In this work, we establish a general theory of phase transitions and quantum entanglement in the equilibrium state at arbitrary temperatures. First, we derived a set of universal functional relations between the matrix elements of two-body reduced density matrix of the canonical density matrix and the Helmholtz free energy of the equilibrium state, which implies that the Helmholtz free energy and its derivatives are directly related to entanglement measures because any entanglement measures are defined as a function of the reduced density matrix. Then we show that the first order phase transitions are signaled by the matrix elements of reduced density matrix while the second order phase transitions are witnessed by the first derivatives of the reduced density matrix elements. Near second order phase transition point, we show that the first derivative of the reduced density matrix elements present universal scaling behaviors. Finally we establish a theorem which connects the phase transitions and entanglement at arbitrary temperatures. Our general results are demonstrated in an experimentally relevant many-body spin model.
Density functional theory (DFT) is shown to provide a novel conceptual and computational framework for entanglement in interacting many-body quantum systems. DFT can, in particular, shed light on the intriguing relationship between quantum phase transitions and entanglement. We use DFT concepts to express entanglement measures in terms of the first or second derivative of the ground state energy. We illustrate the versatility of the DFT approach via a variety of analytically solvable models. As a further application we discuss entanglement and quantum phase transitions in the case of mean fiel…
An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct application of this result provides a closed expression for the Bures measure of entanglement of two qubits. We also prove that the number of elements in an optimal decomposition w.r.t. the geometric measure o…
We show that a von Neumann measurement on a part of a composite quantum system unavoidably creates distillable entanglement between the measurement apparatus and the system if the state has nonzero quantum discord. The minimal distillable entanglement is equal to the one-way information deficit. The quantum discord is shown to be equal to the minimal partial distillable entanglement that is the part of entanglement which is lost, when we ignore the subsystem which is not measured. We then show that any entanglement measure corresponds to some measure of quantum correlations. This powerful corr…
In $3d$ Chern-Simons theory, there is a discrete one-form symmetry, whose symmetry group is isomorphic to the center of the gauge group. We study the 't Hooft anomaly associated to this discrete one-form symmetry in theories with generic gauge groups, $A,B,C,D$-types. We propose to detect the discrete anomaly by computing the Hopf state entanglement in the subspace spanned by the symmetry generators and develop a systematical way based on the truncated modular S matrix. We check our proposal for many examples.
We present an experimentally feasible and efficient method for detecting entangled states with measurements that extend naturally to a tomographically complete set. Our detection criterion is based on measurements from subsets of a quantum 2-design, e.g., mutually unbiased bases or symmetric informationally complete states, and has several advantages over standard entanglement witnesses. First, as more detectors in the measurement are applied, there is a higher chance of witnessing a larger set of entangled states, in such a way that the measurement setting converges to a complete setup for qu…
The paper focuses on aspects of the measurement problem introducing quantum states (q-states) for measured and measuring systems. The link between non-interacting and interacting quantum systems is first look at. For two independent partite systems logical sums A+B stand for non-interacting q-systems; while a direct product space AxB gathers interacting states. However this latter should support physical q-states with base states that do not separately belong to either A nor B; the latter correspond to bridge states, namely entangled states that can perform as links (bridges) between A+B and A…
We analyze the no-cloning theorem in quantum mechanics through the lens of the proposed ER=EPR (Einstein-Rosen = Einstein-Podolsky-Rosen) duality between entanglement and wormholes. In particular, we find that the no-cloning theorem is dual on the gravity side to the no-go theorem for topology change, violating the axioms of which allows for wormhole stabilization and causality violation. Such a duality between important no-go theorems elucidates the proposed connection between spacetime geometry and quantum entanglement.
We compute the entanglement entropy in a 2+1 dimensional topological order in the presence of gapped boundaries. Specifically, we consider entanglement cuts that cut through the boundaries. We argue that based on general considerations of the bulk-boundary correspondence, the "twisted characters" feature in the Renyi entropy, and the topological entanglement entropy is controlled by a "half-linking number" in direct analogy to the role played by the S-modular matrix in the absence of boundaries. We also construct a class of boundary states based on the half-linking numbers that provides a "clo…
Classical and quantum error correction are presented in the form of Maxwell's demon and their efficiency analyzed from the thermodynamic point of view. We explain how Landauer's principle of information erasure applies to both cases. By then extending this principle to entanglement manipulations we rederive upper bounds on purification procedures thereby linking the ''no local increase of entanglement'' principle to the Second Law of thermodynamics.
