We have studied mixed states in the system of three qubits with the property that all their partial transposes are positive; these are called PPT states. We classify a PPT state by the ranks of the state itself and its three single partial transposes. In random numerical searches, we find entangled PPT states with a large variety of rank combinations. For ranks equal to five or higher, we find both extremal and nonextremal PPT states of nearly every rank combination, with the restriction that the square sum of the four ranks of an extremal PPT state can be at most 193. We have studied especially the rank-four entangled PPT states, which are found to have rank four for every partial transpose. These states are all extremal because of the previously known result that every PPT state of rank three or less is separable. We find two distinct classes of rank-4444 entangled PPT states, identified by a real valued quadratic expression invariant under local $\text{SL}(2,\mathbb{C})$ transformations, mathematically equivalent to Lorentz transformations. This quadratic Lorentz invariant is nonzero for one class of states (type I in our terminology) and zero for the other class (type II). The previously known states based on unextendible product bases are a nongeneric subclass of the type-I states. We present analytical constructions of states of both types, general enough to reproduce all the rank-4444 PPT states we have found numerically. We can not exclude the possibility that there exist nongeneric rank-four PPT states that we do not find in our random numerical searches.
We have studied mixed states in the system of three qubits with the property that all their partial transposes are positive; these are called PPT states. We classify a PPT state by the ranks of the state itself and its three single partial transposes. In random numerical searches, we find entangled PPT states with a large variety of rank combinations. For ranks equal to five or higher, we find both extremal and nonextremal PPT states of nearly every rank combination, with the restriction that the square sum of the four ranks of an extremal PPT state can be at most 193. We have studied especially the rank-four entangled PPT states, which are found to have rank four for every partial transpose. These states are all extremal because of the previously known result that every PPT state of rank three or less is separable. We find two distinct classes of rank-4444 entangled PPT states, identified by a real valued quadratic expression invariant under local SL(2,C) transformations, mathematically equivalent to Lorentz transformations. This quadratic Lorentz invariant is nonzero for one class of states (type I in our terminology) and zero for the other class (type II). The previously known states based on unextendible product bases are a nongeneric subclass of the type-I states. We present analytical constructions of states of both types, general enough to reproduce all the rank-4444 PPT states we have found numerically. We can not exclude the possibility that there exist nongeneric rank-four PPT states that we do not find in our random numerical searches.
We have studied mixed quantum states in the system of three qubits with the property that all their partial transposes are positive; these are called PPT states. We classify a PPT state by the ranks of the state itself and its three single partial transposes. We have studied especially the rank 4444 and rank 5555 entangled PPT states.We find two distinct classes of rank four states, identified by a real valued quadratic expression invariant under local sln transformations, mathematically equivalent to continuous Lorentz transformations. This quadratic Lorentz invariant is non-zero for one class of states (type 1) and zero for the other class (type 2). We present analytical constructions of states of both types, general enough to reproduce all the rank four PPT states we have found numerically.There are six product vectors in a generic five dimensional subspace. The product vectors are used to define SL invariants of the state and its partial transposes. We find four distinct classes of states based which invariants are conserved or not under partial transposition.We find that a state is in the equivalence class of a symmetric state when its invariants are conserved under a partial transposition, and we have shown how to transform the state into its symmetric form through Lorentz transformations.The dimensions and geometry of extremal rank 5555 PPT states has also been examined. We find that states of different types lie on surfaces of rank 5555 different dimensions. And the surfaces touch the simplex of separable states in different ways
In this master thesis I study the extremal positive partial transpose (PPT) states of the three qubit (2times2times2) system using numerical methods. Using two algorithms which locate PPT states of a specified rank and extremal PPT states respectively, I have located numerical examples of extremal PPT states with a variety of ranks. These numerical results confirm the analytical result that all PPT states of rank less than four are separable. I also derive an upper limit on the ranks of extremal PPT states. The extremal PPT states of lowest rank, the rank four states, were studied in more detail. These states were confirmed to be biseparable in accordance with both previous analytical and numerical results. The range and kernel of these states were examined for product vectors, but none were found. In an attempt to parametrize the SLotimesSL equivalence classes of these extremal rank four states I have studied an analytical method to construct such states based on unextendible product bases (UPBs). This method can be used to create PPT states from a single equivalence class where, by design, the kernel of all states contain a UPB and the range contains no product vectors. All states where the range is not spanned by a basis of product vectors are necessarily entangled. I also present a numerical method for creating extremal rank four states that are symmetric under various combinations of partial transposes. Numerical examination of these states reveal no product vectors in neither range nor kernel. The existence of rank four states with and without product vectors in their kernel implies the existence of at least two equivalence classes. To get a better impression of these equivalence classes I construct quantities that are invariant under SLotimesSL transformations and must therefore have the same value for all states in the same equivalence class. Calculating the values of these invariants for all the rank four extremal states I have generated gives a seemingly continuous range of values. This indicates that there is an infinite number of equivalence classes likely described by one or more continuous variables. The invariants also revealed an interesting set of states that may belong to a single equivalence class, where one invariant is zero and the others have identical values. This was the only equivalence class where more than one of my states were included. There is obviously something special about this class, but I do not know what it is.
