  A superconductor is a material that conducts electric current with no
resistance. Superconductivity and magnetism are known to be antagonistic
phenomena: superconductors expel weak external magnetic field (the Meissner
effect) while a sufficiently strong magnetic field, in general, destroys
superconductivity. In a seemingly contradictory statement, we show that a very
strong magnetic field can turn an empty space into a superconductor. The
external magnetic field required for this effect should be about 10^{16} Tesla
(eB ~ 1 GeV^2). The physical mechanism of the exotic vacuum superconductivity
is as follows: in strong magnetic field the dynamics of virtual quarks and
antiquarks is effectively one-dimensional because these electrically charged
particles tend to move along the lines of the magnetic field. In one spatial
dimension a gluon-mediated attraction between a quark and an antiquark of
different flavors inevitably leads to formation of a colorless spin-triplet
bound state (a vector analogue of the Cooper pair) with quantum numbers of an
electrically charged rho meson. Such quark-antiquark pairs condense to form an
anisotropic inhomogeneous superconducting state similar to the Abrikosov vortex
lattice in a type-II superconductor. The onset of the superconductivity of the
charged rho mesons should also induce an inhomogeneous superfluidity of the
neutral rho mesons. The vacuum superconductivity should survive at very high
temperatures of typical Quantum Chromodynamics (QCD) scale of 10^{12} K (T ~
100 MeV). We propose the phase diagram of QCD in the plane "magnetic field -
temperature".
