ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

CHAPTER 5
Multimode entanglement under symmetry
In quantum information and computation science, it is of particular relevance to
provide theoretical methods to determine the entanglement of systems susceptible
to encompass many parties. Such an interest does not stem only from pure
intellectual curiosity, but also from practical needs in the implementations of realistic
information protocols. This is especially true as soon as one needs to encode
two-party information in a multipartite structure in order to minimize possible errors
and decoherence effects [163, 111]. The study of the structure of multipartite
entanglement poses many formidable challenges, concerning both its qualification
and quantification, and so far little progress has been achieved for multi-qubit systems
and in general for multi-party systems in finite-dimensional Hilbert spaces.
However, the situation looks somehow more promising in the arena of CV systems,
where some aspects of genuine multipartite entanglement can be, to begin with,
qualitatively understood studying the entanglement of multimode bipartitions.
In the present Chapter, based on Refs. [GA4, GA5] we analyze in detail the
entanglement properties of multimode Gaussian states endowed with particular
symmetry constraints under mode permutations. Their usefulness arises in contexts
like quantum error correction [36], where some redundancy is required for a
fault-tolerant encoding of information. Bisymmetric and, as a special case, fully
symmetric Gaussian states have been introduced in Sec. 2.4.3. An analysis of the
symplectic spectra of (M + N)-mode Gaussian states has revealed that, with respect
to the bipartition across which they exhibit the local permutation invariance
(any bipartition is valid for fully symmetric states), local symplectic diagonalizations
of the M-mode and the N-mode blocks result in a complete reduction of the
multimode state to an equivalent two-mode state, tensor M + N − 2 uncorrelated
thermal single-mode states. The equivalent two-mode state encodes all the information
of the original bisymmetric multimode state for what concerns entropy and
entanglement. As a consequence, the validity of the PPT criterion as a necessary
and sufficient condition for separability has been extended to bisymmetric Gaussian
states in Sec. 3.1.1.
Here, equipped with the powerful theoretical tools for the analysis of two-mode
entanglement in Gaussian states, demonstrated in the previous Chapter, we perform
a close analysis of the multimode entanglement in symmetric and bisymmetric
Gaussian states. In particular, we will investigate how the block entanglement
scales with the number of modes, hinting at the presence of genuine multipartite
entanglement arising among all the modes as their total number increases, at a given
squeezing degree. Motivated by this analysis, in the next Part of this Dissertation
we will face full-force the problem of quantifying the crucial and hideous property
of genuine multipartite CV entanglement in Gaussian states.
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