Multiqubit positive-partial-transpose (PPT) entangled states play an important role in quantum information theory. We characterize such states of minimum rank in three-qubit system, namely rank four. Depending on whether the Lorentz invariant is zero, we classify such states into two types. The PPT entangled states constructed by unextendible product bases (UPB) have nonzero invariants, which belong to type I. We provide a method to effectively determine whether a state can be constructed from UPB. For states with zero invariant, which belong to type II, we provide an explicit expression up to equivalence of stochastic local operations and classical communications (SLOCC). It turns out that we can represent them with only one complex parameter. We further study SLOCC-equivalence relation within the expression. We also investigate the Lorentz invariants of multiqubit states with rank less than three and analyze their range.
Nongeneric positive partial transpose states of rank five in 3×3 dimensions
In 3×3 dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. They are well understood. We say that they have rank (4,4) since a state ρ and its partial transpose ρP both have rank four. The next problem is to understand the extremal PPT states of rank (5,5). We call two states SL⊗SL-equivalent if they are related by a product transformation. A generic rank (5,5) PPT state ρ is extremal, and ρ and ρP both have six product vectors in their ranges, and no product vectors in their kernels. The three numbers {6,6;0} are SL⊗SL-invariants that help us classify the state. We have studied numerically a few types of nongeneric rank five PPT states, in particular states with one or more product vectors in their kernels. We find an interesting new analytical construction of all rank four extremal PPT states, up to SL⊗SL-equivalence, where they appear as boundary states on one single five dimensional face on the set of normalized PPT states. We say that a state ρ is SL⊗SL-symmetric if ρ and ρP are SL⊗SL-equivalent, and is genuinely SL⊗SL-symmetric if it is SL⊗SL-equivalent to a state τ with τ=τP. Genuine SL⊗SL-symmetry implies a special form of SL⊗SL-symmetry. We have produced numerically a random sample of rank (5,5)SL⊗SL-symmetric states. About fifty of these are of type {6,6;0}, among those all are extremal and about half are genuinely SL⊗SL-symmetric.
In [Formula: see text] dimensions, entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. These rank four states are completely understood. We say that they have rank [Formula: see text] since both a state [Formula: see text] and its partial transpose [Formula: see text] have rank four. The next problem is to understand the extremal PPT states of rank [Formula: see text]. We call two states [Formula: see text]-equivalent if they are related by a product transformation. A generic rank [Formula: see text] PPT state [Formula: see text] is extremal, and both [Formula: see text] and [Formula: see text] have six product vectors in their ranges, and no product vectors in their kernels. The three numbers [Formula: see text] are [Formula: see text]-invariants that help us classify the state. There is no analytical understanding of such states. We have studied numerically a few types of nongeneric rank five PPT states, in particular, states with one or more product vectors in their kernels. We find an interesting new analytical construction of all rank four extremal PPT states, up to [Formula: see text]-equivalence, where they appear as boundary states on one single five-dimensional face on the set of normalized PPT states. The interior of the face consists of rank [Formula: see text] states with four common product vectors in their kernels, it is a simplex of separable states surrounded by entangled PPT states. We say that a state [Formula: see text] is [Formula: see text]-symmetric if [Formula: see text] and [Formula: see text] are [Formula: see text]-equivalent, and is genuinely [Formula: see text]-symmetric if it is [Formula: see text]-equivalent to a state [Formula: see text] with [Formula: see text]. Genuine [Formula: see text]-symmetry implies a special form of [Formula: see text]-symmetry. We have produced numerically, by a special method, a random sample of rank [Formula: see text] [Formula: see text]-symmetric states. About 50 of these are of type [Formula: see text], among those all are extremal and about half are genuinely [Formula: see text]-symmetric. All these genuinely [Formula: see text]-symmetric states can be transformed to have a circulant form. We find however that this is not a generic property of genuinely [Formula: see text]-symmetric states. The remaining [Formula: see text]-symmetric states found in the search have product vectors in their kernels, and they inspired us to study such states without regard to [Formula: see text]-symmetry.
From both theoretical and experimental points of view symmetric states constitute an important class of multipartite states. Still, entanglement properties of these states, in particular those with positive partial transposition (PPT), lack a systematic study. Aiming at filling in this gap, we have recently affirmatively answered the open question of existence of four-qubit entangled symmetric states with PPT and thoroughly characterized entanglement properties of such states [J. Tura et al., Phys. Rev. A 85, 060302(R) (2012)]. With the present contribution we continue on characterizing PPT entangled symmetric states. On the one hand, we present all the results of our previous work in a detailed way. On the other hand, we generalize them to systems consisting of an arbitrary number of qubits. In particular, we provide criteria for separability of such states formulated in terms of their ranks. Interestingly, for most of the cases, the symmetric states are either separable or typically separable. Then, edge states in these systems are studied, showing in particular that to characterize generic PPT entangled states with four and five qubits, it is enough to study only those that assume few (respectively, two and three) specific configurations of ranks. Finally, we numerically search for extremal PPT entangled states in such systems consisting of up to 23 qubits. One can clearly notice regularity behind the ranks of such extremal states, and, in particular, for systems composed of odd numbers of qubits we find a single configuration of ranks for which there are extremal states.