3,055 research outputs found
Self-testing multipartite entangled states through projections onto two systems
Finding ways to test the behaviour of quantum devices is a timely enterprise,
especially in the light of the rapid development of quantum technologies.
Device-independent self-testing is one desirable approach, as it makes minimal
assumptions on the devices being tested. In this work, we address the question
of which states can be self-tested. This has been answered recently in the
bipartite case [Nat. Comm. 8, 15485 (2017)], while it is largely unexplored in
the multipartite case, with only a few scattered results, using a variety of
different methods: maximal violation of a Bell inequality, numerical SWAP
method, stabilizer self-testing etc. In this work, we investigate a simple, and
potentially unifying, approach: combining projections onto two-qubit spaces
(projecting parties or degrees of freedom) and then using maximal violation of
the tilted CHSH inequalities. This allows to obtain self-testing of Dicke
states and partially entangled GHZ states with two measurements per party, and
also to recover self-testing of graph states (previously known only through
stabilizer methods). Finally, we give the first self-test of a class
multipartite qudit states: we generalize the self-testing of partially
entangled GHZ states by adapting techniques from [Nat. Comm. 8, 15485 (2017)],
and show that all multipartite states which admit a Schmidt decomposition can
be self-tested with few measurements.Comment: The title is changed and the presentation is slightly restructure
Device-independent tomography of multipartite quantum states
In the usual tomography of multipartite entangled quantum states one assumes
that the measurement devices used in the laboratory are under perfect control
of the experimenter. In this paper, using the so-called SWAP concept introduced
recently, we show how one can remove this assumption in realistic experimental
conditions and nevertheless be able to characterize the produced multipartite
state based only on observed statistics. Such a black box tomography of quantum
states is termed self-testing. As a function of the magnitude of the Bell
violation, we are able to self-test emblematic multipartite quantum states such
as the three-qubit W state, the three- and four-qubit
Greenberger-Horne-Zeilinger states, and the four-qubit linear cluster state.Comment: See also the related work of arXiv:1407.576
Self-testing of binary observables based on commutation
We consider the problem of certifying binary observables based on a Bell
inequality violation alone, a task known as self-testing of measurements. We
introduce a family of commutation-based measures, which encode all the distinct
arrangements of two projective observables on a qubit. These quantities by
construction take into account the usual limitations of self-testing and since
they are "weighted" by the (reduced) state, they automatically deal with
rank-deficient reduced density matrices. We show that these measures can be
estimated from the observed Bell violation in several scenarios and the proofs
rely only on standard linear algebra. The trade-offs turn out to be tight and,
in particular, they give non-trivial statements for arbitrarily small
violations. On the other extreme, observing the maximal violation allows us to
deduce precisely the form of the observables, which immediately leads to a
complete rigidity statement. In particular, we show that for all the
-partite Mermin-Ardehali-Belinskii-Klyshko inequality self-tests the
-partite Greenberger-Horne-Zeilinger state and maximally incompatible qubit
measurements on every party. Our results imply that any pair of projective
observables on a qubit can be certified in a truly robust manner. Finally, we
show that commutation-based measures give a convenient way of expressing
relations among more than two observables.Comment: 5 + 4 pages. v2: published version; v3: formatting errors fixe
Testing axioms for Quantum Mechanics on Probabilistic toy-theories
In Ref. [1] one of the authors proposed postulates for axiomatizing Quantum
Mechanics as a "fair operational framework", namely regarding the theory as a
set of rules that allow the experimenter to predict future events on the basis
of suitable tests, having local control and low experimental complexity. In
addition to causality, the following postulates have been considered: PFAITH
(existence of a pure preparationally faithful state), and FAITHE (existence of
a faithful effect). These postulates have exhibited an unexpected theoretical
power, excluding all known nonquantum probabilistic theories. Later in Ref. [2]
in addition to causality and PFAITH, postulate LDISCR (local discriminability)
and PURIFY (purifiability of all states) have been considered, narrowing the
probabilistic theory to something very close to Quantum Mechanics. In the
present paper we test the above postulates on some nonquantum probabilistic
models. The first model, "the two-box world" is an extension of the
Popescu-Rohrlich model, which achieves the greatest violation of the CHSH
inequality compatible with the no-signaling principle. The second model "the
two-clock world" is actually a full class of models, all having a disk as
convex set of states for the local system. One of them corresponds to the "the
two-rebit world", namely qubits with real Hilbert space. The third model--"the
spin-factor"--is a sort of n-dimensional generalization of the clock. Finally
the last model is "the classical probabilistic theory". We see how each model
violates some of the proposed postulates, when and how teleportation can be
achieved, and we analyze other interesting connections between these postulate
violations, along with deep relations between the local and the non-local
structures of the probabilistic theory.Comment: Submitted to QIP Special Issue on Foundations of Quantum Informatio
Optimal discrimination of single-qubit mixed states
We consider the problem of minimum-error quantum state discrimination for
single-qubit mixed states. We present a method which uses the Helstrom
conditions constructively and analytically; this algebraic approach is
complementary to existing geometric methods, and solves the problem for any
number of arbitrary signal states with arbitrary prior probabilities.Comment: 8 pages, 1 figur
Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations
Schur-Weyl duality is a ubiquitous tool in quantum information. At its heart
is the statement that the space of operators that commute with the tensor
powers of all unitaries is spanned by the permutations of the tensor factors.
In this work, we describe a similar duality theory for tensor powers of
Clifford unitaries. The Clifford group is a central object in many subfields of
quantum information, most prominently in the theory of fault-tolerance. The
duality theory has a simple and clean description in terms of finite
geometries. We demonstrate its effectiveness in several applications:
(1) We resolve an open problem in quantum property testing by showing that
"stabilizerness" is efficiently testable: There is a protocol that, given
access to six copies of an unknown state, can determine whether it is a
stabilizer state, or whether it is far away from the set of stabilizer states.
We give a related membership test for the Clifford group.
(2) We find that tensor powers of stabilizer states have an increased
symmetry group. We provide corresponding de Finetti theorems, showing that the
reductions of arbitrary states with this symmetry are well-approximated by
mixtures of stabilizer tensor powers (in some cases, exponentially well).
(3) We show that the distance of a pure state to the set of stabilizers can
be lower-bounded in terms of the sum-negativity of its Wigner function. This
gives a new quantitative meaning to the sum-negativity (and the related mana)
-- a measure relevant to fault-tolerant quantum computation. The result
constitutes a robust generalization of the discrete Hudson theorem.
(4) We show that complex projective designs of arbitrary order can be
obtained from a finite number (independent of the number of qudits) of Clifford
orbits. To prove this result, we give explicit formulas for arbitrary moments
of random stabilizer states.Comment: 60 pages, 2 figure
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