arXiv subjects

M. M. Wolf

Publications and source records attributed to M. M. Wolf.

At least 19 recordsLinked to original sources

arXiv2024

The $χ^2$-divergence and Mixing times of quantum Markov processes

We introduce quantum versions of the $χ^2$-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in [1-3] for classical Markov chains is taken to bound the trace-distance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes. Furthermore the contractive behavior of the $χ^2$-divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyse different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes.

quant-ph↗

arXiv2016

Fundamental limitations in the purifications of tensor networks

We show a fundamental limitation in the description of quantum many-body mixed states with tensor networks in purification form. Namely, we show that there exist mixed states which can be represented as a translationally invariant (TI) matrix product density operator (MPDO) valid for all system sizes, but for which there does not exist a TI purification valid for all system sizes. The proof is based on an undecidable problem and on the uniqueness of canonical forms of matrix product states. The result also holds for classical states.

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arXiv2012

Precisely timing dissipative quantum information processing

Dissipative engineering constitutes a framework within which quantum information processing protocols are powered by system-environment interaction rather than by unitary dynamics alone. This framework embraces noise as a resource, and consequently, offers a number of advantages compared to one based on unitary dynamics alone, e.g., that the protocols are typically independent of the initial state of the system. However, the time independent nature of this scheme makes it difficult to imagine precisely timed sequential operations, conditional measurements or error correction. In this work, we provide a path around these challenges, by introducing basic dissipative gadgets which allow us to precisely initiate, trigger and time dissipative operations, while keeping the system Liouvillian time-independent. These gadgets open up novel perspectives for thinking of timed dissipative quantum information processing. As an example, we sketch how measurement based computation can be simulated in the dissipative setting.

quant-ph↗

arXiv2010

A quantum version of Wielandt's inequality

In this paper, Wielandt's inequality for classical channels is extended to quantum channels. That is, an upper bound to the number of times a channel must be applied, so that it maps any density operator to one with full rank, is found. Using this bound, dichotomy theorems for the zero--error capacity of quantum channels and for the Matrix Product State (MPS) dimension of ground states of frustration-free Hamiltonians are derived. The obtained inequalities also imply new bounds on the required interaction-range of Hamiltonians with unique MPS ground state.

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arXiv2010

Quantum memory for entangled two-mode squeezed states

A quantum memory for light is a key element for the realization of future quantum information networks. Requirements for a good quantum memory are (i) versatility (allowing a wide range of inputs) and (ii) true quantum coherence (preserving quantum information). Here we demonstrate such a quantum memory for states possessing Einstein-Podolsky-Rosen (EPR) entanglement. These multi-photon states are two-mode squeezed by 6.0 dB with a variable orientation of squeezing and displaced by a few vacuum units. This range encompasses typical input alphabets for a continuous variable quantum information protocol. The memory consists of two cells, one for each mode, filled with cesium atoms at room temperature with a memory time of about 1msec. The preservation of quantum coherence is rigorously proven by showing that the experimental memory fidelity 0.52(2) significantly exceeds the benchmark of 0.45 for the best possible classical memory for a range of displacements.

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arXiv2009

Operator Space theory: a natural framework for Bell inequalities

In this letter we show that the field of Operator Space Theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $\frac{\sqrt{n}}{\log^2n}$ when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates and communication complexity are given.

quant-ph↗

arXiv2009

Unbounded violations of bipartite Bell Inequalities via Operator Space theory

In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $\sqrt{n}$ (up to a logarithmic factor) when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative $L_p$ embedding theory. As a consequence of this result, we obtain better Hilbert space dimension witnesses and quantum violations of Bell inequalities with better resistance to noise.

