We introduce new entanglement monotones which generalize, to the case of many parties, those which give rise to the majorization-based partial ordering of bipartite states' entanglement. We give some examples of restrictions they impose on deterministic and probabilistic conversion between multipartite states via local actions and classical communication. These include restrictions which do not follow from any bipartite considerations. We derive supermultiplicativity relations between each state's monotones and the monotones for collective processing when the parties share several states. We also investigate polynomial invariants under local unitary transformations, and show that a large class of these are invariant under collective unitary processing and also multiplicative, putting restrictions, for example, on the exact conversion of multiple copies of one state to multiple copies of another.We introduce new entanglement monotones which generalize, to the case of many parties, those which give rise to the majorization-based partial ordering of bipartite states' entanglement. We give some examples of restrictions they impose on deterministic and probabilistic conversion between multipartite states via local actions and classical communication. These include restrictions which do not follow from any bipartite considerations. We derive supermultiplicativity relations between each state's monotones and the monotones for collective processing when the parties share several states. We also investigate polynomial invariants under local unitary transformations, and show that a large class of these are invariant under collective unitary processing and also multiplicative, putting restrictions, for example, on the exact conversion of multiple copies of one state to multiple copies of another.

Jan 1, 2001140 citations

Multi-partite entanglement monotones

Abhijit Gadde, Shraiyance Jain, Harshal Kulkarni

If we want to transform the quantum state of a system to another using local measurement processes, what is the probability of success? This probability is bounded by quantifying entanglement in both the states. In this paper, we construct a family of local unitary invariants of multipartite states that are monotonic under local operations and classical communication on average. These monotones are constructed from local unitary invariant polynomials of the state and its conjugate, and hence are easy to compute for pure states. Using these measures we bound the success probability of transforming a given state into another state using local quantum operations and classical communication.If we want to transform the quantum state of a system to another using local measurement processes, what is the probability of success? This probability is bounded by quantifying entanglement in both the states. In this paper, we construct a family of local unitary invariants of multipartite states that are monotonic under local operations and classical communication on average. These monotones are constructed from local unitary invariant polynomials of the state and its conjugate, and hence are easy to compute for pure states. Using these measures we bound the success probability of transforming a given state into another state using local quantum operations and classical communication.

arXiv:2406.17447SciPost PhysicsMar 18, 20260 citations

Multipartite Entangled States: Transformations, Entanglement Monotones, and Applications

Wei Cui

Entanglement is one of the fundamental features of quantum information science.\nThough bipartite entanglement has been analyzed thoroughly in theory and shown to\nbe an important resource in quantum computation and communication protocols, the\ntheory of entanglement shared between more than two parties, which is called multipartite\nentanglement, is still not complete. Specifically, the classification of multipartite\nentanglement and the transformation property between different multipartite states by\nlocal operators and classical communications (LOCC) are two fundamental questions in\nthe theory of multipartite entanglement.\nIn this thesis, we present results related to the LOCC transformation between multipartite\nentangled states. Firstly, we investigate the bounds on the LOCC transformation\nprobability between multipartite states, especially the GHZ class states. By analyzing\nthe involvement of 3-tangle and other entanglement measures under weak two-outcome\nmeasurement, we derive explicit upper and lower bound on the transformation probability\nbetween GHZ class states. After that, we also analyze the transformation between\nN-party W type states, which is a special class of multipartite entangled states that\nhas an explicit unique expression and a set of analytical entanglement monotones. We\npresent a necessary and sufficient condition for a known upper bound of transformation\nprobability between two N-party W type states to be achieved.\nWe also further investigate a novel entanglement transformation protocol, the random\ndistillation, which transforms multipartite entanglement into bipartite entanglement\nii\nshared by a non-deterministic pair of parties. We find upper bounds for the random distillation\nprotocol for general N-party W type states and find the condition for the upper\nbounds to be achieved. What is surprising is that the upper bounds correspond to entanglement\nmonotones that can be increased by Separable Operators (SEP), which gives\nthe first set of analytical entanglement monotones that can be increased by SEP.\nFinally, we investigate the idea of a new class of multipartite entangled states, the\nAbsolutely Maximal Entangled (AME) states, which is characterized by the fact that\nany bipartition of the states would give a maximal entangled state between the two sets.\nThe relationship between AME states and Quantum secret sharing (QSS) protocols is\nexhibited and the application of AME states in novel quantum communication protocols\nis also explored.Entanglement is one of the fundamental features of quantum information science.\nThough bipartite entanglement has been analyzed thoroughly in theory and shown to\nbe an important resource in quantum computation and communication protocols, the\ntheory of entanglement shared between more than two parties, which is called multipartite\nentanglement, is still not complete. Specifically, the classification of multipartite\nentanglement and the transformation property between different multipartite states by\nlocal operators and classical communications (LOCC) are two fundamental questions in\nthe theory of multipartite entanglement.\nIn this thesis, we present results related to the LOCC transformation between multipartite\nentangled states. Firstly, we investigate the bounds on the LOCC transformation\nprobability between multipartite states, especially the GHZ class states. By analyzing\nthe involvement of 3-tangle and other entanglement measures under weak two-outcome\nmeasurement, we derive explicit upper and lower bound on the transformation probability\nbetween GHZ class states. After that, we also analyze the transformation between\nN-party W type states, which is a special class of multipartite entangled states that\nhas an explicit unique expression and a set of analytical entanglement monotones. We\npresent a necessary and sufficient condition for a known upper bound of transformation\nprobability between two N-party W type states to be achieved.\nWe also further investigate a novel entanglement transformation protocol, the random\ndistillation, which transforms multipartite entanglement into bipartite entanglement\nii\nshared by a non-deterministic pair of parties. We find upper bounds for the random distillation\nprotocol for general N-party W type states and find the condition for the upper\nbounds to be achieved. What is surprising is that the upper bounds correspond to entanglement\nmonotones that can be increased by Separable Operators (SEP), which gives\nthe first set of analytical entanglement monotones that can be increased by SEP.\nFinally, we investigate the idea of a new class of multipartite entangled states, the\nAbsolutely Maximal Entangled (AME) states, which is characterized by the fact that\nany bipartition of the states would give a maximal entangled state between the two sets.\nThe relationship between AME states and Quantum secret sharing (QSS) protocols is\nexhibited and the application of AME states in novel quantum communication protocols\nis also explored.

