Bipartite Cut Research Articles

Hilbert space fragmentation and slow dynamics in particle-conserving quantum East models

Quantum kinetically constrained models have recently attracted significant attention due to their anomalous dynamics and thermalization. In this work, we introduce a hitherto unexplored family of kinetically constrained models featuring conserved particle number and strong inversion-symmetry breaking due to facilitated hopping. We demonstrate that these models provide a generic example of so-called quantum Hilbert space fragmentation, that is manifested in disconnected sectors in the Hilbert space that are not apparent in the computational basis. Quantum Hilbert space fragmentation leads to an exponential in system size number of eigenstates with exactly zero entanglement entropy across several bipartite cuts. These eigenstates can be probed dynamically using quenches from simple initial product states. In addition, we study the particle spreading under unitary dynamics launched from the domain wall state, and find faster than diffusive dynamics at high particle densities, that crosses over into logarithmically slow relaxation at smaller densities. Using a classically simulable cellular automaton, we reproduce the logarithmic dynamics observed in the quantum case. Our work suggests that particle conserving constrained models with inversion symmetry breaking realize so far unexplored dynamical behavior and invite their further theoretical and experimental studies.

Superactivating Bound Entanglement in Quantum Networks via Quantum Zeno Dynamics and a Novel Algorithm for Optimized Zeno Evolution

An arbitrary amount of entanglement shared among nodes of a quantum network might be nondistillable if the nodes lack the information on the entangled Bell pairs they share. Making such a system distillable, which is called the superactivation of bound entanglement (BE), was shown to be possible through systematic quantum teleportation between the nodes, requiring the implementation of controlled-gates scaling with the number of nodes. In this work, we show in two scenarios that the superactivation of BE is possible if nodes implement the proposed local quantum Zeno strategies based on only single qubit rotations and simple threshold measurements. In the first scenario we consider, we obtain a two-qubit distillable entanglement system as in the original superactivation proposal. In the second scenario, we show that superactivation can be achieved among the entire network of eight qubits in five nodes. In addition to obtaining all-particle distillable entanglement, the overall entanglement of the system in terms of the sum of bipartite cuts is increased. We also design a general algorithm with variable greediness for optimizing the QZD evolution tasks. Implementing our algorithm for the second scenario, we show that a significant improvement can be obtained by driving the initial BE system into a maximally entangled state. We believe our work contributes to quantum technologies from both practical and fundamental perspectives bridging nonlocality, bound entanglement and the quantum Zeno dynamics among a quantum network.

Entanglement properties of random invariant quantum states

Entanglement properties of random multipartite quantum states which are invariant under global $\textnormal{SU}(d)$ action are investigated. The random states live in the tensor power of an irreducible representation of $\textnormal{SU}(d)$. We calculate and analyze the expectation and fluctuation of the second-order R\'enyi entanglement measure of the random invariant and near-invariant states in high dimension, and reveal the phenomenon of concentration of measure the random states exhibit. We show that with high probability a random SU($d$)-invariant state is close to being maximally entangled with respect to any bipartite cut as the dimension of individual system goes to infinity. We also show that this generic entanglement property of random SU(2)-invariant state is robust to arbitrarily finite disturbation.

Construction of genuinely entangled multipartite subspaces from bipartite ones by reducing the total number of separated parties

Construction of genuinely entangled multipartite subspaces with certain characteristics has become a relevant task in various branches of quantum information. Here we show that such subspaces can be obtained from an arbitrary collection of bipartite entangled subspaces under joining of their adjacent subsystems. In addition, it is shown that direct sums of such constructions under certain conditions are genuinely entangled. These facts are then used in detecting entanglement of tensor products of mixed states and constructing subspaces that are distillable across every bipartite cut, where for the former application we include an example with the analysis of genuine entanglement of a tripartite state obtained from two Werner states.

