Research Articles:

**On the non-locality of tripartite non-signaling boxes emerging from
wirings **
(pp1081-1108)

Jan
Tuziemski and Karol Horodecki

It has been recently shown, that some of the tripartite boxes admitting
bilocal decomposition, lead to non-locality under wiring operation
applied to two of the subsystems [R. Gallego {\it et al.} Physical
Review Letters {\bf 109}, 070401 (2012)]. In the following, we study
this phenomenon quantitatively. Basing on the known classes of boxes
closed under wirings, we introduce multipartite monotones which are
counterparts of bipartite ones - the non-locality cost and robustness of
non-locality. We then provide analytical lower bounds on both the
monotones in terms of the Maximal Non-locality which can be obtained by
Wirings (MWN). We prove also upper bounds for the MWN of a given box,
based on the weight of boxes signaling in a particular direction, that
appear in its fully bilocal decomposition. We study different classes of
partially local boxes (i.e. having local variable model with respect to
some grouping of the parties). For each class the MWN is found, using
the Linear Programming. The wirings which lead to the MWN and exhibit
that some of them can serve as a witness of the certain classes are also
identified. We conclude with example of partially local boxes being
analogue of quantum states that allow to distribute entanglement in
separable manner.

**Characterization and properties of weakly optimal entanglement
witnesses **
(pp1109-1121)

Bang-Hai Wang,
Hai-Ru Xu, Steve Campbell, and Simone Severini

We present an analysis of the properties and characteristics of weakly
optimal entanglement witnesses, that is witnesses whose expectation
value vanishes on at least one product vector. Any weakly optimal
entanglement witness can be written as the form of $W^{wopt}=\sigma-c_{\sigma}^{max}
I$, where $c_{\sigma}^{max}$ is a non-negative number and $I$ is the
identity matrix. We show the relation between the weakly optimal witness
$W^{wopt}$ and the eigenvalues of the separable states $\sigma$. Further
we give an application of weakly optimal witnesses for constructing
entanglement witnesses in a larger Hilbert space by extending the result
of [P. Badzi\c{a}g {\it et al}, Phys. Rev. A {\bf 88}, 010301(R)
(2013)], and we examine their geometric properties.

**Monte Carlo simulation of stoquastic Hamiltonians**
(pp1122-1140)

Sergey Bravyi

Stoquastic Hamiltonians are characterized by the property that their
off-diagonal matrix elements in the standard product basis are real and
non-positive. Many interesting quantum models fall into this class
including the Transverse field Ising Model (TIM), the Heisenberg model
on bipartite graphs, and the bosonic Hubbard model. Here we consider the
problem of estimating the ground state energy of a local stoquastic
Hamiltonian $H$ with a promise that the ground state of $H$ has a
non-negligible correlation with some `guiding' state that admits a
concise classical description. A formalized version of this problem
called Guided Stoquastic Hamiltonian is shown to be complete for the
complexity class $\MA$ (a probabilistic analogue of $\NP$). To prove
this result we employ the Projection Monte Carlo algorithm with a
variable number of walkers. Secondly, we show that the ground state and
thermal equilibrium properties of the ferromagnetic TIM can be simulated
in polynomial time on a classical probabilistic computer. This result is
based on the approximation algorithm for the classical ferromagnetic
Ising model due to Jerrum and Sinclair (1993).

**Entanglement in a linear coherent feedback chain of nondegenerate
optical parametric amplifiers**
(pp1141-1164)

Zhan Shi and
Hendra I. Nurdin

This paper is concerned with linear quantum networks of $N$
nondegenerate optical parametric amplifiers (NOPAs), with $N$ up to 6,
which are interconnected in a coherent feedback chain. Each network
connects two communicating parties (Alice and Bob) over two transmission
channels. In previous work we have shown that a dual-NOPA coherent
feedback network generates better Einstein-Podolsky-Rosen (EPR)
entanglement (i.e., more two-mode squeezing) between its two outgoing
Gaussian fields than a single NOPA, when the same total pump power is
consumed and the systems undergo the same transmission losses over the
same distance. This paper aims to analyze stability, EPR entanglement
between two outgoing fields of interest, and bipartite entanglement of
two-mode Gaussian states of cavity modes of the $N$-NOPA networks under
the effect of transmission and amplification losses, as well as the
effect of time delays on the outgoing fields. It is numerically shown
that, in the absence of losses and delays, the network with more NOPAs
in the chain requires less total pump power to generate the same degree
of EPR entanglement. Moreover, we report on the internal entanglement
synchronization that occurs in the steady state between certain pairs of
Gaussian oscillator modes inside the NOPA cavities of the networks.

