Quantum control has been a cornerstone of quantum information science, driving major advances in quantum computing, quantum communication, and quantum sensing. Over the years, it has enabled the implementation of arbitrary completely positive and trace-preserving (CPTP) maps; an important next step is to extend control to Hermitian-preserving and trace-preserving (HPTP) maps, which underpin applications such as entanglement detection, quantum error mitigation, quantum simulation, and quantum machine learning. Here we present an efficient and fully constructive method for implementing arbitrary HPTP maps. Unlike existing methods that decompose an HPTP map into multiple CPTP maps or approximate it using bipartite Hamiltonians with large Hilbert spaces, our approach compiles a target HPTP map into a single executable CPTP map whose Kraus rank is guaranteed to be no larger than the intrinsic rank of the target HPTP map plus one, followed by simple classical post-processing. Numerical results for inverse noise channels used in quantum error mitigation, including bosonic photon loss, confirm substantial reductions in resources and highlight scalability in higher-dimensional settings. Together with our numerical benchmarks, these results validate the efficiency and versatility of the proposed framework, opening a route to broader quantum-information applications enabled by HPTP processing.
Quantum control has been a cornerstone of quantum information science, driving major advances in quantum computing, quantum communication, and quantum sensing. Over the years, it has enabled the implementation of arbitrary completely positive and trace-preserving (CPTP) maps; an important next step is to extend control to Hermitian-preserving and trace-preserving (HPTP) maps, which underpin applications such as entanglement detection, quantum error mitigation, quantum simulation, and quantum machine learning. Here we present an efficient and fully constructive method for implementing arbitrary HPTP maps. Unlike existing methods that decompose an HPTP map into multiple CPTP maps or approximate it using bipartite Hamiltonians with large Hilbert spaces, our approach compiles a target HPTP map into a single executable CPTP map whose Kraus rank is guaranteed to be no larger than the intrinsic rank of the target HPTP map plus one, followed by simple classical post-processing. Numerical results for inverse noise channels used in quantum error mitigation, including bosonic photon loss, confirm substantial reductions in resources and highlight scalability in higher-dimensional settings. Together with our numerical benchmarks, these results validate the efficiency and versatility of the proposed framework, opening a route to broader quantum-information applications enabled by HPTP processing.
Quantum mechanics features a variety of distinct properties such as coherence and entanglement, which could be explored to showcase potential advantages over classical counterparts in information processing. In general, legitimate quantum operations must adhere to principles of quantum mechanics, particularly the requirements of complete positivity and trace preservation. Nonetheless, non-physical maps, especially Hermitian-preserving maps, play a crucial role in quantum information science. To date, there exists no effective method for implementing these non-physical maps with quantum devices. In this work, we introduce the Hermitian-preserving map exponentiation algorithm, which can effectively realize the action of an arbitrary Hermitian-preserving map by encoding its output into a quantum process. We analyze the performances of this algorithm, including its sample complexity and robustness, and prove its optimality in certain cases. When combined with algorithms such as the Hadamard test and quantum phase estimation, it allows for the extraction of information and generation of states from outputs of Hermitian-preserving maps, enabling various applications. Utilizing positive but not completely positive maps, this algorithm provides exponential advantages in entanglement detection and quantification compared to protocols based on single-copy operations. In addition, it facilitates the recovery of noiseless quantum states from multiple copies of noisy states by implementing the inverse map of the corresponding noise channel, offering an intriguing approach to handling quantum errors. Our findings present a pathway for systematically and efficiently implementing non-physical actions with quantum devices, thereby boosting the exploration of potential quantum advantages across a wide range of information processing tasks.
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 Λ1,Λ2 and an arbitrary unknown input state, we show that the output state of the quantum channel Λ2 from the pair can be uniquely identified from the output statistics of the other channel Λ1 from the pair, using some POVM, iff Λ2 can be obtained from Λ1 by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, Λ2 may not always be obtained from Λ1 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 Λ2 given that Λ1 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.
Given an open quantum system S with environment E, we establish a one-to-one correspondence between completely positive trace-preserving (CPTP) maps from S to E and assignment maps initializing the composite system SE in a possibly correlated state prior to unitary evolution. We show that every possible joint state ρSE can be realized by such assignment maps, which are inherently Hermiticity-preserving and trace-preserving (HPTP). While these maps sacrifice global positivity, we derive a simple necessary and sufficient condition for any local state of S to belong to their domain of validity. Furthermore, we prove that the geometry of this domain is governed by the joint state assigned to the maximally mixed input, ensuring that a non-zero volume of admissible local states can readily coexist with initial system-environment entanglement. Finally, we provide an operational protocol for witnessing such limits of validity.
For an open quantum system to evolve under CPTP maps, assumptions are made on the initial correlations between the system and the environment. Hermitian-preserving trace-preserving (HPTP) maps are considered as the local dynamic maps beyond CPTP. In this paper, we provide a succinct answer to the question of what physical maps are in the HPTP realm by two approaches. The first is by taking one step out of the CPTP set, which provides us with Semi-Positivity (SP) TP maps. The second way is by examining the physicality of HPTP maps, which leads to Semi-Nonnegative (SN) TP maps. Physical interpretations and geometrical structures are studied for these maps. The non-CP SPTP maps Ψ correspond to the quantum non-Markovian process under the CP-divisibility definition (Ψ=Ξ∘Φ−1, where Ξ and Φ are CPTP). When removing the invertibility assumption on Φ, we land in the set of SNTP maps. A by-product of set relations is an answer to the following question -- what kind of dynamics the system will go through when the previous dynamic Φ is non-invertible. In this case, the only locally well-defined maps are in SN\SP, they live on the boundary of SN. Otherwise, the non-local information will be irreplaceable in the system's dynamic. With the understanding of physical maps beyond CPTP, we prove that the current quantum error correction scheme is still sufficient to correct quantum non-Markovian errors. In some special cases, lack of complete positivity could provide us with more error correction methods with less overhead.
Abstract Legitimate quantum operations must adhere to principles of quantum mechanics, particularly the requirements of complete positivity and trace preservation. Yet, non-physical maps, especially Hermitian-preserving maps, play a crucial role in quantum information science. Here, we introduce the Hermitian-preserving map exponentiation algorithm, which can effectively simulate the action of an arbitrary Hermitian-preserving map by exponentiating its output, N(ρ), into a quantum process, e-iN(ρ)t. We analyze the sample complexity of this algorithm and prove its optimality in certain cases. Utilizing positive but not completely positive maps, this algorithm provides exponential speedups in entanglement detection and quantification compared to protocols based on single-copy operations. In addition, it facilitates the recovery of noiseless quantum states from multiple copies of noisy ones by simulating the inverse map of the corresponding noise channel, providing a new approach to handling quantum noises. This algorithm acts as a building block of large-scale quantum algorithms and presents a pathway for exploring potential quantum speedups across a wide range of information-processing tasks.
Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general quantum maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. It motivates how well one can approximate a general quantum map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. More specifically, we decompose a target quantum map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several quantum maps of practical interests. Furthermore, we endow this measure an operational meaning within the quantum error mitigation scenario, quantifying the ultimate sampling cost achievable using the full expressibility of quantum computers. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.