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Physical Review A

From limits of quantum operations to multicopy entanglement witnesses and state-spectrum estimation

Paweł Horodecki

Published Nov 3, 2003• 97 citations
Publisher

Abstract

Limits of state transformations in quantum mechanics are studied. Impossibility of physical implementation of the transformation ${\ensuremath{\varrho}}^{\ensuremath{\bigotimes}n}\ensuremath{\rightarrow}{\ensuremath{\varrho}}^{n}$ in quantum mechanics is proved. The most natural notions of structural completely positive approximation and structural physical approximations of nonphysical map are introduced. It is shown that these always exist for linear Hermitian maps and can be optimized under natural conditions. It is pointed out that it is physically possible to measure in a simple way the traces of $n\mathrm{th}$ power of quantum state $\mathrm{Tr}({\ensuremath{\varrho}}^{n})$ if only joint measurements on n copies of the system are allowed. This gives the interpretation of Tsallis entropies as values of ``multicopy'' observables. Following this observations, the notion of multicopy entanglement witnesses is defined and examples are provided. Finally, using notion of multicopy observable, a simple method of spectrum state estimation is discussed.

DOI
10.1103/physreva.68.052101

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Limits of state transformations in quantum mechanics are studied. Impossibility of physical implementation of the transformation \ensuremathϱ\ensuremath⨂n\ensuremath→\ensuremathϱn{\ensuremath{\varrho}}^{\ensuremath{\bigotimes}n}\ensuremath{\rightarrow}{\ensuremath{\varrho}}^{n}\ensuremathϱ\ensuremath⨂n\ensuremath→\ensuremathϱn in quantum mechanics is proved. The most natural notions of structural completely positive approximation and structural physical approximations of nonphysical map are introduced. It is shown that these always exist for linear Hermitian maps and can be optimized under natural conditions. It is pointed out that it is physically possible to measure in a simple way the traces of nthn\mathrm{th}nth power of quantum state Tr(\ensuremathϱn)\mathrm{Tr}({\ensuremath{\varrho}}^{n})Tr(\ensuremathϱn) if only joint measurements on n copies of the system are allowed. This gives the interpretation of Tsallis entropies as values of ``multicopy'' observables. Following this observations, the notion of multicopy entanglement witnesses is defined and examples are provided. Finally, using notion of multicopy observable, a simple method of spectrum state estimation is discussed.
Limits of nonlinear in quantum mechanics are studied. Impossibility of physical implementation of the transformation ϱ⊗n→ϱn\varrho^{\otimes n} \to \varrho^{n}ϱ⊗n→ϱn in quantum mechanics is proved. For sake of further analysis the simplest notion of structural completely positive approximation (SCPA) and structural physical approximations (SPA) of unphysical map are introduced. Both always exist for linear hermitian maps and can be optimised under natural assumptions. However it is shown that some intuitively natural SPA of the nonlinear operation ϱ⊗2→ϱ2\varrho^{\otimes 2} \to \varrho^{2}ϱ⊗2→ϱ2 that was already proven to be unphysical is impossible. It is conjectured that there exist no SPA of the operation ϱ⊗n→ϱn\varrho^{\otimes n} \to \varrho^{n}ϱ⊗n→ϱn at all. It is pointed out that, on the other hand, it is physically possible to measure the trace of the second power of the state Tr(ϱ2)Tr(\varrho^{2})Tr(ϱ2) if only two copies of the system are available. This gives the interpretation of one of Tsallis entropy as mean value of some ``multicopy'' observable. The (partial) generalisation of this idea shows that each of higher order Tsallis entropies can be measured with help of only two multicopy observables. Following this observations the notion of multicopy entanglement witnesses is defined and first example is provided. Finally, with help of multicopy observables simple method of spectrum state estimation is pointed out and discussed.
In quantum information processing it may be possible to have efficient computation and secure communication beyond the limitations of classical systems. In a fundamental point of view, however, evolution of quantum systems by the laws of quantum mechanics is more restrictive than classical systems, identified to a specific form of dynamics, that is, unitary transformations and, consequently, positive and completely positive maps to subsystems. This also characterizes classes of disallowed transformations on quantum systems, among which positive but not completely maps are of particular interest as they characterize entangled states, a general resource in quantum information processing. Structural physical approximation offers a systematic way of approximating those non-physical maps, positive but not completely positive maps, with quantum channels. Since it has been proposed as a method of detecting entangled states, it has stimulated fundamental problems on classifications of positive maps and the structure of Hermitian operators and quantum states, as well as on quantum measurement such as quantum design in quantum information theory. It has developed efficient and feasible methods of directly detecting entangled states in practice, for which proof-of-principle experimental demonstrations have also been performed with photonic qubit states. Here, we present a comprehensive review on quantum information processing with structural physical approximations and the related progress. The review mainly focuses on properties of structural physical approximations and their applications toward practical information applications.
Structural approximations to positive, but not completely positive maps are approximate physical realizations of these nonphysical maps. They find applications in the design of direct entanglement-detection methods. We show that many of these approximations, in the relevant case of optimal positive maps, define an entanglement breaking channel and, consequently, can be implemented via a measurement and state-preparation protocol. We also show how our findings can be useful for the design of better and simpler direct entanglement detection methods.
Entanglement detection is essential in quantum information science and quantum many-body physics. It has been proved that entanglement exists almost surely for a random quantum state, while the realizations of effective entanglement criteria usually consume exponentially many resources with regard to system size or qubit number, and efficient criteria often perform poorly without prior knowledge. This fact implies a fundamental limitation might exist in the detectability of entanglement. In this work, we formalize this limitation as a fundamental trade-off between the efficiency and effectiveness of entanglement criteria via a systematic method to evaluate the detection capability of entanglement criteria theoretically. For a system coupled to an environment, we prove that any entanglement criterion needs exponentially many observables to detect the entanglement effectively when restricted to single-copy operations. Otherwise, the detection capability of the criterion will decay double exponentially. Furthermore, if multicopy joint measurements are allowed, the effectiveness of entanglement detection can be exponentially improved, which implies a quantum advantage in entanglement detection problems. Our results may shed light on why quantum phenomena are difficult to observe in large noisy systems.
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.
We present a technique for enhancing the estimation of quantum state properties by incorporating approximate prior knowledge about the quantum state. This consists in performing randomized measurements on a quantum processor and comparing the results with those obtained from a classical computer that stores an approximation of the quantum state. We provide unbiased estimators for expectation values of multicopy observables and present performance guarantees in terms of variance bounds that depend on the prior knowledge accuracy. We demonstrate the effectiveness of our approach through experimental and numerical examples detecting mixed-state entanglement, and estimating polynomial approximations of the von Neumann entropy and state fidelities. Published by the American Physical Society 2024