Mixed Entangled States Research Articles

Capacity of entanglement for a nonlocal Hamiltonian

The notion of capacity of entanglement is the quantum information theoretic counterpart of the heat capacity which is defined as the second cumulant of the entanglement spectrum. Given any bipartite pure state, we can define the capacity of entanglement as the variance of the modular Hamiltonian in the reduced state of any of the subsystems. Here, we study the dynamics of this quantity under a nonlocal Hamiltonian. Specifically, we address the following question: Given an arbitrary nonlocal Hamiltonian, what is the capacity of entanglement that the system can possess? As a useful application, we show that the quantum speed limit for creating the entanglement is not only governed by the fluctuation in the nonlocal Hamiltonian, but also depends inversely on the time average of the square root of the capacity of entanglement. Furthermore, we discuss this quantity for a general self-inverse Hamiltonian and provide a bound on the rate of capacity of entanglement. Towards the end, we generalize the capacity of entanglement for bipartite mixed states based on the relative entropy of entanglement and show that the above definition reduces to the capacity of entanglement for pure bipartite states. Our results can have several applications in diverse areas of physics.

Optimal two-qubit gates in recurrence protocols of entanglement purification

We propose and investigate a method to optimize recurrence entanglement purification protocols. The approach is based on a numerical search in the whole set of SU(4) matrices with the aid of a quasi-Newton algorithm. Our method evaluates average concurrences where the probabilistic occurrence of mixed entangled states is also taken into account. We show for certain families of states that optimal protocols are not necessarily achieved by bilaterally applied controlled-NOT gates. As we discover several optimal solutions, the proposed method offers some flexibility in experimental implementations of entanglement purification protocols and interesting perspectives in quantum information processing.

Experimental witnessing for entangled states with limited local measurements

We experimentally demonstrate a method for detection of entanglement via construction of entanglement witnesses from a limited fixed set of local measurements ( M ). Such a method does not require a priori knowledge about the form of the entanglement witnesses. It is suitable for a scenario where a full state tomography is not available, but the only resource is a limited set of M . We demonstrate the method on pure two-qubit entangled states and mixed two-qubit entangled states, which emerge from photonic implementation of controllable quantum noisy channels. The states we select are motivated by realistic experimental conditions, and we confirm it works well for both cases. Furthermore, possible generalizations to higher-dimensional bipartite systems have been considered, which can potentially detect both decomposable and indecomposable entanglement witnesses. Our experimental results show perfect validity of the method, which indicates that even a limited set of local measurements can be used for quick entanglement detection and further provide a practical test bed for experiments with entanglement witnesses.

Bound Entanglement in Thermalized States and Black Hole Radiation.

We study the mixed-state entanglement structure of chaotic quantum many-body systems at late times using the recently developed equilibrium approximation. A rich entanglement phase diagram emerges when we generalize this technique to evaluate the logarithmic negativity for various universality classes of macroscopically thermalized states. Unlike in the infinite-temperature case, when we impose energy constraints at finite temperature, the phase diagrams for the logarithmic negativity and the mutual information become distinct. In particular, we identify a regime where the negativity is extensive but the mutual information is subextensive, indicating a large amount of bound entanglement. When applied to evaporating black holes, these results imply that there is quantum entanglement within the Hawking radiation long before the Page time, although this entanglement may not be distillable into Einstein-Podolsky-Rosen pairs. We claim that at this earlier time, rather than the Page time, information about diaries thrown into the black hole first starts to leak out.

Erratum: Entanglement negativity, reflected entropy, and anomalous gravitation [Phys. Rev. D 105 , 086013 (2022)

We investigate mixed state entanglement measures of entanglement negativity and reflected entropy for bipartite states in two dimensional conformal field theories with an anomaly through appropriate replica techniques. Furthermore we propose holographic constructions for these measures from the corresponding bulk dual geometries involving topologically massive gravity in AdS$_3$ and find exact agreement with the field theory results. In this connection we extend an earlier holographic proposal for the entanglement negativity to the bulk action with a gravitational Chern-Simons term and compute its contribution to the entanglement wedge cross section dual to the reflected entropy.

