123 research outputs found
N-representability is QMA-complete
We study the computational complexity of the N-representability problem in
quantum chemistry. We show that this problem is QMA-complete, which is the
quantum generalization of NP-complete. Our proof uses a simple mapping from
spin systems to fermionic systems, as well as a convex optimization technique
that reduces the problem of finding ground states to N-representability
Quantum cryptography based on qutrit Bell inequalities
We present a cryptographic protocol based upon entangled qutrit pairs. We analyze the scheme under a symmetric incoherent attack and plot the region for which the protocol is secure and compare this with the region of violations of certain Bell inequalities
Upper bound on the secret key rate distillable from effective quantum correlations with imperfect detectors
We provide a simple method to obtain an upper bound on the secret key rate
that is particularly suited to analyze practical realizations of quantum key
distribution protocols with imperfect devices. We consider the so-called
trusted device scenario where Eve cannot modify the actual detection devices
employed by Alice and Bob. The upper bound obtained is based on the available
measurements results, but it includes the effect of the noise and losses
present in the detectors of the legitimate users.Comment: 9 pages, 1 figure; suppress sifting effect in the figure, final
versio
Tomographic Quantum Cryptography: Equivalence of Quantum and Classical Key Distillation
The security of a cryptographic key that is generated by communication
through a noisy quantum channel relies on the ability to distill a shorter
secure key sequence from a longer insecure one. For an important class of
protocols, which exploit tomographically complete measurements on entangled
pairs of any dimension, we show that the noise threshold for classical
advantage distillation is identical with the threshold for quantum entanglement
distillation. As a consequence, the two distillation procedures are equivalent:
neither offers a security advantage over the other.Comment: 4 pages, 1 figur
The Uncertainty Principle in the Presence of Quantum Memory
The uncertainty principle, originally formulated by Heisenberg, dramatically
illustrates the difference between classical and quantum mechanics. The
principle bounds the uncertainties about the outcomes of two incompatible
measurements, such as position and momentum, on a particle. It implies that one
cannot predict the outcomes for both possible choices of measurement to
arbitrary precision, even if information about the preparation of the particle
is available in a classical memory. However, if the particle is prepared
entangled with a quantum memory, a device which is likely to soon be available,
it is possible to predict the outcomes for both measurement choices precisely.
In this work we strengthen the uncertainty principle to incorporate this case,
providing a lower bound on the uncertainties which depends on the amount of
entanglement between the particle and the quantum memory. We detail the
application of our result to witnessing entanglement and to quantum key
distribution.Comment: 5 pages plus 12 of supplementary information. Updated to match the
journal versio
Secure certification of mixed quantum states with application to two-party randomness generation
We investigate sampling procedures that certify that an arbitrary quantum
state on subsystems is close to an ideal mixed state
for a given reference state , up to errors on a few positions. This
task makes no sense classically: it would correspond to certifying that a given
bitstring was generated according to some desired probability distribution.
However, in the quantum case, this is possible if one has access to a prover
who can supply a purification of the mixed state.
In this work, we introduce the concept of mixed-state certification, and we
show that a natural sampling protocol offers secure certification in the
presence of a possibly dishonest prover: if the verifier accepts then he can be
almost certain that the state in question has been correctly prepared, up to a
small number of errors.
We then apply this result to two-party quantum coin-tossing. Given that
strong coin tossing is impossible, it is natural to ask "how close can we get".