Tensor Products of Entangled Subspaces and Their Properties from Diagrammatic Constructions

Construction of genuinely entangled multipartite subspaces with certain characteristics has become a relevant task in various branches of quantum information. Here we show that such subspaces can be obtained from an arbitrary collection of bipartite entangled subspaces under joining of their adjacent subsystems. In addition, it is shown that direct sums of such constructions under certain conditions are genuinely entangled. These facts are then used in detecting entanglement of tensor products of mixed states and constructing subspaces that are distillable across every bipartite cut, where for the former application we include example with the analysis of genuine entanglement of a tripartite state obtained from two Werner states.

Master integrals for bipartite cuts of three-loop photon self energy

We calculate the master integrals for bipartite cuts of the three-loop propagator QED diagrams. These master integrals determine the spectral density of the photon self energy. Our results are expressed in terms of the iterated integrals, which, apart from the 4m cut (the cut of 4 massive lines), reduce to Goncharov’s polylogarithms. The master integrals for 4m cut have been calculated in our previous paper in terms of the one-fold integrals of harmonic polylogarithms and complete elliptic integrals. We provide the threshold and high-energy asymptotics of the master integrals found, including those for 4m cut.

The Non-m-Positive Dimension of a Positive Linear Map

We introduce a property of a matrix-valued linear map Φ that we call its ``non-m-positive dimension'' (or ``non-mP dimension'' for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of Im⊗Φ. Equivalently, the non-mP dimension of Φ tells us the maximal number of negative eigenvalues that the adjoint map Im⊗Φ∗ can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good Φ is at detecting entanglement in quantum states. We derive non-trivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer--Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.

Characterization of the quantumness of unsteerable tripartite correlations

Quantumness for a bipartite unsteerable quantum correlation is operationally characterized by the notion of super-unsteerability. Super-unsteerability refers to the requirement of a larger dimension of the random variable that the steering party has to preshare with the party to be steered in the classical simulation protocol to generate an unsteerable correlation than the local Hilbert space dimension of the quantum states (reproducing the given unsteerable correlation) at the steering party’s side. In the present study, this concept of super-unsteerability is generalized by defining the notion of super-bi-unsteerability for tripartite correlations, which is unsteerable across a bipartite cut. Genuine super-bi-unsteerability is defined as the occurrence of super-bi-unsteerability across all possible bipartite cuts. Specific example of genuine super-bi-unsteerability for tripartite correlations has been presented. This study provides a tool to characterize the genuine quantumness of tripartite quantum correlations which are unsteerable across every bipartite cut.

Constructing genuinely entangled multipartite states with applications to local hidden variables and local hidden states models

Building upon the results of [R. Augusiak et al., Phys. Rev. Lett. 115, 030404 (2015)] we develop a general approach to the generation of genuinely entangled multipartite states of any number of parties from genuinely entangled states of a fixed number of parties, in particular, the bipartite entangled ones. In our approach, certain isometries whose output subspaces are either symmetric or genuinely entangled in some multipartite Hilbert spaces are applied to local subsystems of bipartite entangled or multipartite genuinely entangled quantum states. To prove that entanglement of the resulting states is indeed genuine we then introduce novel criteria allowing to decide it efficiently. The construction is then exploited to provide examples of multipartite states that are genuinely entangled but not genuinely nonlocal, giving further illustration for the inequivalence between entanglement and nonlocality in the multiparticle scenario. It is also shown how to construct genuinely entangled states which are unsteerable across certain bipartite cuts.

An algorithm to explore entanglement in small systems.

A quantum state’s entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels.

Entanglement rates for bipartite open systems

We provide upper bound on the maximal rate at which irreversible quantum dynamics can generate entanglement in a bipartite system. The generator of irreversible dynamics consists of a Hamiltonian and dissipative terms in Lindblad form. The relative entropy of entanglement is chosen as a measure of entanglement in an ancilla-free system. We provide an upper bound on the entangling rate which has a logarithmic dependence on a dimension of a smaller system in a bipartite cut. We also investigate the rate of change of quantum mutual information in an ancilla-assisted system and provide an upper bound independent of dimension of ancillas.

