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.
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.
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.
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.
Linear maps that are not completely positive play a crucial role in the study of quantum information, yet their non-completely positive nature renders them challenging to realize physically. The core difficulty lies in the fact that when acting such a map
N on a state
ρ,
N(ρ) may not correspond to a valid density matrix, making it difficult to prepare directly in a physical system. We introduce Block Encoding of Linear Transformation (BELT), a systematic protocol that simulates arbitrary linear maps by embedding the output
N(ρ) into a block of a unitary operator. BELT enables the manipulation and extraction of information about
N(ρ) through coherent quantum evolution. Notably, BELT accommodates maps that fall outside the scope of quantum singular value transformation, such as the transpose map. BELT finds applications in entanglement detection, quantum channel inversion, and simulating pseudo-differential operators, and demonstrates improved sample complexity compared to protocols based on Hermitian-preserving map exponentiation.
For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian inputs or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem.