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.
A foundational result in relativistic quantum information theory due to Peres, Scudo, and Terno, is that von Neumann entropy is not Lorentz invariant. Motivated by the "It from Qubit" paradigm, here we show that Lorentzian symmetries of quantum information emerge naturally in a pre-spacetime setting, without any reference to external variables such as position or momentum. In particular, we derive the natural action of the restricted Lorentz group
SO+(1,3) on the internal degrees of freedom of a single qubit from a simple, information-theoretic principle we refer to as preservation of linear entropy. It is then shown that the Lorentz invariance of the linear entropy of a relativistic qubit is a special case of a much more general phenomenon, namely, that any spectral invariant of an operator we term the '
W-matrix' is an
SL(2,C)⊗n invariant scalar. Consequently, the linear
n-partite quantum mutual information is shown to be an
SL(2,C)⊗n invariant for all
n-qubit states. Finally, we show that the correlation function associated with a pair of qubits in the singlet state yields the Minkowski metric on the space of qubit observables, whose symmetry group is the full Lorentz group
SO(1,3). In accordance with the "It from Qubit" paradigm, our results thus establish the natural emergence of relativistic spacetime structure from intrinsic properties of quantum information.
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