This question has been well studied and is nowadays well understood from the
perspective of the bias of individual coin tosses. We approach and answer this
question from a different---and somewhat orthogonal---perspective, where we do
not look at individual coin tosses but at the global entropy instead. We show
how two distrusting parties can produce a common high-entropy source, where the
entropy is an arbitrarily small fraction below the maximum (except with
negligible probability)
Robustness of quantum Markov chains
If the conditional information of a classical probability distribution of
three random variables is zero, then it obeys a Markov chain condition. If the
conditional information is close to zero, then it is known that the distance
(minimum relative entropy) of the distribution to the nearest Markov chain
distribution is precisely the conditional information. We prove here that this
simple situation does not obtain for quantum conditional information. We show
that for tri-partite quantum states the quantum conditional information is
always a lower bound for the minimum relative entropy distance to a quantum
Markov chain state, but the distance can be much greater; indeed the two
quantities can be of different asymptotic order and may even differ by a
dimensional factor.Comment: 14 pages, no figures; not for the feeble-minde
On asymptotic continuity of functions of quantum states
A useful kind of continuity of quantum states functions in asymptotic regime
is so-called asymptotic continuity. In this paper we provide general tools for
checking if a function possesses this property. First we prove equivalence of
asymptotic continuity with so-called it robustness under admixture. This allows
us to show that relative entropy distance from a convex set including maximally
mixed state is asymptotically continuous. Subsequently, we consider it arrowing
- a way of building a new function out of a given one. The procedure originates
from constructions of intrinsic information and entanglement of formation. We
show that arrowing preserves asymptotic continuity for a class of functions
(so-called subextensive ones). The result is illustrated by means of several
examples.Comment: Minor corrections, version submitted for publicatio
Gaussian quantum marginal problem
The quantum marginal problem asks what local spectra are consistent with a
given spectrum of a joint state of a composite quantum system. This setting,
also referred to as the question of the compatibility of local spectra, has
several applications in quantum information theory. Here, we introduce the
analogue of this statement for Gaussian states for any number of modes, and
solve it in generality, for pure and mixed states, both concerning necessary
and sufficient conditions. Formally, our result can be viewed as an analogue of
the Sing-Thompson Theorem (respectively Horn's Lemma), characterizing the
relationship between main diagonal elements and singular values of a complex
matrix: We find necessary and sufficient conditions for vectors (d1, ..., dn)
and (c1, ..., cn) to be the symplectic eigenvalues and symplectic main diagonal
elements of a strictly positive real matrix, respectively. More physically
speaking, this result determines what local temperatures or entropies are
consistent with a pure or mixed Gaussian state of several modes. We find that
this result implies a solution to the problem of sharing of entanglement in
pure Gaussian states and allows for estimating the global entropy of
non-Gaussian states based on local measurements. Implications to the actual
preparation of multi-mode continuous-variable entangled states are discussed.
We compare the findings with the marginal problem for qubits, the solution of
which for pure states has a strikingly similar and in fact simple form.Comment: 18 pages, 1 figure, material added, references updated, except from
figure identical with version to appear in Commun. Math. Phy
Faithful Squashed Entanglement
Squashed entanglement is a measure for the entanglement of bipartite quantum
states. In this paper we present a lower bound for squashed entanglement in
terms of a distance to the set of separable states. This implies that squashed
entanglement is faithful, that is, strictly positive if and only if the state
is entangled. We derive the bound on squashed entanglement from a bound on
quantum conditional mutual information, which is used to define squashed
entanglement and corresponds to the amount by which strong subadditivity of von
Neumann entropy fails to be saturated. Our result therefore sheds light on the
structure of states that almost satisfy strong subadditivity with equality. The
proof is based on two recent results from quantum information theory: the
operational interpretation of the quantum mutual information as the optimal
rate for state redistribution and the interpretation of the regularised
relative entropy of entanglement as an error exponent in hypothesis testing.
The distance to the set of separable states is measured by the one-way LOCC
norm, an operationally-motivated norm giving the optimal probability of
distinguishing two bipartite quantum states, each shared by two parties, using
any protocol formed by local quantum operations and one-directional classical
communication between the parties. A similar result for the Frobenius or
Euclidean norm follows immediately. The result has two applications in
complexity theory. The first is a quasipolynomial-time algorithm solving the
weak membership problem for the set of separable states in one-way LOCC or
Euclidean norm. The second concerns quantum Merlin-Arthur games. Here we show
that multiple provers are not more powerful than a single prover when the
verifier is restricted to one-way LOCC operations thereby providing a new
characterisation of the complexity class QMA.Comment: 24 pages, 1 figure, 1 table. Due to an error in the published
version, claims have been weakened from the LOCC norm to the one-way LOCC
nor
- …