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arXiv:1903.05303 · Physical Review A

Analytic semi-device-independent entanglement quantification for bipartite quantum states

Zhaohui Wei, Lijinzhi Lin

Published Mar 17, 2021• 4 citations
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Abstract

We define a property called nondegeneracy for Bell inequalities, which describes the situation that in a Bell setting, if a Bell inequality and involved local measurements are fixed, any quantum state with a given dimension and its orthogonal quantum state cannot violate remarkably the inequality simultaneously. By choosing a proper nondegenerate Bell inequality, we prove that for an unknown bipartite quantum state of given dimension, based on the measurement statistics only, we can provide an analytic lower bound for the entanglement of formation or even the distillable entanglement, making the whole process semi device independent. We characterize the mathematical structure of nondegeneracy, and prove that quite a lot of well-known Bell inequalities are nondegenerate. We demonstrate our approach by quantifying entanglement for qutrit-qutrit states based on their violations of the Collins-Gisin-Linden-Masser-Popescu inequality.

DOI
10.1103/physreva.103.032215

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We define a property called nondegeneracy for Bell inequalities, which describes the situation that in a Bell setting, if a Bell inequality and involved local measurements are fixed, any quantum state with a given dimension and its orthogonal quantum state cannot violate remarkably the inequality simultaneously. By choosing a proper nondegenerate Bell inequality, we prove that for an unknown bipartite quantum state of given dimension, based on the measurement statistics only, we can provide an analytic lower bound for the entanglement of formation or even the distillable entanglement, making the whole process semi device independent. We characterize the mathematical structure of nondegeneracy, and prove that quite a lot of well-known Bell inequalities are nondegenerate. We demonstrate our approach by quantifying entanglement for qutrit-qutrit states based on their violations of the Collins-Gisin-Linden-Masser-Popescu inequality.
We propose two semi-device-independent approaches that are able to quantify unknown multipartite quantum entanglement experimentally, where the only information that has to be known beforehand is quantum dimension, and the concept that plays a key role is nondegenerate Bell inequalities. Specifically, using the nondegeneracy of multipartite Bell inequalities, we obtain useful information on the purity of target quantum state. Combined with an estimate of the maximal overlap between the target state and pure product states, and a continuous property of the geometric measure of entanglement, we shall prove the information on purity allows us to give a lower bound for this entanglement measure. In addition, we show that a different combination of the above results also converts to a lower bound for the relative entropy of entanglement. As a demonstration, we apply our approach on 5-partite qubit systems with the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality, and show that useful lower bounds for the geometric measure of entanglement can be obtained if the Bell expression value is larger than 3.60, and those for the relative entropy of entanglement can be given if the Bell expression value is larger than 3.80, where the Tsirelson bound is 4.
FINDING: Semi-device-independent entanglement quantification reduces to Bell-inequality violation bounds under fixed local measurements, with a "nondegeneracy" condition separating orthogonal quantum states by dimension. | MATH: Nondegeneracy property — for a Bell inequality BBB and fixed local measurements {MAa,MBb}\{M_A^a, M_B^b\}{MAa​,MBb​}, if a state ρ\rhoρ of dimension ddd violates BBB (i.e., ⟨B⟩ρ>βlocal\langle B\rangle_\rho > \beta_{\text{local}}⟨B⟩ρ​>βlocal​), then its orthogonal complement ρ⊥\rho^\perpρ⊥ (with Tr(ρρ⊥)=0\text{Tr}(\rho\rho^\perp)=0Tr(ρρ⊥)=0) cannot also violate BBB under the same measurements. This yields a dimension-witness bound: Ent(ρ)≥f(⟨B⟩ρ,d)\text{Ent}(\rho) \geq f(\langle B\rangle_\rho, d)Ent(ρ)≥f(⟨B⟩ρ​,d) where fff is monotonic in the Bell violation. | CONNECTION: The orthogonal-state separation mirrors root-system orthogonality (e.g., A2A_2A2​ root lattice angles at 60° and 120°), and the dimension constraint ddd acts like a lattice rank — violation strength maps to a radial coordinate in a convex set of quantum corre Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Detection and quantification of entanglement in quantum resources are two key steps in the implementation of various quantum-information processing tasks. Here, we show that Bell-type inequalities are not only useful in verifying the presence of entanglement but can also be used to bound the entanglement of the underlying physical system. Our main tool consists of a family of Clauser-Horne-like Bell inequalities that cannot be violated maximally by any finite-dimensional maximally entangled state. Using these inequalities, we demonstrate the explicit construction of both lower and upper bounds on the concurrence for two-qubit states. The fact that these bounds arise from Bell-type inequalities also allows them to be obtained in a semi-device-independent manner, that is, with assumption of the dimension of the Hilbert space but without resorting to any knowledge of the actual measurements being performed on the individual subsystems.
We present a general method to quantify both bipartite and multipartite entanglement in a device-independent manner, meaning that we put a lower bound on the amount of entanglement present in a system based on observed data only but independently of any quantum description of the employed devices. Some of the bounds we obtain, such as for the Clauser-Horne-Shimony-Holt Bell inequality or the Svetlichny inequality, are shown to be tight. Besides, device-independent entanglement quantification can serve as a basis for numerous tasks. We show in particular that our method provides a rigorous way to construct dimension witnesses, gives new insights into the question whether bound entangled states can violate a Bell inequality, and can be used to construct device independent entanglement witnesses involving an arbitrary number of parties.
The semi-device-independent framework allows one to draw conclusions about properties of an unknown quantum system under weak assumptions. Here we present a semi-device-independent scheme for the characterization of multipartite entanglement based around a game played by several isolated parties whose devices are uncharacterized beyond an assumption about the dimension of their Hilbert spaces. Our scheme can simultaneously certify that an nnn-partite high-dimensional quantum state features genuine multipartite entanglement, and that a joint measurement on nnn subsystems is entangled. Moreover, it provides a lower bound on the number of entangled measurement operators. These tests are strongly robust to noise, and even optimal for certain classes of states and measurements, as we demonstrate with illustrative examples. Notably, our scheme allows the certification of many entangled states admitting a local model, which therefore cannot violate any Bell inequality.
FINDING: Buscemi's semi-quantum games generalize Bell inequalities by replacing local measurements with quantum channels, unifying entanglement and classical communication in a single statistical comparison framework. | MATH: Bell inequality violation condition: ∑a,b,x,yP(a,b∣x,y) μ(a,b,x,y)≤1 \sum_{a,b,x,y} P(a,b|x,y) \, \mu(a,b,x,y) \leq 1 ∑a,b,x,y​P(a,b∣x,y)μ(a,b,x,y)≤1 (classical bound); Buscemi's extension replaces P(a,b∣x,y)P(a,b|x,y)P(a,b∣x,y) with quantum channel correlations Tr[(Eax⊗Fby)ρ] \mathrm{Tr}[(E_a^x \otimes F_b^y) \rho] Tr[(Eax​⊗Fby​)ρ] where Eax,FbyE_a^x, F_b^yEax​,Fby​ are POVMs, and the "free" operations become arbitrary quantum instruments, yielding a hierarchy of semi-device-independent bounds. The nondegeneracy property (arXiv:1903.05303) defines when a Bell inequality is violated only by a state ρ\rhoρ and not by its orthogonal complement ρ⊥\rho^\perpρ⊥, requiring Tr[ρρ⊥]=0\mathrm{Tr}[\rho \rho^\perp]=0Tr[ρρ⊥]=0 in the fixed measurement basis. | CONNECTION: The statistical comparison framework maps to convex cones — the set of classical correlations forms a polytope whose facets are Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com