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arXiv:1002.1949 · Physical Review A

Low-rank extremal positive-partial-transpose states and unextendible product bases

Jon Magne Leinaas, Jan Myrheim, Per Øyvind Sollid

Published Jun 21, 2010• 31 citations
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Abstract

It is known how to construct, in a bipartite quantum system, a unique low-rank entangled mixed state with positive partial transpose (a PPT state) from an unextendible product basis (UPB), defined as an unextendible set of orthogonal product vectors. We point out that a state constructed in this way belongs to a continuous family of entangled PPT states of the same rank, all related by nonsingular unitary or nonunitary product transformations. The characteristic property of a state $\ensuremath{\rho}$ in such a family is that its kernel $\text{Ker} \ensuremath{\rho}$ has a generalized UPB, a basis of product vectors, not necessarily orthogonal, with no product vector in $\mathrm{Im} \ensuremath{\rho}$, the orthogonal complement of $\text{Ker} \ensuremath{\rho}$. The generalized UPB in $\text{Ker} \ensuremath{\rho}$ has the special property that it can be transformed to orthogonal form by a product transformation. In the case of a system of dimension $3\ifmmode\times\else\texttimes\fi{}3$, we give a complete parametrization of orthogonal UPBs. This is then a parametrization of families of rank 4 entangled (and extremal) PPT states, and we present strong numerical evidence that it is a complete classification of such states. We speculate that the lowest rank entangled and extremal PPT states also in higher dimensions are related to generalized, nonorthogonal UPBs in similar ways.

DOI
10.1103/physreva.81.062330

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It is known how to construct, in a bipartite quantum system, a unique low-rank entangled mixed state with positive partial transpose (a PPT state) from an unextendible product basis (UPB), defined as an unextendible set of orthogonal product vectors. We point out that a state constructed in this way belongs to a continuous family of entangled PPT states of the same rank, all related by nonsingular unitary or nonunitary product transformations. The characteristic property of a state \ensuremathρ\ensuremath{\rho}\ensuremathρ in such a family is that its kernel Ker\ensuremathρ\text{Ker} \ensuremath{\rho}Ker\ensuremathρ has a generalized UPB, a basis of product vectors, not necessarily orthogonal, with no product vector in Im\ensuremathρ\mathrm{Im} \ensuremath{\rho}Im\ensuremathρ, the orthogonal complement of Ker\ensuremathρ\text{Ker} \ensuremath{\rho}Ker\ensuremathρ. The generalized UPB in Ker\ensuremathρ\text{Ker} \ensuremath{\rho}Ker\ensuremathρ has the special property that it can be transformed to orthogonal form by a product transformation. In the case of a system of dimension 3\ifmmode×\else\texttimes\fi33\ifmmode\times\else\texttimes\fi{}33\ifmmode×\else\texttimes\fi3, we give a complete parametrization of orthogonal UPBs. This is then a parametrization of families of rank 4 entangled (and extremal) PPT states, and we present strong numerical evidence that it is a complete classification of such states. We speculate that the lowest rank entangled and extremal PPT states also in higher dimensions are related to generalized, nonorthogonal UPBs in similar ways.
In bipartite quantum systems of dimension $3\ifmmode\times\else\texttimes\fi{}3$, entangled states that are positive under partial transposition (PPT) can be constructed with the use of unextendible product bases (UPBs). As discussed in a previous publication, all the lowest rank entangled PPT states of this system seem to be equivalent, under $\mathrm{SL}\ensuremath{\bigotimes}\mathrm{SL}$ transformations, to states that are constructed in this way. Here we consider a possible generalization of the UPB constuction to low-rank entangled PPT states in higher dimensions. The idea is to give up the condition of orthogonality of the product vectors, while keeping the relation between the density matrix and the projection on the subspace defined by the UPB. We examine first this generalization for the $3\ifmmode\times\else\texttimes\fi{}3$ system where numerical studies indicate that one-parameter families of such generalized states can be found. Similar numerical searches in higher dimensional systems show the presence of extremal PPT states of similar form. Based on these results we suggest that the UPB construction of the lowest rank entangled states in the $3\ifmmode\times\else\texttimes\fi{}3$ system can be generalized to higher dimensions, with the use of nonorthogonal UPBs.
In bipartite quantum systems of dimension 3× 3 the lowest rank entangled states that are positive under partial transposition (PPT) can be related, by product transformations, to states that are constructed by the use of unextendible product bases (UPB) in the Hilbert space of the system. Here we consider a possible generalization of the product bases, which can be related in a similar way to low-rank entangled density matrices in higher dimensions. The idea is to give up the condition of orthogonality of the product vectors, while keeping the relation between the density matrix and the projection on the associated UPB. We examine first this generalization for the 3 × 3 system where numerical studies indicate that one-parameter families of such generalized states can be found. Similar numerical searches in higher dimensional systems show the presence of extremal PPT states of the same form. Based on these results we indicate that the UPB construction of the lowest rank entangled states in the 3×3 system can be generalized to higher dimensions, with the use of non-orthogonal UPBs of the suggested form.
Deciding whether a mixed quantum state is separable or entangled is a difficult problem in general. Separable states are positive under partial transposition [they are positive-partial-transpose (PPT) states], but this simple test does not exclude all entangled states. In order to understand the entangled PPT states, having so-called bound entanglement, we want to study the extremal PPT states. An extremal PPT state is either a pure product state, then it is separable, or it is entangled, in which case the state and its partial transpose must still be rather low-rank density matrices, although it is known that the rank must be at least 4. In a previous paper we presented a complete classification of the rank-4 entangled PPT states in dimension $3\ifmmode\times\else\texttimes\fi{}3$, generalizing the construction by Bennett et al. from unextendible sets of orthogonal product vectors. In the present paper we continue our investigations of the low-rank entangled PPT states, mostly in dimension $3\ifmmode\times\else\texttimes\fi{}3$, using a combination of analytical and numerical methods. We use perturbation theory in order to construct rank-5 entangled PPT states close to the known rank-4 states, and in order to compute dimensions and study the geometry of surfaces of low-rank PPT states. We exploit the close connection between low-rank PPT states and product vectors. In particular, we show how to reconstruct a low-rank PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.
In this master thesis I study the extremal positive partial transpose (PPT) states of the three qubit (2times2times2)(2times2times2)(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 SLotimesotimesotimesSL 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 SLotimesotimesotimesSL 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.
Constructing 2×2×42\times2\times42×2×4 and 4×44\times44×4 unextendible product bases and positive-partial-transpose entangled states
The 4-qubit unextendible product basis (UPB) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. From this result we show that there is only one UPB of size 666 and six UPBs of size 999 in $\cH=\bbC^2\ox\bbC^2\ox\bbC^4$, three UPBs of size 999 in $\cK=\bbC^4\ox\bbC^4$, and no UPB of size 777 in $\cH$ and $\cK$. Furthermore we construct a 4-qubit positive-partial-transpose (PPT) entangled state $\r$ of rank seven, and show that it is also a PPT entangled state in $\cH$ and $\cK$, respectively. We analytically derive the geometric measure of entanglement of a special $\r$.
The 4-qubit unextendible product bases (UPBs) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. Based on this study we show that there is only one UPB of size 6 and six UPBs of size 9 in H=C2⊗C2⊗C4, three UPBs of size 9 in K=C4⊗C4, and no UPB of size 7 in H and K. Furthermore we construct a new family of 4-qubit positive-partial-transpose (PPT) entangled states of rank seven, and show that they are also PPT entangled states in H and K, respectively. We analytically derive a geometric measure of a special PPT entangled states.