Entanglement distillation is a key primitive for distributing high-quality entanglement between remote locations. Probabilistic noiseless linear amplification based on the quantum scissors is a candidate for entanglement distillation from noisy conti
nuous-variable (CV) entangled states. Being a non-Gaussian operation, quantum scissors is challenging to analyze. We present a derivation of the non-Gaussian state heralded by multiple quantum scissors in a pure loss channel with two-mode squeezed vacuum input. We choose the reverse coherent information (RCI)---a proven lower bound on the distillable entanglement of a quantum state under one-way local operations and classical communication (LOCC), as our figure of merit. We evaluate a Gaussian lower bound on the RCI of the heralded state. We show that it can exceed the unlimited two-way LOCCassisted direct transmission entanglement distillation capacity of the pure loss channel. The optimal heralded Gaussian RCI with two quantum scissors is found to be significantly more than that with a single quantum scissors, albeit at the cost of decreased success probability. Our results fortify the possibility of a quantum repeater scheme for CV quantum states using the quantum scissors.
Quantum repeaters are indispensable for high-rate, long-distance quantum communications. The vision of a future quantum internet strongly hinges on realizing quantum repeaters in practice. Numerous repeaters have been proposed for discrete-variable (
DV) single-photon-based quantum communications. Continuous variable (CV) encodings over the quadrature degrees of freedom of the electromagnetic field mode offer an attractive alternative. For example, CV transmission systems are easier to integrate with existing optical telecom systems compared to their DV counterparts. Yet, repeaters for CV have remained elusive. We present a novel quantum repeater scheme for CV entanglement distribution over a lossy bosonic channel that beats the direct transmission exponential rate-loss tradeoff. The scheme involves repeater nodes consisting of a) two-mode squeezed vacuum (TMSV) CV entanglement sources, b) the quantum scissors operation to perform nondeterministic noiseless linear amplification of lossy TMSV states, c) a layer of switched, mode multiplexing inspired by second-generation DV repeaters, which is the key ingredient apart from probabilistic entanglement purification that makes DV repeaters work, and d) a non-Gaussian entanglement swap operation. We report our exact results on the rate-loss envelope achieved by the scheme.
We analyze the fundamental quantum limit of the resolution of an optical imaging system from the perspective of the detection problem of deciding whether the optical field in the image plane is generated by one incoherent on-axis source with brightne
ss $epsilon$ or by two $epsilon/2$-brightness incoherent sources that are symmetrically disposed about the optical axis. Using the exact thermal-state model of the field, we derive the quantum Chernoff bound for the detection problem, which specifies the optimum rate of decay of the error probability with increasing number of collected photons that is allowed by quantum mechanics. We then show that recently proposed linear-optic schemes approach the quantum Chernoff bound---the method of binary spatial-mode demultiplexing (B-SPADE) is quantum-optimal for all values of separation, while a method using image-inversion interferometry (SLIVER) is near-optimal for sub-Rayleigh separations. We then simplify our model using a low-brightness approximation that is very accurate for optical microscopy and astronomy, derive quantum Chernoff bounds conditional on the number of photons detected, and show the optimality of our schemes in this conditional detection paradigm. For comparison, we analytically demonstrate the superior scaling of the Chernoff bound for our schemes with source separation relative to that of spatially-resolved direct imaging. Our schemes have the advantages over the quantum-optimal (Helstrom) measurement in that they do not involve joint measurements over multiple modes, and that they do not require the angular separation for the two-source hypothesis to be given emph{a priori} and can offer that information as a bonus in the event of a successful detection.
Recent work has shown that quantum computers can compute scattering probabilities in massive quantum field theories, with a run time that is polynomial in the number of particles, their energy, and the desired precision. Here we study a closely relat
ed quantum field-theoretical problem: estimating the vacuum-to-vacuum transition amplitude, in the presence of spacetime-dependent classical sources, for a massive scalar field theory in (1+1) dimensions. We show that this problem is BQP-hard; in other words, its solution enables one to solve any problem that is solvable in polynomial time by a quantum computer. Hence, the vacuum-to-vacuum amplitude cannot be accurately estimated by any efficient classical algorithm, even if the field theory is very weakly coupled, unless BQP=BPP. Furthermore, the corresponding decision problem can be solved by a quantum computer in a time scaling polynomially with the number of bits needed to specify the classical source fields, and this problem is therefore BQP-complete. Our construction can be regarded as an idealized architecture for a universal quantum computer in a laboratory system described by massive phi^4 theory coupled to classical spacetime-dependent sources.