Jan 7, 20140 citations

Multipartite-entanglement monotones and polynomial invariants

Christopher Eltschka, Thierry Bastin, Andreas Osterloh +1 more

We show that a positive homogeneous function that is invariant under determinant-1 stochastic local operations and classical communication (SLOCC) transformations defines an $N$ -qubit entanglement monotone if and only if the homogeneous degree is not larger than 4. We then describe a common basis and formalism for the $N$ -tangle and other known invariant polynomials of degree 4. This allows us to elucidate the relation of the four-qubit invariants defined by Luque and Thibon [Phys. Rev. A 67, 042303 (2003)] and the reduced two-qubit density matrices of the states under consideration, thus giving a physical interpretation for those invariants. We demonstrate that this is a special case of a completely general law that holds for any multipartite system with bipartitions of equal dimension, e.g., for an even number of qudits.We show that a positive homogeneous function that is invariant under determinant-1 stochastic local operations and classical communication (SLOCC) transformations defines an $N$-qubit entanglement monotone if and only if the homogeneous degree is not larger than 4. We then describe a common basis and formalism for the $N$-tangle and other known invariant polynomials of degree 4. This allows us to elucidate the relation of the four-qubit invariants defined by Luque and Thibon [Phys. Rev. A 67, 042303 (2003)] and the reduced two-qubit density matrices of the states under consideration, thus giving a physical interpretation for those invariants. We demonstrate that this is a special case of a completely general law that holds for any multipartite system with bipartitions of equal dimension, e.g., for an even number of qudits.

arXiv:1105.1556Physical Review AFeb 3, 201238 citations

Local transformations of multiple multipartite states

Antoine Neven, David Gunn, Martin Hebenstreit +1 more

Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.

arXiv:2007.06256SciPost PhysicsAug 30, 20219 citations

Invariant-based entanglement monotones as expectation values and their experimental detection

Andreas Osterloh, Jens Siewert

Characterization and quantification of multipartite entanglement is one of the challenges in state-of-the-art experiments in quantum-information processing. According to theory, this is achieved via entanglement monotones, that is, functions that do not increase under stochastic local operations and classical communication (SLOCC). Typically such monotones include the wave function and its time reversal (antilinear-operator formalism) or they are based on not completely positive maps (e.g., partial transpose). Therefore, they are not directly accessible to experimental observations. We show how entanglement monotones derived from polynomial local SL$(2,\phantom{\rule{2.40005pt}{0ex}}\mathrm{width}.04\mathrm{em}\mathrm{height}1.46\mathrm{ex}\mathrm{depth}\ensuremath{-}.07\mathrm{ex}\mathrm{C})$ invariants can be re-written in terms of expectation values of observables. Consequently, the amount of entanglement---of specific SLOCC classes---in a given state can be extracted from the measurement of correlation functions of local operators.Characterization and quantification of multipartite entanglement is one of the challenges in state-of-the-art experiments in quantum-information processing. According to theory, this is achieved via entanglement monotones, that is, functions that do not increase under stochastic local operations and classical communication (SLOCC). Typically such monotones include the wave function and its time reversal (antilinear-operator formalism) or they are based on not completely positive maps (e.g., partial transpose). Therefore, they are not directly accessible to experimental observations. We show how entanglement monotones derived from polynomial local SL$(2,\phantom{\rule{2.40005pt}{0ex}}\mathrm{width}.04\mathrm{em}\mathrm{height}1.46\mathrm{ex}\mathrm{depth}\ensuremath{-}.07\mathrm{ex}\mathrm{C})$ invariants can be re-written in terms of expectation values of observables. Consequently, the amount of entanglement---of specific SLOCC classes---in a given state can be extracted from the measurement of correlation functions of local operators.

arXiv:1003.4862Physical Review AOct 1, 201217 citations

We show that a positive homogeneous function that is invariant under determinant-1 stochastic local operations and classical communication (SLOCC) transformations defines an

N

-qubit entanglement monotone if and only if the homogeneous degree is not larger than 4. We then describe a common basis and formalism for the

N

-tangle and other known invariant polynomials of degree 4. This allows us to elucidate the relation of the four-qubit invariants defined by Luque and Thibon [Phys. Rev. A 67, 042303 (2003)] and the reduced two-qubit density matrices of the states under consideration, thus giving a physical interpretation for those invariants. We demonstrate that this is a special case of a completely general law that holds for any multipartite system with bipartitions of equal dimension, e.g., for an even number of qudits.