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Register a new userInterpolatability distinguishes LOCC from separable von Neumann measurements
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Local operations with classical communication (LOCC) and separable operations are two classes of quantum operations that play key roles in the study of quantum entanglement. Separable operations are strictly more powerful than LOCC, but no simple explanation of this phenomenon is known. We show that, in the case of von Neumann measurements, the ability to interpolate measurements is an operational principle that sets apart LOCC and separable operations.
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A breakthrough took place in the von Neumann algebra theory when the Tomita-Takesaki theory was established around 1970. Since then, many important issues in the theory were developed through 1970s by Araki, Connes, Haagerup, Takesaki and others, which are already very classics of the von Neumann algebra theory. Nevertheless, it seems still difficult for beginners to access them, though a few big volumes on the theory are available. These lecture notes are delivered as an intensive course in 2019, April at Department of Mathematical Analysis, Budapest University of Technology and Economics. The course was aimed at giving a fast track study of those main classics of the theory, from which people gain an enough background knowledge so that they can consult suitable volumes when more details are needed.
Reproducing with elementary resources the correlations that arise when a quantum system is measured (quantum state simulation), allows one to get insight on the operational and computational power of quantum correlations. We propose a family of models that can simulate von Neumann measurements in the x-y plane of the Bloch sphere on n-partite GHZ states using only bipartite nonlocal boxes. For the tripartite and fourpartite states, the models use only bipartite nonlocal boxes; they can be translated into classical communication schemes with finite average communication cost.
Quantum error correcting codes with finite-dimensional Hilbert spaces have yielded new insights on bulk reconstruction in AdS/CFT. In this paper, we give an explicit construction of a quantum error correcting code where the code and physical Hilbert spaces are infinite-dimensional. We define a von Neumann algebra of type II$_1$ acting on the code Hilbert space and show how it is mapped to a von Neumann algebra of type II$_1$ acting on the physical Hilbert space. This toy model demonstrates the equivalence of entanglement wedge reconstruction and the exact equality of bulk and boundary relative entropies in infinite-dimensional Hilbert spaces.
Is is shown here that the simple test of quantumness for a single system of arXiv:0704.1962 (for a recent experimental realization see arXiv:0804.1646) has exactly the same relation to the discussion of to the problem of describing the quantum system via a classical probabilistic scheme (that is in terms of hidden variables, or within a realistic theory) as the von Neumann theorem (1932). The latter one was shown by Bell (1966) to stem from an assumption that the hidden variable values for a sum of two non-commuting observables (which is an observable too) have to be, for each individual system, equal to sums of eigenvalues of the two operators. One cannot find a physical justification for such an assumption to hold for non-commeasurable variables. On the positive side. the criterion may be useful in rejecting models which are based on stochastic classical fields. Nevertheless the example used by the Authors has a classical optical realization.
For an arbitrary open, nonempty, bounded set $Omega subset mathbb{R}^n$, $n in mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider the closed, strictly positive, higher-order differential operator $A_{Omega, 2m} (a,b,q)$ in $L^2(Omega)$ defined on $W_0^{2m,2}(Omega)$, associated with the higher-order differential expression $$ tau_{2m} (a,b,q) := bigg(sum_{j,k=1}^{n} (-i partial_j - b_j) a_{j,k} (-i partial_k - b_k)+qbigg)^m, quad m in mathbb{N}, $$ and its Krein--von Neumann extension $A_{K, Omega, 2m} (a,b,q)$ in $L^2(Omega)$. Denoting by $N(lambda; A_{K, Omega, 2m} (a,b,q))$, $lambda > 0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K, Omega, 2m} (a,b,q)$, we derive the bound $$ N(lambda; A_{K, Omega, 2m} (a,b,q)) leq C v_n (2pi)^{-n} bigg(1+frac{2m}{2m+n}bigg)^{n/(2m)} lambda^{n/(2m)} , quad lambda > 0, $$ where $C = C(a,b,q,Omega)>0$ (with $C(I_n,0,0,Omega) = |Omega|$) is connected to the eigenfunction expansion of the self-adjoint operator $widetilde A_{2m} (a,b,q)$ in $L^2(mathbb{R}^n)$ defined on $W^{2m,2}(mathbb{R}^n)$, corresponding to $tau_{2m} (a,b,q)$. Here $v_n := pi^{n/2}/Gamma((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $mathbb{R}^n$. Our method of proof relies on variational considerations exploiting the fundamental link between the Krein--von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of $widetilde A_{2} (a,b,q)$ in $L^2(mathbb{R}^n)$. We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension $A_{F,Omega, 2m} (a,b,q)$ in $L^2(Omega)$ of $A_{Omega, 2m} (a,b,q)$. No assumptions on the boundary $partial Omega$ of $Omega$ are made.
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