Replica wormholes and holographic entanglement negativity

Recent work has shown how to understand the Page curve of an evaporating black hole from replica wormholes. However, more detailed information about the structure of its quantum state is needed to fully understand the dynamics of black hole evaporation. Here we study entanglement negativity, an important measure of quantum entanglement in mixed states, in a couple of toy models of evaporating black holes. We find four phases dominated by different types of geometries: the disconnected, cyclically connected, anti-cyclically connected, and pairwise connected geometries. The last of these geometries are new replica wormholes that break the replica symmetry spontaneously. We also analyze the transitions between these four phases by summing more generic replica geometries using a Schwinger-Dyson equation. In particular, we find enhanced corrections to various negativity measures near the transition between the cyclic and pairwise phase.

Aspects of entanglement in non-local field theories with fractional Laplacian

In recent years, various aspects of theoretical models with long range interactions have attracted attention, ranging from out-of-time-ordered correlators to entanglement. In the present paper, entanglement properties of a simple non-local model with long-range interactions in the form of a fractional Laplacian is investigated in both static and a quantum quench scenario. Logarithmic negativity, which is a measure for entanglement in mixed states is calculated numerically. In the static case, it is shown that the presence of long-range interaction ensures that logarithmic negativity decays much slower with distance compared to short-range models. For a sudden quantum quench, the temporal evolution of the logarithmic negativity reveals that, in contrast to short-range models, logarithmic negativity exhibits no revivals for long-range interactions for the time intervals considered. To further support this result, a simpler measure of entanglement, namely the entanglement entropy is also studied for this class of models.

Improved proof-by-contraction method and relative homologous entropy inequalities

The celebrated holographic entanglement entropy triggered investigations on the connections between quantum information theory and quantum gravity. An important achievement is that we have gained more insights into the quantum states. It allows us to diagnose whether a given quantum state is a holographic state, a state whose bulk dual admits semiclassical geometrical description. The effective tool of this kind of diagnosis is holographic entropy cone (HEC), an entropy space bounded by holographic entropy inequalities allowed by the theory. To fix the HEC and to prove a given holographic entropy inequality, a proof-by-contraction technique has been developed. This method heavily depends on a contraction map f, which is very difficult to construct especially for more-region (n ≥ 4) cases. In this work, we develop a general and effective rule to rule out most of the cases such that f can be obtained in a relatively simple way. In addition, we extend the whole framework to relative homologous entropy, a generalization of holographic entanglement entropy that is suitable for characterizing the entanglement of mixed states.

Asymptotic Survival of Genuine Multipartite Entanglement in Noisy Quantum Networks Depends on the Topology

The study of entanglement in multipartite quantum states plays a major role in quantum information theory and genuine multipartite entanglement signals one of its strongest forms for applications. However, its characterization for general (mixed) states is a highly nontrivial problem. We introduce a particularly simple subclass of multipartite states, which we term pair-entangled network (PEN) states, as those that can be created by distributing exclusively bipartite entanglement in a connected network. We show that genuine multipartite entanglement in a PEN state depends on both the level of noise and the network topology and, in sharp contrast to the case of pure states, it is not guaranteed by the mere distribution of mixed bipartite entangled states. Our main result is a markedly drastic feature of this phenomenon: the amount of connectivity in the network determines whether genuine multipartite entanglement is robust to noise for any system size or whether it is completely washed out under the slightest form of noise for a sufficiently large number of parties. This latter case implies fundamental limitations for the application of certain networks in realistic scenarios, where the presence of some form of noise is unavoidable. To illustrate the applicability of PEN states to study the complex phenomenology behind multipartite entanglement, we also use them to prove superactivation of genuine multipartite nonlocality for any number of parties.

