We describe a direct method to determine the negativity of an arbitrary two-qubit state in experiments. The method is derived by analyzing the relation between the purity, negativity, and a universal entanglement witness for two-qubit entanglement. We show how the negativity of a two-qubit state can be calculated from just three experimentally accessible moments of the partially transposed density matrix of a two-photon state. Moreover, we show that the negativity can be given as a function of only six invariants, which are linear combinations of nine invariants from the complete set of 21 fundamental and independent two-qubit invariants. We analyze the relation between these moments and the concurrence for some classes of two-qubit states (including the
X states, as well as pure states affected by the amplitude-damping and phase-damping channels). We also discuss the possibility of using the universal entanglement witness as an entanglement measure for various classes of two-qubit states. Moreover, we analyze how noise affects the estimation of entanglement via this witness.
Motivated by recent developments in conformal field theory (CFT), we devise a quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. These quantities can serve as witnesses of criticality. In particular, we study several scale invariant combinations of the moments for the one-dimensional (1D) hard-core boson model. For two adjacent intervals unusual finite-size corrections are present, showing parity effects that oscillate with a filling dependent period. These are more pronounced in the presence of boundaries. For large chains we find perfect agreement with CFT. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. Remarkably, the finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderately large sizes.
Can the entanglement of an arbitrary many-body state be quantified without reconstructing the state? We answer the question affirmatively by showing how the measure of logarithmic negativity can be estimated using a finite number of even partial transpose (PT) moments of a many-body state. Considering the squared PT spectrum as a probability measure, we solve exactly the task of a truncated Hausdorff moment problem with Gauss and Gauss-Radau quadrature. We obtain a hierarchy of lower bounds on the logarithmic negativity, which improves with increasing order of moments. Convergence to the exact value of the measure can be achieved for every state of finite dimension. We illustrate the bounds for ground states of three different interacting spin models, as well as for a thermal state. An error analysis demonstrates the efficacy of our scheme towards handling the limitation of finite statistics encountered in actual experiments with randomized measurements. Moreover, an example of density matrix renormalization group based numerical estimation is provided for handling system sizes beyond the scope of exact diagonalization.
Abstract We propose an ordered set of experimentally accessible conditions for detecting entanglement in mixed states. The k -th condition involves comparing moments of the partially transposed density operator up to order k . Remarkably, the union of all moment inequalities reproduces the Peres-Horodecki criterion for detecting entanglement. Our empirical studies highlight that the first four conditions already detect mixed state entanglement reliably in a variety of quantum architectures. Exploiting symmetries can help to further improve their detection capabilities. We also show how to estimate moment inequalities based on local random measurements of single state copies (classical shadows) and derive statistically sound confidence intervals as a function of the number of performed measurements. Our analysis includes the experimentally relevant situation of drifting sources, i.e. non-identical, but independent, state copies.
The
p3-PPT criterion is an experimentally viable relaxation of the well-known positive partial transposition (PPT) criterion for the certification of quantum entanglement. Recently, it has been generalized to various families of entanglement criteria based on the PT moments
pk=Tr
[(ρΓ)k], where
ρΓ denotes the partially transposed density matrix of a quantum state
ρ. While most of these generalizations are strictly more powerful than the
p3-PPT criterion, their
m-th level versions usually rely on the availability of
pk for all moment orders
k≤m. Here, we show that one can alternatively compare any three PT moments of orders
k<l<m, which can significantly reduce experimental overheads. More precisely, we show that any state satisfying $p_l>p_k^xp_m^{1-x} $ must be entangled, where
x=(m−l)/(m−k). Using the example of locally depolarized GHZ states, we identify the most promising versions of these three-moment criteria and compare their performance with a broad range of entanglement criteria. In the case of globally depolarized stabilizer states, we prove that having access to
pk for
k≤5 is sufficient to reproduce the full PPT criterion. More generally, we show that the Stieltjes-
m criterion is as powerful as the PPT criterion whenever
ρΓ has no more than
(m+1)/2 distinct eigenvalues. Finally, we introduce a notion of quantum weight enumerators that capture the decay of
pk under local white noise for arbitrary quantum states and illustrate this concept for an AME state. Our results contribute to the growing body of literature on higher-moment PPT relaxations and modern applications of weight enumerators in quantum error correction and information theory.
For the certification and benchmarking of medium-size quantum devices, efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partially transposed density matrix. The first method is based on the Hankel matrices and provides a family of entanglement criteria, of which the lowest order reduces to the known p_{3}-positive-partial-transpose criterion proposed in A. Elben et al. [Phys. Rev. Lett. 125, 200501 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.200501]. The second method is optimal and gives necessary and sufficient conditions for entanglement based on some moments of the partially transposed density matrix.