Local unitary invariants allow one to test whether multipartite states are equivalent up to local basis changes. Equivalently, they specify the geometry of the obtained by factoring out local unitary action from the state space. This space is of interest because of its intimate relationship to entanglement. Unfortunately, the dimension of the orbit space grows exponentially with the number of subsystems, and the number of invariants needed to characterise orbits grows at least as fast. This makes the study of entanglement via local unitary invariants seem very daunting. I point out here that there is a simplifying principle: Invariants fall into families related by the tracing-out of subsystems, and these families grow exponentially with the number of subsystems. In particular, in the case of pure qubit systems, there is a family whose size is about half the dimension of orbit space. These invariants are closely related to cumulants and to multipartite separability. Members of the family have been repeatedly discovered in the literature, but the fact that they are related to cumulants and constitute a family has apparently not been observed.
Local unitary invariants allow one to test whether multipartite states are equivalent up to local basis changes. Equivalently, they specify the geometry of the obtained by factoring out local unitary action from the state space. This space is of interest because of its intimate relationship to entanglement. Unfortunately, the dimension of the orbit space grows exponentially with the number of subsystems, and the number of invariants needed to characterise orbits grows at least as fast. This makes the study of entanglement via local unitary invariants seem very daunting. I point out here that there is a simplifying principle: Invariants fall into families related by the tracing-out of subsystems, and these families grow exponentially with the number of subsystems. In particular, in the case of pure qubit systems, there is a family whose size is about half the dimension of orbit space. These invariants are closely related to cumulants and to multipartite separability. Members of the family have been repeatedly discovered in the literature, but the fact that they are related to cumulants and constitute a family has apparently not been observed.
Local unitary invariants allow one to test whether multipartite states are equivalent up to local basis changes. Equivalently, they specify the geometry of the "orbit space" obtained by factoring out local unitary action from the state space. This space is of interest because of its intimate relationship to entanglement. Unfortunately, the dimension of the orbit space grows exponentially with the number of subsystems, and the number of invariants needed to characterise orbits grows at least as fast. This makes the study of entanglement via local unitary invariants seem very daunting. I point out here that there is a simplifying principle: Invariants fall into families related by the tracing-out of subsystems, and these families grow exponentially with the number of subsystems. In particular, in the case of pure qubit systems, there is a family whose size is about half the dimension of orbit space. These invariants are closely related to cumulants and to multipartite separability. Members of the family have been repeatedly discovered in the literature, but the fact that they are related to cumulants and constitute a family has apparently not been observed.
Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.
Research on quantum states often focuses on the correlation between nonlocal effects and local unitary invariants, among which local unitary equivalence plays a significant role in quantum state classification and resource theories. This paper focuses on the local unitary equivalence of multipartite quantum states in quantum information theory, aiming to determine a complete set of invariants that identify their local unitary orbits; these invariants are crucial for deriving polynomial invariants and describing the physical properties preserved under local unitary transformations. The study deeply explores the characterization of local unitary equivalence and the method of detecting entanglement using local unitary Bargmann invariants. Taking two-qubit systems as an example, it verifies the measurability of invariants that determine equivalence and establishes a connection between Makhlin fundamental invariants (a complete set of 18 local unitary invariants for two-qubit states) and local unitary Bargmann invariants. These Bargmann invariants, related to the traces of products of density operators and marginal states, can be measured through cycle tests (an extended form of swap tests).
We investigate the equivalence of quantum states under local unitary transformations. A complete set of invariants under local unitary transformations is presented for a class of non-generic three-qubit mixed states. It is shown that two such states in this class are locally equivalent if and only if all these invariants have equal values for them. PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w Key words: Invariant, quantum state, local unitary transformation Quantum entangled states are playing very important roles in quantum information processing and quantum computation [1]. As the properties of entanglement of a multipartite quantum system remain invariant under local unitary transformations on the subsystems, the entanglement can be characterized by all the invariants of the local unitary transformations. The trace norms of realigned or partial transposed density matrices in entanglement measure and separability criteria are some of these invariants [2]. Invariants are also relevant in the discussions of Bell inequalities [3, 4] and teleportation, etc. [5]. A complete
1. For mathematical point of view, Quantum Information Theory deals with finite dimensional Hilbert spaces, the state spaces of finite k-bipartite systems. In the case of pure k-bipartite systems, this space has the special form 2. H=V1 ⊗ ... ⊗ Vk, 3. where Vi denotes a finite dimensional state space of the ith particle of the system (most of the time, one assumes that it is a qubit, i.e. dim Vi = 2). The interesting non-classical behavior already occurs for two-qubit systems, for the so-called entangled states: the states which cannot be written in the form v1 ⊗ v2. There is no general agreement on the definition of entanglement for system with more than two parts. Klyachko has proposed to regard as entangled the states which are semi-stable for the action of the SLOCC group (G=SL(V1) × ... × SL(Vk)) in the sense of geometric invariant theory. Geometric invariant theory provides 4. methods for characterizing such states without explicitly computing the invariants. In order to test the significance of this property, we have explicitly computed the invariants in the simplest cases. 5. The non-locality properties of an entangled states does not change under unitary operations acting independently on each of its subsystem. The idea to construct entanglement measure by means of unitary invariant have been explored for example by Grassl et al. (1998), Schlienz and Mahler (1996). We have proved in 2006, that such invariants can be deduced from SLOCC-covariants. The knowledge of all the covariants and hence, the classification of the SLOCC orbits seems to be relevant in this context. However, except for simplest systems there are far too many orbits and the complete classification is out of reach. 6. 7. The aim of the series of four lectures given by Jean-Gabriel Luque and Jean-Yves Thibon is to explain a subproblem: how to compute the Hilbert series of the algebraic invariants for the pure k-qubit systems. 8. 9. In the two first lectures, Jean-Gabriel Luque will give a brief account on the symmetric functions and the characters of the symmetric groups. The first two talks will be meant for an audience of beginner in the area of symmetric functions. In the 10. two remaining lectures, Jean-Yves Thibon will explore the links between symmetric functions and Kronecker product (Lecture 3) in order to explain how to compute the Hilbert series (Lecture 4).
Invariance under local unitary operations is a fundamental property that must be obeyed by every proper measure of quantum entanglement. However, this is not the only aspect of entanglement theory where local unitary operations play a relevant role. In the present work we show that the application of suitable local unitary operations defines a family of bipartite entanglement monotones, collectively referred to as ``mirror entanglement.'' They are constructed by first considering the (squared) Hilbert-Schmidt distance of the state from the set of states obtained by applying to it a given local unitary operator. To the action of each different local unitary operator there corresponds a different distance. We then minimize these distances over the sets of local unitary operations with different spectra, obtaining an entire family of different entanglement monotones. We show that these mirror-entanglement monotones are organized in a hierarchical structure, and we establish the conditions that need to be imposed on the spectrum of a local unitary operator for the associated mirror entanglement to be faithful, i.e., to vanish in and only in separable pure states. We analyze in detail the properties of one particularly relevant member of the family, the ``stellar mirror entanglement'' associated with the traceless local unitary operations with nondegenerate spectra and equispaced eigenvalues in the complex plane. This particular measure generalizes the original analysis of S. M. Giampaolo and F. Illuminati [Phys. Rev. A 76, 042301 (2007)], valid for qubits and qutrits. We prove that the stellar entanglement is a faithful bipartite entanglement monotone in any dimension and that it is bounded from below by a function proportional to the linear entropy and from above by the linear entropy itself, coinciding with it in two- and three-dimensional spaces.