Entanglement of formation is an important measure of quantum entanglement. We present an experimental way to measure the entanglement of formation for arbitrary dimensional pure states. The measurement only evolves local quantum mechanical observables.
We present in detail a statistical approach for the reference-frame-independent detection and characterization of multipartite entanglement based on moments of randomly measured correlation functions. We start by discussing how the corresponding moments can be evaluated with designs, linking methods from group and entanglement theory. Then, we illustrate the strengths of the presented framework with a focus on the multipartite scenario. We discuss a condition for characterizing genuine multipartite entanglement for three qubits, and we prove criteria that allow for a discrimination of $W$-type…
Two measures of entanglement, negativity and concurrence are studied for two arbitrary qudits. We obtain negativity as an expectation value of an operator. The differences of the squares of negativity and concurrence are invariants of multilevel entanglement. Explicit results for qutrits and quadrits are obtained.
In this paper, we define a cross product operator and construct the cross Bell basis, by use this basis and Bell measurements we give a simple scheme of the teleportation of arbitrary multipartite qubit entanglement.
We present an inequality for detecting entanglement and distillability of arbitrary dimensional bipartite systems. This inequality provides a sufficient condition of entanglement for bipartite mixed states, and a necessary and sufficient condition of entanglement for bipartite pure states. Moreover, the inequality also gives a necessary and sufficient condition for distillability.
We investigate polygamy relations of multipartite entanglement in arbitrary-dimensional quantum systems. By improving an inequality and using the $β$th ($0\leqβ\leq1$) power of entanglement of assistance, we provide a new class of weighted polygamy inequalities of multipartite entanglement in arbitrary-dimensional quantum systems. We show that these new polygamy relations are tighter than the ones given in [Phys. Rev. A 97, 042332 (2018)].
After carrying out a protocol for quantum key agreement over a noisy quantum channel, the parties Alice and Bob must process the raw key in order to end up with identical keys about which the adversary has virtually no information. In principle, both classical and quantum protocols can be used for this processing. It is a natural question which type of protocols is more powerful. We prove for general states but under the assumption of incoherent eavesdropping that Alice and Bob share some so-called intrinsic information in their classical random variables, resulting from optimal measurements, …
Based on the generators of $SU(n)$ we present inequalities for detecting quantum entanglement for $2 \otimes d$ and $M \otimes N$ systems. These inequalities provide a sufficient condition of entanglement for bipartite mixed states and give rise to an experimental way of entanglement detection.
We provide a generalization for the polygamy constraint of multiparty entanglement in arbitrary dimensional quantum systems. By using the $β$th-power of entanglement of assistance for $0\leq β\leq1$ and the Hamming weight of the binary vector related with the distribution of subsystems, we establish a class of weighted polygamy inequalities of multiparty entanglement in arbitrary dimensional quantum systems. We further show that our class of weighted polygamy inequalities can even be improved to be tighter inequalities with some conditions on the assisted entanglement of bipartite subsystems.
We write the optimal pure-state decomposition of any two-mode Gaussian state and show that its entanglement of formation coincides with the Gaussian one. This enables us to develop an insightful approach of evaluating the exact entanglement of formation. Its additivity is finally proven.
We consider entanglement for quantum states defined in vector spaces over the real numbers. Such real entanglement is different from entanglement in standard quantum mechanics over the complex numbers. The differences provide insight into the nature of entanglement in standard quantum theory. Wootters [Phys. Rev. Lett. 80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. We give a contrasting formula for the entanglement of formation of an arbitrary state of two ``rebits,'' a rebi…
The information spectrum approach gives general formulae for optimal rates of various information theoretic protocols, under minimal assumptions on the nature of the sources, channels and entanglement resources involved. This paper culminates in the derivation of the dense coding capacity for a noiseless quantum channel, assisted by arbitrary shared entanglement, using this approach. We also review the currently known coding theorems, and their converses, for protocols such as data compression for arbitrary quantum sources and transmission of classical information through arbitrary quantum cha…
In Phys. Rev. Lett. 80, 2245 (1998) an explicit expression for entanglement of formation for any two qubit state was given. Based on this result we present an expression for the Bures measure of entanglement for two qubit states. This measure was proposed in Phys. Rev. A 57, 1619 (1998), where the authors showed that it satisfies all properties every entanglement measure must have.