**Thresholds for reduction-related entanglement criteria in quantum
information theory**
(pp1165-1184)

Maria A. Jivulescu,
Nicolae Lupa, and Ion Nechita

We consider random bipartite quantum states obtained by tracing out one
subsystem from a random, uniformly distributed, tripartite pure quantum
state. We compute thresholds for the dimension of the system being
traced out, so that the resulting bipartite quantum state satisfies the
reduction criterion in different asymptotic regimes. We consider as well
the basis-independent version of the reduction criterion (the absolute
reduction criterion), computing thresholds for the corresponding
eigenvalue sets. We do the same for other sets relevant in the study of
absolute separability, using techniques from random matrix theory.
Finally, we gather and compare the known values for the thresholds
corresponding to different entanglement criteria, and conclude with a
list of open questions.

**Quantum Gaussian channels with weak measurements**
(pp1185-1196)

Boaz Tamir and
Eliahu Cohen

In this paper we perform a novel analysis of quantum Gaussian channels
in the context of weak measurements. Suppose Alice sends classical
information to Bob using a quantum channel. Suppose Bob is allowed to
use only weak measurements, what would be the channel capacity? We
formulate weak measurement theory in these terms and discuss the above
question.

**Perturbative gadgets without strong interactions**
(pp1197-1222)

Yudong Cao and
Daniel Nagaj

Perturbative gadgets are used to construct a quantum Hamiltonian whose
low-energy subspace approximates a given quantum $k$-local Hamiltonian
up to an absolute error $\epsilon$. Typically, gadget constructions
involve terms with large interaction strengths of order $\text{poly}(\epsilon^{-1})$.
Here we present a 2-body gadget construction and prove that it
approximates a Hamiltonian of interaction strength $\gamma = O(1)$ up to
absolute error $\epsilon\ll\gamma$ using interactions of strength
$O(\epsilon)$ instead of the usual inverse polynomial in $\epsilon$. A
key component in our proof is a new condition for the convergence of the
perturbation series, allowing our gadget construction to be applied in
parallel on multiple many-body terms. We also discuss how to apply this
gadget construction for approximating 3- and $k$-local Hamiltonians. The
price we pay for using much weaker interactions is a large overhead in
the number of ancillary qubits, and the number of interaction terms per
particle, both of which scale as $O(\text{poly}(\epsilon^{-1}))$. Our
strong-from-weak gadgets have their primary application in complexity
theory (QMA hardness of restricted Hamiltonians, a generalized area law
counterexample, gap amplification), but could also motivate practical
implementations with several weak interactions simulating a much
stronger quantum many-body interaction.

**The impact of resource on remote quantum correlation Preparation**
(pp1223-1232)

Chengjun Wu, Bin
Luo, and Hong Guo

When Alice and Bob share two pairs of quantum correlated states, Alice
can remotely prepare quantum entanglement and quantum discord in Bob's
side by measuring the parts in her side and telling Bob the measurement
results by classical communication. For remote entanglement preparation,
entanglement is necessary . We find that for some shared resources
having the same amount of entanglement, when Bell measurement is used,
the entanglement remotely prepared can be different, and more discord in
the resources actually decreases the entanglement prepared. We also find
that for some resources with more entanglement, the entanglement
remotely prepared may be less. Therefore, we conclude that entanglement
is a necessary resource but may not be the only resource responsible for
the entanglement remotely prepared, and discord does not likely to
assist this process. Also, for the preparation of discord, we find that
some states with no entanglement could outperform entangled states.

**Spatial search on grids with minimum memory **
(pp1233-1247)

Andris Ambainis,
Renato Portugal, and Nikolay Nahimov

We study quantum algorithms for spatial search on finite dimensional
grids. Patel \textit{et al.}~and Falk have proposed algorithms based on
a quantum walk without a coin, with different operators applied at even
and odd steps. Until now, such algorithms have been studied only using
numerical simulations. In this paper, we present the first rigorous
analysis for an algorithm of this type, showing that the optimal number
of steps is $O(\sqrt{N\log N})$ and the success probability is $O(1/\log
N)$, where $N$ is the number of vertices. This matches the performance
achieved by algorithms that use other forms of quantum walks.

**Limit distributions for different forms of four-state quantum walks
on a two-dimensional lattice**
(pp1248-1258)

Takuya Machida,
C.M. Chandrashekar, Norio Konno, and Thomas Busch

Long-time limit distributions are key quantities for understanding the
asymptotic dynamics of quantum walks, and they are known for most forms
of one-dimensional quantum walks using two-state coin systems. For
two-dimensional quantum walks using a four-state coin system, however,
the only known limit distribution is for a walk using a parameterized
Grover coin operation and analytical complexities have been a major
obstacle for obtaining long-time limit distributions for other coins. In
this work however, we present two new types of long-time limit
distributions for walks using different forms of coin-flip operations in
a four-state coin system. This opens the road towards understanding the
dynamics and asymptotic behaviour for higher state coin system from a
mathematical view point.