Narrow Results

Motivated by the notions of $k$-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state.

Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys.15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.

In this work, we study quantum teleportation, a fundamental protocol of quantum physics. In particular, we present a mathematical methodology to study the combined effects of noisy resource state and noisy classical communication on teleportation fidelity and its deviation. We describe a teleportation protocol where an arbitrary two-qubit state in canonical form is used as a resource. Thereafter, to teleport an unknown qubit, Alice measures her qubits in Bell basis and conveys the measurement outcome to Bob via noisy classical channel(s). We derive the exact formulae of optimal teleportation fidelity and corresponding fidelity deviation where the resource state and the classical communication, both can be noisy. To provide the proof of optimality, we provide a systematic method. We further find conditions for non-classical fidelity and dispersion-free teleportation for our teleportation protocol. In this way, we identify noisy environments where it is possible to achieve dispersion-free teleportation without compromising non-classical fidelity. We also demonstrate scenarios where the increase of entanglement in the resource state, may degrade the quality of teleportation — a counter-intuitive instance. Finally, we discuss about the minimum classical communication cost required to achieve non-classical fidelity for our protocol. Here we mainly focus on a possible way to optimize the classical communication cost.

The momentum and position observables in an $n$-mode boson Fock space $\mathrm{\Gamma }({ℂ}^n)$ have the whole real line $ℝ$ as their spectrum. But the total number operator $N$ has a discrete spectrum ${\mathbb{Z}}_{+}=\{0,1,2,\dots \}$. An $n$-mode Gaussian state in $\mathrm{\Gamma }({ℂ}^n)$ is completely determined by the mean values of momentum and position observables and their covariance matrix which together constitute a family of $n(2n+3)$ real parameters. Starting with $N$ and its unitary conjugates by the Weyl displacement operators and operators from a representation of the symplectic group $\mathrm{\text{Sp}}(2n)$ in $\mathrm{\Gamma }({ℂ}^n)$, we construct $n(2n+3)$ observables with spectrum ${\mathbb{Z}}_{+}$ but whose expectation values in a Gaussian state determine all its mean and covariance parameters. Thus measurements of discrete-valued observables enable the tomography of the underlying Gaussian state and it can be done by using five one-mode and four two-mode Gaussian symplectic gates in single and pair mode wires of $\mathrm{\Gamma }({ℂ}^n)=\mathrm{\Gamma }{(ℂ)}^{\otimes n}$. Thus the tomography protocol admits a simple description in a language similar to circuits in quantum computation theory. Such a Gaussian tomography applied to outputs of a Gaussian channel with coherent input states permit a tomography of the channel parameters. However, in our procedure the number of counting measurements exceeds the number of channel parameters slightly. Presently, it is not clear whether a more efficient method exists for reducing this tomographic complexity.

As a byproduct of our approach an elementary derivation of the probability generating function of $N$ in a Gaussian state is given. In many cases the distribution turns out to be infinitely divisible and its underlying Lévy measure can be obtained. However, we are unable to derive the exact distribution in all cases. Whether this property of infinite divisibility holds in general is left as an open problem.

Finite-dimensional entanglement for pure states has been used extensively in quantum information theory. Depending on the tensor product structure, even a set of separable states can show non-intuitive characters. Two situations are well studied in the literature, namely, the unextendible product basis (UPB) by Bennett et al.⁴, and completely entangled subspaces explicitly given by Parthasarathy ²². More recently, Boyer et al.⁶, Boyer and Mor⁷ and Liss et al.²¹ studied spaces which have only finitely many pure product states. We carry this further and consider the problem of perturbing different spaces, such as the orthogonal complement of an UPB and also Parthasarathy’s completely entangled spaces, by taking linear spans with specified product vectors. To this end, we develop methods and theory of variations and perturbations of the linear spans of certain UPBs, their orthogonal complements, and also Parthasarathy’s completely entangled subspaces. Finally, we give examples of perturbations with infinitely many pure product states.