking.We obtain an explicit characterization of linear maps, in particular, quantum channels, which are covariant with respect to an irreducible representation (U) of a finite group (G), whenever U⊗Uc is simply reducible (with Uc being the contragradient representation). Using the theory of group representations, we obtain the spectral decomposition of any such linear map. The eigenvalues and orthogonal projections arising in this decomposition are expressed entirely in terms of representation characteristics of the group G. This in turn yields necessary and sufficient conditions on the eigenvalues of any such linear map for it to be a quantum channel. We also obtain a wide class of quantum channels which are irreducibly covariant by construction. For two-dimensional irrreducible representations of the symmetric group S(3), and the quaternion group Q, we also characterize quantum channels which are both irreducibly covariant and entanglement breaking.

arXiv:1610.05657Journal of Mathematical PhysicsMay 1, 201731 citations

Λ_1, Λ_2

and an arbitrary unknown input state, we show that the output state of the quantum channel

Λ_2

from the pair can be uniquely identified from the output statistics of the other channel

Λ_1

from the pair, using some POVM, iff

Λ_2

can be obtained from

Λ_1

by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases,

Λ_2

may not always be obtained from

Λ_1

through post-processing. The relation between these channels is a preorder, and we study its characterization in this work. In other words, we provide an operational interpretation of this preorder induced by HPTP linear maps in a communication scenario. Furthermore, we characterize the difficulty of implementing

Λ_2

given that

Λ_1

has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example. In short, we provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.