Two-message quantum interactive proofs and the quantum separability problem

Suppose that a polynomial-time mixed-state quantum circuit, described as a sequence of local unitary interactions followed by a partial trace, generates a quantum state shared between two parties. One might then wonder, does this quantum circuit produce a state that is separable or entangled? Here, we give evidence that it is computationally hard to decide the answer to this question, even if one has access to the power of quantum computation. We begin by exhibiting a two-message quantum interactive proof system that can decide the answer to a promise version of the question. We then prove that the promise problem is hard for the class of promise problems with "quantum statistical zero knowledge" QSZK proof systems by demonstrating a polynomial-time Karp reduction from the QSZK-complete promise problem "quantum state distinguishability" to our quantum separability problem. By exploiting Knill's efficient encoding of a matrix description of a state into a description of a circuit to generate the state, we can show that our promise problem is NP-hard with respect to Cook reductions. Thus, the quantum separability problem (as phrased above) constitutes the first nontrivial promise problem decidable by a two-message quantum interactive proof system while being hard for both NP and QSZK. We also consider a variant of the problem, in which a given polynomial-time mixed-state quantum circuit accepts a quantum state as input, and the question is to decide if there is an input to this circuit which makes its output separable across some bipartite cut. We prove that this problem is a complete promise problem for the class QIP of problems decidable by quantum interactive proof systems. Finally, we show that a two-message quantum interactive proof system can also decide a multipartite generalization of the quantum separability problem.

Two-message quantum interactive proofs and the quantum separability problem

Suppose that a polynomial-time mixed-state circuit, described as a sequence of local unitary interactions followed by a partial trace, generates a state shared between two parties. One might then wonder, does this circuit produce a state that is separable or entangled? Here, we give evidence that it is computationally hard to decide the answer to this question, even if one has access to the power of computation. We begin by exhibiting a two-message interactive proof system that can decide the answer to a promise version of the question. We then prove that the promise problem is hard for the class of promise problems with quantum statistical zero knowledge (QSZK) proof systems by demonstrating a polynomial-time Karp reduction from the QSZK-complete promise problem quantum state distinguishability to our separability problem. By exploiting Knill's efficient encoding of a matrix description of a state into a description of a circuit to generate the state, we can show that our promise problem is NP-hard with respect to Cook reductions. Thus, the separability problem (as phrased above) constitutes the first nontrivial promise problem decidable by a two-message interactive proof system while being hard for both NP and QSZK. We also consider a variant of the problem, in which a given polynomial-time mixed-state circuit accepts a state as input, and the question is to decide if there is an input to this circuit which makes its output separable across some bipartite cut. We prove that this problem is a complete promise problem for the class QIP of problems decidable by interactive proof systems. Finally, we show that a two-message interactive proof system can also decide a multipartite generalization of the separability problem.

All maximally entangled four-qubit states

We find an operational interpretation for the 4-tangle as a type of residual entanglement, somewhat similar to the interpretation of the 3-tangle. Using this remarkable interpretation, we are able to find the class of maximally entangled four-qubits states which is characterized by four real parameters. The states in the class are maximally entangled in the sense that their average bipartite entanglement with respect to all possible bipartite cuts is maximal. We show that while all the states in the class maximize the average tangle, there are only a few states in the class that maximize the average Tsillas or Renyi α-entropy of entanglement. Quite remarkably, we find that up to local unitaries, there exists two unique states, one maximizing the average α-Tsallis entropy of entanglement for all α ⩾ 2, while the other maximizing it for all 0 < α ⩽ 2 (including the von-Neumann case of α = 1). Furthermore, among the maximally entangled four qubits states, there are only three maximally entangled states that have the property that for two, out of the three bipartite cuts consisting of two-qubits verses two-qubits, the entanglement is 2 ebits and for the remaining bipartite cut the entanglement between the two groups of two qubits is 1 ebit. The unique three maximally entangled states are the three cluster states that are related by a swap operator. We also show that the cluster states are the only states (up to local unitaries) that maximize the average α-Renyi entropy of entanglement for all α ⩾ 2.