The entanglement cost of arbitrary sequences of bipartite states is shown to be expressible as the minimization of a conditional spectral entropy rate over sequences of separable extensions of the states in the sequence. The expression is shown to reduce to the regularized entanglement of formation when the n-th state in the sequence consists of n copies of a single bipartite state.
Dicke states represent a class of multipartite entangled states that can be generated experimentally with many applications in quantum information. We propose a method to experimentally detect genuine multipartite entanglement in the vicinity of arbitrary Dicke states. The detection scheme can be used to experimentally quantity the entanglement depth of many-body systems and is easy to implement as it requires to measure only three collective spin operators. The detection criterion is strong as it heralds multipartite entanglement even in cases where the state fidelity goes down exponentially …
Based on the complementarity relation between entanglement of a composite system and the purity of a subsystem, we propose a simple method to measure the amount of entanglement. The method can be applied to a bipartite system in a pure state of any arbitrary dimension. It requires only single qudit rotations and straightforward probability measurements performed on one of the subsystems, and can thus be easily implemented experimentally using linear optical devices.
We have found a quantum cloning machine that optimally duplicates the entanglement of a pair of $d$-dimensional quantum systems. It maximizes the entanglement of formation contained in the two copies of any maximally-entangled input state, while preserving the separability of unentangled input states. Moreover, it cannot increase the entanglement of formation of all isotropic states. For large $d$, the entanglement of formation of each clone tends to one half the entanglement of the input state, which corresponds to a classical behavior. Finally, we investigate a local entanglement cloner, whi…
Incoherent scattering of photons off two remote atoms with a Lambda-level structure is used as a basic Young-type interferometer to herald long-lived entanglement of an arbitrary degree. The degree of entanglement, as measured by the concurrence, is found to be tunable by two easily accessible experimental parameters. Fixing one of them to certain values unveils an analog to the Malus' law. An estimate of the variation in the degree of entanglement due to uncertainties in an experimental realization is given.
An entanglement purification scheme for arbitrary unknown(mixed and pure non-maximally) entangled ionic states is proposed by using linear optical elements. The main advantage of the scheme is that not only two-ion maximally entangled pairs but also four-ion maximally entangled pairs can be extracted from the less entangled pairs. The scheme is within current technology.
The entanglement quantification and classification of multipartite quantum states are two important research fields in quantum information. In this work, we study the entanglement of arbitrary-dimensional multipartite pure states by looking at the averaged partial entropies of various bipartite partitions of the system, namely, the so-called Manhattan distance ($l_1$ norm) of averaged partial entropies (MAPE), and it is proved to be an entanglement measure for pure states. We connected the MAPE with the coefficient matrices, which are important tools in entanglement classification and reexpres…
We investigate the thermal entanglement in two superconducting qubits for arbitrary interaction strength and ground state frequencies. We calculate the concurrence of the system to quantify the thermal entanglement. We suggest a scheme, where an external tunable coupler qubit sandwich between two superconducting qubits generates entanglement. The behavior of concurrence is analyzed for three different cases, in which we consider the effects of the temperature, the qubit-qubit effective coupling strength, and the qubit frequencies on the thermal entanglement. What deserves mentioning here is th…
The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state is defined as the minimum average entanglement of an ensemble of pure states that represents the given mixed state. An earlier paper [Phys. Rev. Lett. 78, 5022 (1997)] conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula to be true for a special class of mixed states. The present paper extends the proof to arbitra…
Given an arbitrary statistical theory, different from quantum mechanics, how to decide which are the nonclassical correlations? We present a formal framework which allows for a definition of nonclassical correlations in such theories, alternative to the current one. This enables one to formulate extrapolations of some important quantum mechanical features via adequate extensions of reciprocal maps relating states of a system with states of its subsystems. These extended maps permit one to generalize i) separability measures to any arbitrary statistical model as well as ii) previous entanglemen…
We establish a unified view to the polygamy of multi-party quantum entanglement in arbitrary dimensions. Using quantum Tsallis-$q$ entropy, we provide a one-parameter class of polygamy inequalities of multi-party quantum entanglement. This class of polygamy inequalities reduces to the known polygamy inequalities based on tangle and entanglement of assistance for a selective choice of the parameter $q$. We further provide one-parameter generalizations of various quantum correlations based on Tsallis-$q$ entropy. By investigating the properties of the generalized quantum correlations, we provide…
We give a security proof of quantum cryptography based entirely on entanglement purification. Our proof applies to all possible attacks (individual and coherent). It implies the security of cryptographic keys distributed with the help of entanglement-based quantum repeaters. We prove the security of the obtained quantum channel which may not only be used for quantum key distribution, but also for secure, albeit noisy, transmission of quantum information.