Entanglement negativity, reflected entropy, and anomalous gravitation

We investigate mixed state entanglement measures of entanglement negativity and reflected entropy for bipartite states in two dimensional conformal field theories with an anomaly through appropriate replica techniques. Furthermore we propose holographic constructions for these measures from the corresponding bulk dual geometries involving topologically massive gravity in ${\mathrm{AdS}}_{3}$ and find exact agreement with the field theory results. In this connection we extend an earlier holographic proposal for the entanglement negativity to the bulk action with a gravitational Chern-Simons term and compute its contribution to the entanglement wedge cross section dual to the reflected entropy.

Symmetric inseparability and number entanglement in charge-conserving mixed states

We explore sufficient conditions for inseparability in mixed states with a globally conserved charge, such as a particle number. We argue that even separable states may contain entanglement in fixed charge sectors, as long as the state can not be separated into charge conserving components. As a witness of symmetric inseparability we study the number entanglement (NE), $\Delta S_m$, defined as the entropy change due to a subsystem's charge measurement. Whenever $\Delta S_m > 0$, there exist inseparable charge sectors, having finite (logarithmic) negativity, even when the full state is either separable or has vanishing negativity. We demonstrate that the NE is not only a witness of symmetric inseparability, but also an entanglement monotone. Finally, we study the scaling of $\Delta S_m$ in thermal 1D systems combining high temperature expansion and conformal field theory.

Negativity Hamiltonian: An Operator Characterization of Mixed-State Entanglement.

In the context of ground states of quantum many-body systems, the locality of entanglement between connected regions of space is directly tied to the locality of the corresponding entanglement Hamiltonian: the latter is dominated by local, few-body terms. In this work, we introduce the negativity Hamiltonian as the (non-Hermitian) effective Hamiltonian operator describing the logarithm of the partial transpose of a many-body system. This allows us to address the connection between entanglement and operator locality beyond the paradigm of bipartite pure systems. As a first step in this direction, we study the structure of the negativity Hamiltonian for fermionic conformal field theories and a free-fermion chain: in both cases, we show that the negativity Hamiltonian assumes a quasilocal functional form, that is captured by simple functional relations.

Demonstration of entanglement and coherence in GHZ-like state when exposed to classical environments with power-law noise

We investigate the dynamics and protection of entanglement and coherence in three non-interacting qubits that are initially prepared as mixed entangled GHZ-like state when coupled with external classical fields. A type of Gaussian noise, namely power-law (PL) noise, controls the external local fields in three different configurations with single, double, and triple noise sources. When influenced by a single PL noise source, the GHZ-class state remains partially entangled and coherent indefinitely, in contrast to the multiple PL noise sources. However, long-term non-local correlations and coherence are still possible in the presence of multiple noise sources. In classical fluctuating environments, the PL noise restricts the entanglement sudden death and birth phenomenon, thereby preventing the conversion of free states to resource states. In addition to the noise phase, parameter optimization in local fields regulates disorders, noise effects, and memory properties. Unlike the bipartite and tripartite W-type states, the GHZ-like state has shown positive traits of preserving entanglement and coherence in the presence of PL noise.

There exist infinitely many kinds of partial separability/entanglement

In tri-partite systems, there are three basic biseparability, A-BC, B-CA, and C-AB, according to bipartitions of local systems. We begin with three convex sets consisting of these basic biseparable states in the three-qubit system, and consider arbitrary iterations of intersections and/or convex hulls of them to get convex cones. One natural way to classify tri-partite states is to consider those convex sets to which they belong or do not belong. This is especially useful to classify partial entanglement of mixed states. We show that the lattice generated by those three basic convex sets with respect to convex hull and intersection has infinitely many mutually distinct members to see that there are infinitely many kinds of three-qubit partial entanglement. To do this, we consider an increasing chain of convex sets in the lattice and exhibit three-qubit Greenberger–Horne–Zeilinger diagonal states distinguishing those convex sets in the chain.