Entanglement measure and dynamics of multiqubit systems: non-Markovian versus Markovian and generalized monogamy relations

In this paper, we first present a simple measure for multiqubit entanglement based on the strategy of bipartite cuts and the measure of negativity. Then, we establish generalized monogamy inequalities and associated partition-dependent residual entanglement (PRE) accounting for arbitrary partitions of a multiqubit system. By virtue of the defined quantities, we investigate the entanglement dynamics of a system of N qubits, either in the Greenberger–Horne–Zeilinger (GHZ)-type state or in the W state, interacting with N independent reservoirs in both Markovian and non-Markovian regimes. We observe entanglement revivals of qubits at instantaneous points of disappearance or after a finite interval of abrupt vanishing due to the memory effect of non-Markovian reservoirs. We also follow the whole entanglement evolution in terms of the PRE to demonstrate the process of transition between the bipartite entanglement of all possible bipartitions and the multipartite entanglement. In particular, we show that the change in time of entanglement formats differs qualitatively for the GHZ-type and W states.

Class of positive-partial-transpose bound entangled states associated with almost any set of pure entangled states

We analyze a class of entangled states for bipartite $d \otimes d$ systems, with $d$ non-prime. The entanglement of such states is revealed by the construction of canonically associated entanglement witnesses. The structure of the states is very simple and similar to the one of isotropic states: they are a mixture of a separable and a pure entangled state whose supports are orthogonal. Despite such simple structure, in an opportune interval of the mixing parameter their entanglement is not revealed by partial transposition nor by the realignment criterion, i.e. by any permutational criterion in the bipartite setting. In the range in which the states are Positive under Partial Transposition (PPT), they are not distillable; on the other hand, the states in the considered class are provably distillable as soon as they are Nonpositive under Partial Transposition (NPT). The states are associated to any set of more than two pure states. The analysis is extended to the multipartite setting. By an opportune selection of the set of multipartite pure states, it is possible to construct mixed states which are PPT with respect to any choice of bipartite cuts and nevertheless exhibit genuine multipartite entanglement. Finally, we show that every $k$-positive but not completely positive map is associated to a family of nondecomposable maps.

Entanglement probability distribution of bi-partite randomised stabilizer states

We study the entanglement properties of random pure stabilizer states in spin-1/2 particles. We obtain a compact and exact expression for the probability distribution of the entanglement values across any bipartite cut. This allows for exact derivations of the average entanglement and the degree of concentration of measure around this average. We also give simple bounds on these quantities. We find that for large systems the average entanglement is near maximal and the measure is concentrated around it.

Maximum cuts and judicious partitions in graphs without short cycles

We consider the bipartite cut and the judicious partition problems in graphs of girth at least 4. For the bipartite cut problem we show that every graph G with m edges, whose shortest cycle has length at least r⩾4, has a bipartite subgraph with at least m 2 +c(r)m r r+1 edges. The order of the error term in this result is shown to be optimal for r=5 thus settling a special case of a conjecture of Erdős. (The result and its optimality for another special case, r=4, were already known.) For judicious partitions, we prove a general result as follows: if a graph G=( V, E) with m edges has a bipartite cut of size m 2 +δ , then there exists a partition V= V 1∪ V 2 such that both parts V 1, V 2 span at most m 4 −(1−o(1)) δ 2 +O( m ) edges for the case δ= o( m), and at most ( 1 4 −Ω(1))m edges for δ=Ω(m). This enables one to extend results for the bipartite cut problem to the corresponding ones for judicious partitioning.