The Groverian entanglement measure of pure quantum states of $n$ qubits is generalized to the case in which the qubits are divided into any $m \le n$ parties and the entanglement between these parties is evaluated. To demonstrate this measure we apply it to general states of three qubits and to symmetric states with any number of qubits such as the Greenberg-Horne-Zeiliner state and the W state.
We study the entanglement entropy of connected bipartitions in free fermion gases of N particles in arbitrary dimension d. We show that the von Neumann and Renyi entanglement entropies grow asymptotically as N^(1-1/d) ln N, with a prefactor that is analytically computed using the Widom conjecture both for periodic and open boundary conditions. The logarithmic correction to the power-law behavior is related to the area-law violation in lattice free fermions. These asymptotic large-N behaviors are checked against exact numerical calculations for N-particle systems.
Universal two-particle entanglement processes are analyzed in arbitrary dimensional Hilbert spaces. On the basis of this analysis the class of possible optimal universal entanglement processes is determined whose resulting output states do not contain any separable states. It is shown that these processes form a one-parameter family. For all Hilbert space dimensions larger than two the resulting optimally entangled output states are mixtures of anti-symmetric states which are freely entangled and which also preserve information about input states. Within this one-parameter family there is only…
In this paper, we define the return operator, the cross product operator and the cross Bell bases. Using the cross Bell bases, we give a protocol of teleportation of arbitrary multipartite qubit entanglement, this scheme is a quite natural generalization of the BBCJPW scheme. We find that this teleportation, in fact, is essentially determined by the teleportation of every single unknown qubit state as in the original scheme of BBCJPW. The calculation in detail is given for the case of tripartite qubit.
We present an efficient quantum entanglement distribution over an arbitrary collective-noise channel. The basic idea in the present scheme is that two parties in quantum communication first transmit the entangled states in the frequency degree of freedom which suffers little from the noise in an optical fiber. After the two parties share the photon pairs, they add some operations and equipments to transfer the frequency entanglement of pairs into the polarization entanglement with the success probability of 100\%. Finally, they can get maximally entangled polarization states with polarization …
We consider the case of a pair of particles initially in a superposition state corresponding to a separated pair of wave packets. We calculate \emph{exactly} the time development of this non-Gaussian state due to interaction with an \emph{arbitrary} heat bath. We find that coherence decays continuously, as expected. We then investigate entanglement and find that at a finite time the system becomes separable (not entangled). Thus, we see that entanglement sudden death is also prevalent in continuous variable systems which should raise concern for the designers of entangled systems.
We propose a protocol ${\cal D}_n$ for faithfully teleporting an arbitrary $n$-qudit state with the tensor product state (TPS) of $n$ generalized Bell states (GBSs) as the quantum channel. We also put forward explicit protocol ${\cal D}'_n$ and ${\cal D}''_n$ for faithfully teleporting an arbitrary $n$-qudit state with two classes of $2n$-qudit GESs as the quantum channel, where the GESs are a kind of genuine entangled states we construct and can not be reducible to the TPS of $n$ GBSs.