Knot and link invariants naturally arise from any braided Hopf algebra. We consider the computational complexity of the invariants arising from an elementary family of finite-dimensional Hopf algebras: quantum doubles of finite groups (denoted D(G),
for a group G). Regarding algorithms for these invariants, we develop quantum circuits for the quantum Fourier transform over D(G); in general, we show that when one can uniformly and efficiently carry out the quantum Fourier transform over the centralizers Z(g) of the elements of G, one can efficiently carry out the quantum Fourier transform over D(G). We apply these results to the symmetric groups to yield efficient circuits for the quantum Fourier transform over D(S_n). With such a Fourier transform, it is straightforward to obtain additive approximation algorithms for the related link invariant. Additionally, we show that certain D(G) invariants (such as D(A_n) invariants) are BPP-hard to additively approximate, SBP-hard to multiplicatively approximate, and #P-hard to exactly evaluate. Finally, we make partial progress on the question of simulating anyonic computation in groups uniformly as a function of the group size. In this direction, we provide efficient quantum circuits for the Clebsch-Gordan transform over D(G) for fluxon irreps, i.e., irreps of D(G) characterized by a conjugacy class of G. For general irreps, i.e., those which are associated with a conjugacy class of G and an irrep of a centralizer, we present an efficient implementation under certain conditions such as when there is an efficient Clebsch-Gordan transform over the centralizers. We remark that this also provides a simulation of certain anyonic models of quantum computation, even in circumstances where the group may have size exponential in the size of the circuit.
We define the hitting (or absorbing) time for the case of continuous quantum walks by measuring the walk at random times, according to a Poisson process with measurement rate $lambda$. From this definition we derive an explicit formula for the hittin
g time, and explore its dependence on the measurement rate. As the measurement rate goes to either 0 or infinity the hitting time diverges; the first divergence reflects the weakness of the measurement, while the second limit results from the Quantum Zeno effect. Continuous-time quantum walks, like discrete-time quantum walks but unlike classical random walks, can have infinite hitting times. We present several conditions for existence of infinite hitting times, and discuss the connection between infinite hitting times and graph symmetry.
A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an endi
ng condition. We derive an expression for hitting time using superoperators, and numerically evaluate it for the walk on the hypercube for various coins and decoherence models. We show that, by contrast to classical walks, quantum walks can have infinite hitting times for some initial states. We seek criteria to determine if a given walk on a graph will have infinite hitting times, and find a sufficient condition for their existence. The phenomenon of infinite hitting times is in general a consequence of the symmetry of the graph and its automorphism group. Symmetries of a graph, given by its automorphism group, can be inherited by the evolution operator. Using the irreducible representations of the automorphism group, we derive conditions such that quantum walks defined on this graph must have infinite hitting times for some initial states. Symmetry can also cause the walk to be confined to a subspace of the original Hilbert space for certain initial states. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph and we give an explicit construction of the quotient graph. We conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up. Finally, we use symmetry and the theory of decoherence-free subspaces to determine when the subspace of the quotient graph is a decoherence-free subspace of the dynamics.
We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the syste
m qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed post-Markovian master equation performs comparably to the time-convolutionless master equation for a properly chosen memory kernel, and outperforms all the approximation methods considered here at long times. Our findings shed light on the applicability of master equations to the description of reduced system dynamics in the presence of spin-baths.
Local pure states are an important resource for quantum computing. The problem of distilling local pure states from mixed ones can be cast in an information theoretic paradigm. The bipartite version of this problem where local purity must be distille
d from an arbitrary quantum state shared between two parties, Alice and Bob, is closely related to the problem of separating quantum and classical correlations in the state and in particular, to a measure of classical correlations called the one-way distillable common randomness. In Phys. Rev. A 71, 062303 (2005), the optimal rate of local purity distillation is derived when many copies of a bipartite quantum state are shared between Alice and Bob, and the parties are allowed unlimited use of a unidirectional dephasing channel. In the present paper, we extend this result to the setting in which the use of the channel is bounded. We demonstrate that in the case of a classical-quantum system, the expression for the local purity distilled is efficiently computable and provide examples with their tradeoff curves.
