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arXiv:2305.05686 · Journal of Physics A Mathematical and Theoretical

Separable ball around any full-rank multipartite product state

Robin Y. Wen, Achim Kempf

Published Jul 17, 2023• 3 citations
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Abstract

Abstract We show that around any m-partite product state ρ p r o d = ρ 1 ⊗ … ⊗ ρ m of full rank (that is d e t ( ρ p r o d ) ≠ 0 ) , there exists a finite-sized closed ball of separable states centered around ρ p r o d whose radius is β := 2 1 − m / 2 λ m i n ( ρ p r o d ) . Here, λ m i n ( ρ p r o d ) is the smallest eigenvalue of ρ p r o d . We are assuming that the total Hilbert space is finite dimensional and we use the notion of distance induced by the Frobenius norm. Applying a scaling relation, we also give a new and simple sufficient criterion for multipartite separability based on trace: T r [ ρ ρ p r o d ] 2 / T r

DOI
10.1088/1751-8121/ace810

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arXiv:quant-ph/0409095•Physical Review A•Sep 16, 2005•44 citations
Abstract We show that around any m-partite product state ρ p r o d = ρ 1 ⊗ … ⊗ ρ m of full rank (that is d e t ( ρ p r o d ) ≠ 0 ) , there exists a finite-sized closed ball of separable states centered around ρ p r o d whose radius is β := 2 1 − m / 2 λ m i n ( ρ p r o d ) . Here, λ m i n ( ρ p r o d ) is the smallest eigenvalue of ρ p r o d . We are assuming that the total Hilbert space is finite dimensional and we use the notion of distance induced by the Frobenius norm. Applying a scaling relation, we also give a new and simple sufficient criterion for multipartite separability based on trace: T r [ ρ ρ p r o d ] 2 / T r
We show that, in finite dimensions, around any mmm-partite product state ρprod=ρ1⊗...⊗ρmρ_{\rm prod}=ρ_1\otimes...\otimesρ_mρprod​=ρ1​⊗...⊗ρm​, there exists an ellipsoid of separable states centered around ρprodρ_{\rm prod}ρprod​. This separable ellipsoid contains the separable ball proposed in previous works, and the volume of the ellipsoid is typically exponentially larger than that of the ball, due to the hierarchy of eigenvalues in typical states. We further generalize this ellipsoidal criterion to a trace formula that yields separable region around all separable states, and further study biseparability. Our criteria not only help numerical procedures to rigorously detect separability, but they also lead to a nested hierarchy of SLOCC-stable subsets that cover the separable set. We apply the procedure for separability detection to 3-qubit X states, genuinely entangled 4-qubit states mixed with noise, and the 1d transverse field Ising model at finite temperature to illustrate the power of our procedure for understanding entanglement in physical systems.
Let H 1 , H 2 , …, H n , K ( n ⩾ 2) be complex Hilbert spaces of finite or infinite dimension. A necessary and sufficient criterion of full separability for multipartite states is established, which says that a state ρ in H 1 ⊗ H 2 ⊗⋅⋅⋅⊗ H n ⊗ K is fully separable if and only if (Λ⊗Id)ρ ⩾ 0 for all (finite-rank) LPP elementary operators Λ from into . Furthermore, a characterization of LPP elementary operators and a method of constructing LPP elementary operators by positive elementary operators are given. An example is also presented to demonstrate how to apply the criterion to detect entanglement of tripartite states.
We show that for an m-partite quantum system, there is a ball of radius ${2}^{\ensuremath{-}(m/2\ensuremath{-}1)}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices. This can be used to derive an \ensuremath{\epsilon} below which mixtures of \ensuremath{\epsilon} of any density matrix with $1\ensuremath{-}\ensuremath{\epsilon}$ of the maximally mixed state will be separable. The \ensuremath{\epsilon} thus obtained is exponentially better (in the number of systems) than existing results. This gives a number of qubits below which nuclear magnetic resonance with standard pseudopure-state preparation techniques can access only unentangled states; with parameters realistic for current experiments, this is 23 qubits (compared to 13 qubits via earlier results). A ball of radius 1 is obtained for multipartite states separable over the reals.
We investigate the separable states ρ of an arbitrary multi-partite quantum system with Hilbert space H of dimension d . The length L ( ρ ) of ρ is defined as the smallest number of pure product states having ρ as their mixture. The length filtration of the set of separable states, S , is the increasing chain ∅ ⊊ S 1 ′ ⊆ S 2 ′ ⊆ ⋯ , where S i ′ = { ρ ∈ S : L ( ρ ) ≤ i } . We define the maximum length, L max = max ρ ∈ S L ( ρ ) , critical length, L crit , and yet another special length, L c , which was defined by a simple formula in one of our previous papers. The critical length indicates the first term in the length filtration whose dimension is equal to Dim S . We show that in general d ≤ L c ≤ L crit ≤ L max ≤ d 2 . We conjecture that the equality L crit = L c holds for all finite-dimensional multi-partite quantum systems. Our main result is that L crit = L c for the bipartite systems having a single qubit as one of the parties. This is accomplished by computing the rank of the Jacobian matrix of a suitable map having S as its range.
We show that for an mmm-qubit quantum system, there is a ball of radius asymptotically approaching $\ensuremath{\kappa}{2}^{\ensuremath{-}\ensuremath{\gamma}m}$ in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent $\ensuremath{\gamma}=0.5(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}3∕\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2\ensuremath{-}1)\ensuremath{\approx}0.292\phantom{\rule{0.2em}{0ex}}481\phantom{\rule{0.2em}{0ex}}25$ much smaller in magnitude than the best previously known exponent, from our earlier work, of 1∕21∕21∕2. For normalized mmm-qubit states, we get a separable ball of radius $\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}(1+\ensuremath{\gamma})m}\ensuremath{\equiv}\sqrt{{3}^{m+1}∕({3}^{m}+3)}\ifmmode\times\else\texttimes\fi{}{6}^{\ensuremath{-}m∕2}$ (note that $\ensuremath{\kappa}=\sqrt{3}$), compared to the previous $2\ifmmode\times\else\texttimes\fi{}{2}^{\ensuremath{-}3m∕2}$. This implies that with parameters realistic for current experiments, nuclear magnetic resonance (NMR) with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for mmm-partite systems of fixed local dimension d0{d}_{0}d0​, although approaching our earlier exponent as ${d}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$.