We generalize entanglement detection with covariance matrices for an arbitrary set of observables. A generalized uncertainty relation is constructed using the covariance and commutation matrices, then a criterion is established by performing a partial transposition on the operators. The method is highly efficient and versatile in the sense that the set of measurement operators can be freely chosen, do not need to be complete, and there is no constraint on the commutation relations. The method is particularly suited for systems with higher dimensionality since the computations do not scale with…
We propose a method to generate arbitrary symmetric states of N qubits, which can be easily associated with their entanglement classes. It is particularly suited to quantum optics systems like trapped ions or superconducting circuits. We encode each qubit in two metastable levels of the system and use a bosonic quantum bus for creating the states. The method is deterministic and relies on a sequence of selective unitary gates upon the qubits within the system coherence time.
We obtain the general formula for the optimal rate at which singlets can be distilled from any given noisy and arbitrarily correlated entanglement resource, by means of local operations and classical communication (LOCC). Our formula, obtained by employing the quantum information spectrum method, reduces to that derived by Devetak and Winter, in the special case of an i.i.d. resource. The proofs rely on a one-shot version of the so-called "hashing bound," which in turn provides bounds on the one-shot distillable entanglement under general LOCC.
Entanglement and Bell nonlocality are used to describe quantum inseparabilities. Bell-nonlocal states form a strict subset of entangled states. A natural question arises concerning how much territory Bell nonlocality occupies entanglement for a general two-qubit entangled state. In this work, we investigate the relation between entanglement and Bell nonlocality by using lots of randomly generated two-qubit states, and give out a constraint inequality relation between the two quantum resources. For studying the upper or lower boundary of the inequality relation, we discover maximally (minimally…
We present a way of experimentally determining the concurrence in terms of the expectation values of local observables for arbitrary multipartite pure states. In stead of the joint measurements on two copies of a state in the experiment for two-qubit systems [S. P. Walborn et al. Nature (London)440, 20(2006)], we only need one copy of the state in every measurement for any arbitrary dimensional multipartite systems, avoiding the preparation of twin states or the imperfect copy of the state.
We study the process of heat transfer through an entangled pair of two-level system, demonstrating the role of quantum correlations in this nonequilibrium process. While quantum correlations generally degrade with increasing the temperature bias, introducing spatial asymmetry leads to an intricate behavior: Connecting the qubits unequally to the reservoirs one finds that quantum correlations persist and increase with the temperature bias when the system is more weakly linked to the hot reservoir. In the reversed case, linking the system more strongly to the hot bath, the opposite, more natural…
The dynamics of mixedness and entanglement is examined by solving the time-dependent Schrödinger equation for three coupled harmonic oscillator system with arbitrary time-dependent frequency and coupling constants parameters. We assume that part of oscillators is inaccessible and remaining oscillators accessible. We compute the dynamics of entanglement between inaccessible and accessible oscillators. In order to show the dynamics pictorially we introduce three quenched models. In the quenched models both mixedness and entanglement exhibit oscillatory behavior in time with multi-frequencies. It…
We introduce a new multipartite communication scheme, with the aim to enable the senders to remotely and obliviously provide the receivers with an arbitrary amount of multipartite entanglement. The scheme is similar to Remote State Preparation (RSP). However, we show that even though the receivers are restricted to local unitary operations, the required resources for remote entanglement preparation are less than for RSP. In order to do so we introduce a novel canonical form of arbitrary multipartite pure states describing an arbitrary number of qubits. Moreover, we show that if the receivers a…
We generalise the study of fermionic and bosonic entanglement in non-inertial frames to fields of arbitrary spin and beyond the single mode approximation. After the general analysis we particularise for two interesting cases: entanglement between an inertial and an accelerated observer for massless fields of spin 1 (electromagnetic) and 3/2 (Rarita-Schwinger). We show that in the limit of infinite acceleration, no significant differences appear between the different spin fields for the states considered.
Fidelity estimation is essential for the quality control of entanglement distribution networks. Because measurements collapse quantum states, we consider a setup in which nodes randomly sample a subset of the entangled qubit pairs to measure and then estimate the average fidelity of the unsampled pairs conditioned on the measurement outcome. The proposed estimation protocol achieves the lowest mean squared estimation error in a difficult scenario with arbitrary noise and no prior information. Moreover, this protocol is implementation friendly because it only performs local Pauli operators acco…
