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Physical Review A

Quantum Lorentz-group invariants ofn-qubit systems

Mihail Teodorescu-Frumosu, Gregg S. Jaeger

Published May 21, 2003• 36 citations
Publisher

Abstract

We consider the behavior of quantum states under stochastic local quantum operations and classical communication (SLOCC) for an arbitrary fixed number of qubits. We use a real (Lorentz) group to describe the action of SLOCC operations on n-qubit states. We discuss the natural quantum Lorentz-group group-invariant length for an arbitrary number of qubits. We relate this approach to that based on local operations and classical communication and provide an example of how the invariant length can be used to describe entanglement. We also note that this invariant length is the Minkowskian analog to the quantum state purity, which is the corresponding Euclidean length.

DOI
10.1103/physreva.67.052305

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We consider the behavior of quantum states under stochastic local quantum operations and classical communication (SLOCC) for an arbitrary fixed number of qubits. We use a real (Lorentz) group to describe the action of SLOCC operations on n-qubit states. We discuss the natural quantum Lorentz-group group-invariant length for an arbitrary number of qubits. We relate this approach to that based on local operations and classical communication and provide an example of how the invariant length can be used to describe entanglement. We also note that this invariant length is the Minkowskian analog to the quantum state purity, which is the corresponding Euclidean length.
The behavior of quantum states under local unitary transformations (LUTs) and stochastic local quantum operations and classical communication (SLOCC) has proven central to the understanding of entanglement in multipartite quantum systems. In particular, invariants under these operations have provided insight into quantum entanglement in multiple‐qubit states. Relationships between entanglement, mixedness and spin symmetry in multiple‐qubit quantum states can be found by exploiting these properties. For example, concurrence and n‐tangle are naturally expressed in terms of spin‐flip transformations. In the case of specialized entanglement measures and/or special families of states, complementarity relations involving these lengths and state transformations are derivable. Here, the role of geometry in such investigations is explored. Minkowskian geometry is seen to provide elegant representations of entanglement in multiple‐qubit systems, particularly the language of twistors.
We find an invariant for nnn qubits and propose residual entanglement for nnn qubits by means of the invariant. Thus, we establish a relation between stochastic local operations and classical communication (SLOCC) entanglement and residual entanglement. Invariant and residual entanglement can be used for SLOCC entanglement classification of nnn qubits.
We show there are at least 28 distinct true stochastic local operations and classical communication (SLOCC) entanglement classes for four qubits by means of SLOCC invariant and semi-invariants and derive the number of degenerate SLOCC classes for nnn qubits.
We consider the Minkowskian norm of the n-photon Stokes tensor, a scalar invariant under the group realized by the transformations of stochastic local quantum operations and classical communications (SLOCC). This invariant is offered as a candidate entanglement measure for n-qubit states and discussed in relation to measures of quantum state entanglement for certain important classes of two-qubit and three-qubit systems. This invariant can be directly estimated via a quantum network, obviating the need to perform laborious quantum state tomography. We also show that this invariant directly captures the extent of entanglement purification due to SLOCC filters.
It is a recent observation that entanglement classification for qubits is closely related to stochastic local operations and classical communication (SLOCC) invariants. Verstraete et al.[ Phys. Rev. A 65 (2002) 052112 ] showed that for pure states of four qubits there are nine different degenerate SLOCC entanglement classes. Li et al.[ Phys. Rev. A 76 (2007) 052311 ] showed that there are at feast 28 distinct true SLOCC entanglement classes for four qubits by means of the SLOCC invariant and semi-invariant. We give 16 different entanglement classes for four qubits by means of basic SLOCC invariants.
We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such a SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions.