Compressed entanglement testing

Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.

quant-pharXiv:2606.16712Jun 16, 20260 citations

Detecting entanglement with partial state information

Federico M. Spedalieri

We introduce a sequence of numerical tests that can determine the entanglement or separability of a state even when there is not enough information to completely determine its density matrix. Given partial information about the state in the form of linear constraints on the density matrix, the sequence of tests can prove that either all states satisfying the constraints are entangled or there is at least one separable state that satisfies them. The algorithm works even if the values of the constraints are only known to fall in a certain range. If the states are entangled, an entanglement witness is constructed and lower bounds on entanglement measures and related quantities are provided; if a separable state satisfies the constraints, a separable decomposition is provided to certify this fact.We introduce a sequence of numerical tests that can determine the entanglement or separability of a state even when there is not enough information to completely determine its density matrix. Given partial information about the state in the form of linear constraints on the density matrix, the sequence of tests can prove that either all states satisfying the constraints are entangled or there is at least one separable state that satisfies them. The algorithm works even if the values of the constraints are only known to fall in a certain range. If the states are entangled, an entanglement witness is constructed and lower bounds on entanglement measures and related quantities are provided; if a separable state satisfies the constraints, a separable decomposition is provided to certify this fact.

arXiv:1208.0860Physical Review ADec 13, 20129 citations

Entanglement and Quantumness - New Numerical Approaches -

Fabian Bohnet-Waldraff

The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.

Sep 29, 20170 citations

Constructing all entanglement witnesses from density matrices

Bang-Hai Wang, Dongyang Long

We demonstrate a general procedure to construct entanglement witnesses for any entangled state. This procedure is based on the trace inequality and a general form of entanglement witnesses, which is in the form $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$, where $\ensuremath{\rho}$ is a density matrix, ${c}_{\ensuremath{\rho}}$ is a non-negative number related to $\ensuremath{\rho}$, and $I$ is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio\l{}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$ in operation, as well as in theory.We demonstrate a general procedure to construct entanglement witnesses for any entangled state. This procedure is based on the trace inequality and a general form of entanglement witnesses, which is in the form $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$, where $\ensuremath{\rho}$ is a density matrix, ${c}_{\ensuremath{\rho}}$ is a non-negative number related to $\ensuremath{\rho}$, and $I$ is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio\l{}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$ in operation, as well as in theory.

arXiv:1111.5255Physical Review AJul 25, 20115 citations

Witnessing multipartite entanglement

Stefan Gerke

This cumulative dissertation introduces new methods of the topic of entanglement testing based on the separability eigenvalue equations. In principle, the separability eigenvalue equations represent a method to construct entanglement tests via witnessing approaches. An approach is presented that enables finding reliable entanglement tests even for experimental states. A newly introduced algorithm, the Separability Power Iteration, enables finding solutions to the separability eigenvalue equations numerically, thereby broadening the range on the introduced approach.This cumulative dissertation introduces new methods of the topic of entanglement testing based on the separability eigenvalue equations. In principle, the separability eigenvalue equations represent a method to construct entanglement tests via witnessing approaches. An approach is presented that enables finding reliable entanglement tests even for experimental states. A newly introduced algorithm, the Separability Power Iteration, enables finding solutions to the separability eigenvalue equations numerically, thereby broadening the range on the introduced approach.

Jan 1, 20180 citations

Entanglement witnesses and semidefinite programming

Pablo A. Parrilo, Andrew C. Doherty, Federico M. Spedalieri

We study the application of sums of squares decompositions and semidefinite programming in the formulation of computational criteria for distinguishing entangled from separable quantum states. A hierarchy of tests is obtained, the simplest of which corresponds to the well-known positive partial transpose (PPT) sufficient criterion, with the more complicated tests being strictly stronger. The duality structure of the problem allows us to provide an explicit construction of the entanglement witnesses corresponding to our tests.We study the application of sums of squares decompositions and semidefinite programming in the formulation of computational criteria for distinguishing entangled from separable quantum states. A hierarchy of tests is obtained, the simplest of which corresponds to the well-known positive partial transpose (PPT) sufficient criterion, with the more complicated tests being strictly stronger. The duality structure of the problem allows us to provide an explicit construction of the entanglement witnesses corresponding to our tests.

IEEE Conference on Decision and Control (CDC)Jan 23, 20042 citations

I

is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio\l{}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=\ensuremath{\rho}\ensuremath{-}{c}_{\ensuremath{\rho}}I$ in operation, as well as in theory.