A bipartite state ${\ensuremath{\rho}}^{AB}$ has a
k-symmetric extension if there exists a (
k+1)-partite state ${\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}$ with marginals ${\ensuremath{\rho}}^{A{B}_{i}}={\ensuremath{\rho}}^{AB},\ensuremath{\forall}i$. The
k-symmetric extension is called bosonic if ${\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}$ is supported on the symmetric subspace of
B1B2...Bk. Understanding the structure of symmetric and bosonic extension has various applications in the theory of quantum entanglement, quantum key distribution, and the quantum marginal problem. In particular, bosonic extension gives a tighter bound for the quantum marginal problem based on separability. In general, it is known that a ${\ensuremath{\rho}}^{AB}$ admitting symmetric extension may not have bosonic extension. In this work, we show that, when the dimension of the subsystem
B is 2 (i.e., a qubit), ${\ensuremath{\rho}}^{AB}$ admits a
k-symmetric extension if and only if it has a
k-bosonic extension. Our result has an immediate application to the quantum marginal problem and indicates a special structure for qubit systems based on group representation theory.
This work is a technical examination of concepts in Arveson’s paper Probability of Entanglement [3]. For finite dimensional Hilbert spaces H with dimension n and K with dimension m such that 2 ≤ n ≤ m, every extension of a faithful state of B(H) to a state of B(K ⊗ H) is parameterized by a complex nm2 × n matrix v with the property that v∗v is the identity on B(H). The set of all such nm2 × n matrices is a Steifel manifold V mn(H,K) which corresponds to the set of isometries from H to K⊗mn. We show that the set of extensions of a faithful state E(!) is isomorphic to V mn(H,K)/U(Cmn) where U(Cmn) is the set of unitary linear operators from Cmn → Cmn. We can construct a filtration of extensions: E1(!) ⊆ · · · ⊆ Er(!) ⊆ · · · ⊆ Emn(!) where Er(!) is the set of extensions with rank at most r. Each Er(!) is isomorphic to V r(H,K)/U(Cr). We show that there is a unique measure μ on V r(H,K) defined by the inner product on B(H, r · K) where hu, vi = Tr(v∗u) for u, v ∈ B(H, r · K). We show that entangled extensions of every rank exist and for r < √dimH every extension in Er(!) is entangled. With respect to μ, for a fixed r with 1 ≤ r ≤ mn the set of extensions with rank strictly less than r in Er(!) has measure zero. For r < dimH almost every extension is entangled, in other words the probability that an extension of a faithful state with rank less than dimH is not entangled is zero.