Tripartite Quantum Correlations under Power‐Law and Random Telegraph Noises: Collective Effects of Markovian and Non‐Markovian Classical Fields

AbstractThe collective effects of Markovian and non‐Markovian classical fields on the dynamics of entanglement, coherence, and quantum state mixing in a three‐qubit mixed entangled state are examined. Three independent local fields are considered to be influenced by the decoherence effects of two different mixed noisy models: Markovian maximized and non‐Markovian maximized local random configurations. Power‐law noise characterizes two classical environments in the first case, whereas random telegraph noise governs the third. Random telegraph noise rules two classical environments in the latter example, whereas power‐law noise governs the third. In classical fields, non‐Markovian effects are more robust and have a stronger character than Markovian effects, yet both are driven by relevant noises. Non‐Markovianity vanishes at the upper bound of the Markovian noise parameters. Markovian effects are found to be vulnerable to the environments non‐Markovianity and longer memory features. The strength of Markovianity and non‐Markovianity is only moderately affected by the type of environment, but noise parameter optimization has a significant impact. Initial and final limits are provided for the restriction and relaxation in Markovianity and non‐Markovianity of the classical fields for the actual deployment of non‐local protocols.

Experimental Investigation of the Robustness of a New Bell-Type Inequality of Triphoton GHZ States in Open Systems.

Knowing the level of entanglement robustness against quantum bit loss or decoherence mechanisms is an important issue for any application of quantum information. Fidelity of states can be used to judge whether there is entanglement in multi-particle systems. It is well known that quantum channel security in QKD can be estimated by measuring the robustness of Bell-type inequality against noise. We experimentally investigate a new Bell-type inequality (NBTI) in the three-photon Greenberger–Horne–Zeilinger (GHZ) states with different levels of spin-flip noise. The results show that the fidelity and the degree of violation of the NBTI decrease monotonically with the increase of noise intensity. They also provide a method to judge whether there is entanglement in three-particle mixed states.

Symmetry-resolved entanglement detection using partial transpose moments

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.

Disentangling (2+1)D topological states of matter with entanglement negativity

We use the entanglement negativity, a bipartite measure of entanglement in mixed quantum states, to study how multipartite entanglement constrains the real-space structure of the ground state wavefunctions of $(2+1)$-dimensional topological phases. We focus on the (Abelian) Laughlin and (non-Abelian) Moore-Read states at filling fraction $\nu=1/m$. We show that a combination of entanglement negativities, calculated with respect to specific cylinder and torus geometries, determines a necessary condition for when a topological state can be disentangled, i.e., factorized into a tensor product of states defined on cylinder subregions. This condition, which requires the ground state to lie in a definite topological sector, is sufficient for the Laughlin state. On the other hand, we find that a general Moore-Read ground state cannot be disentangled even when the disentangling condition holds.

Trade-off between Squashed Entanglement and Concurrence in Bipartite Quantum States

In this article we investigate the unitary dynamics of squashed entanglement and concurrence measures in Werner state and maximally entangled mixed states (MEMS) under two different Hamiltonians. The aim of the present study is two fold. The first part of the study deals with the dynamics under Heisenberg Hamiltonian and the second part deals under bi-linear bi-quadratic Hamiltonian which is the extension of first Hamiltonian. In both the parts we investigate the dynamical trade off and balancing points for squashed entanglement and concurrence. During the study, we also found the results of entanglement sudden death (ESD) with Heisenberg Hamiltonian in Werner state under concurrence measure. In second part, we investigate the special result for bi-linear bi-quadratic Hamiltonian; it does not disturb squashed entanglement and concurrence in both the states and exhibit the robust character for both of the states.

Mixed state entanglement and thermal phase transitions

We study the relationship between mixed state entanglement and thermal phase transitions. As a typical example, we compute the holographic entanglement entropy (HEE), holographic mutual information (MI), and the holographic entanglement wedge minimum cross section (EWCS) over the superconductivity phase transition. We find that HEE, MI, and EWCS can all diagnose the superconducting phase transition. They are continuous at the critical point, but their first derivative with respect to temperature is discontinuous. MI decreases with increasing temperature and exhibits a convex behavior, while HEE increases with increasing temperature and exhibits a concave behavior. However, EWCS can exhibit either the same or the opposite behavior as MI, depending on the size of the specific configuration. These results show that EWCS captures more abundant information than HEE and MI. We also provide a new algorithm to compute the EWCS for general configurations.