Alladi Ramakrishnan Hall

Lorentzian Geometry of Qubit Entanglement

Joseph Samuel

Raman Research Institute, Bangalore

This talk will deal with the problem of two qubit entanglement. As is well known, the Poincare-Bloch sphere gives an elegant visualisation of
the pure states of a qubit and its interior describes the impure states. After an elementary introduction, this talk will give a characterisation of the impure states
of the two qubit system. The approach is based on Lorentzian geometry and gives a new way to detect qubit entanglement. The treatment is based physically, on the causal structure of Minkowski spacetime, and mathematically, on a Lorentzian Singular Value Decomposition. A surprising feature is the natural emergence of ‘‘Energy conditions’’ used in Relativity. All states satisfy a ‘‘Dominant Energy Condition’’ (DEC) and separable states satisfy the Strong Energy Condition(SEC), while entangled states violate the SEC. We thus propose a test for two qubit entanglement which is an alternative to the positive partial transpose (PPT) test. This test is based on the partial Lorentz transformation (PLT) on individual qubits. Apart from testing for entanglement, our approach also enables us to construct a separable form for the density matrix in those cases where it exists. Our approach leads to a simple graphical three dimensional representation of the state space which shows the entangled states within the set of all states.

This talk will deal with the problem of two qubit entanglement. As is well known, the Poincare-Bloch sphere gives an elegant visualisation of
the pure states of a qubit and its interior describes the impure states. After an elementary introduction, this talk will give a characterisation of the impure states
of the two qubit system. The approach is based on Lorentzian geometry and gives a new way to detect qubit entanglement. The treatment is based physically, on the causal structure of Minkowski spacetime, and mathematically, on a Lorentzian Singular Value Decomposition. A surprising feature is the natural emergence of ‘‘Energy conditions’’ used in Relativity. All states satisfy a ‘‘Dominant Energy Condition’’ (DEC) and separable states satisfy the Strong Energy Condition(SEC), while entangled states violate the SEC. We thus propose a test for two qubit entanglement which is an alternative to the positive partial transpose (PPT) test. This test is based on the partial Lorentz transformation (PLT) on individual qubits. Apart from testing for entanglement, our approach also enables us to construct a separable form for the density matrix in those cases where it exists. Our approach leads to a simple graphical three dimensional representation of the state space which shows the entangled states within the set of all states.