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arXiv2009

A canonical form for Projected Entangled Pair States and applications

We show that two different tensors defining the same translational invariant injective Projected Entangled Pair State (PEPS) in a square lattice must be the same up to a trivial gauge freedom. This allows us to characterize the existence of any local or spatial symmetry in the state. As an application of these results we prove that a SU(2) invariant PEPS with half-integer spin cannot be injective, which can be seen as a Lieb-Shultz-Mattis theorem in this context. We also give the natural generalization for U(1) symmetry in the spirit of Oshikawa-Yamanaka-Affleck, and show that a PEPS with Wilson loops cannot be injective.

quant-ph↗

arXiv2009

Matrix Product States: Symmetries and Two-Body Hamiltonians

We characterize the conditions under which a translationally invariant matrix product state (MPS) is invariant under local transformations. This allows us to relate the symmetry group of a given state to the symmetry group of a simple tensor. We exploit this result in order to prove and extend a version of the Lieb-Schultz-Mattis theorem, one of the basic results in many-body physics, in the context of MPS. We illustrate the results with an exhaustive search of SU(2)--invariant two-body Hamiltonians which have such MPS as exact ground states or excitations.

cond-mat.str-el↗

arXiv2008

Assessing non-Markovian dynamics

We investigate what a snapshot of a quantum evolution - a quantum channel reflecting open system dynamics - reveals about the underlying continuous time evolution. Remarkably, from such a snapshot, and without imposing additional assumptions, it can be decided whether or not a channel is consistent with a time (in)dependent Markovian evolution, for which we provide computable necessary and sufficient criteria. Based on these, a computable measure of `Markovianity' is introduced. We discuss how the consistency with Markovian dynamics can be checked in quantum process tomography. The results also clarify the geometry of the set of quantum channels with respect to being solutions of time (in)dependent master equations.

quant-ph↗

arXiv2008

Squeezing the limit: Quantum benchmarks for the teleportation and storage of squeezed states

We derive fidelity benchmarks for the quantum storage and teleportation of squeezed states of continuous variable systems, for input ensembles where the degree of squeezing $s$ is fixed, no information about its orientation in phase space is given, and the distribution of phase space displacements is a Gaussian. In the limit where the latter becomes flat, we prove analytically that the maximal classical achievable fidelity (which is 1/2 without squeezing, for $s=1$) is given by $\sqrt{s}/(1+s)$, vanishing when the degree of squeezing diverges. For mixed states, as well as for general distributions of displacements, we reduce the determination of the benchmarks to the solution of a finite-dimensional semidefinite program, which yields accurate, certifiable bounds thanks to a rigorous analysis of the truncation error. This approach may be easily adapted to more general ensembles of input states.

quant-ph↗

arXiv2008

Area laws in quantum systems: mutual information and correlations

The holographic principle states that on a fundamental level the information content of a region should depend on its surface area rather than on its volume. This counterintuitive idea which has its roots in the nonextensive nature of black-hole entropy serves as a guiding principle in the search for the fundamental laws of Planck-scale physics. In this paper we show that a similar phenomenon emerges from the established laws of classical and quantum physics: the information contained in part of a system in thermal equilibrium obeys an area law. While the maximal information per unit area depends classically only on the number of microscopic degrees of freedom, it may diverge as the inverse temperature in quantum systems. A rigorous relation between area laws and correlations is established and their explicit behavior is revealed for a large class of quantum many-body states beyond equilibrium systems.

quant-ph↗

arXiv2008

String order and symmetries in quantum spin lattices

We show that the existence of string order in a given quantum state is intimately related to the presence of a local symmetry by proving that both concepts are equivalent within the framework of finitely correlated states. Once this connection is established, we provide a complete characterization of local symmetries in these states. The results allow to understand in a straightforward way many of the properties of string order parameters, like their robustness/fragility under perturbations and their typical disappearance beyond strictly one-dimensional lattices. We propose and discuss an alternative definition, ideally suited for detecting phase transitions, and generalizations to two and more spatial dimensions.

cond-mat.str-el↗

arXiv2007

Matrix Product State Representations

This work gives a detailed investigation of matrix product state (MPS) representations for pure multipartite quantum states. We determine the freedom in representations with and without translation symmetry, derive respective canonical forms and provide efficient methods for obtaining them. Results on frustration free Hamiltonians and the generation of MPS are extended, and the use of the MPS-representation for classical simulations of quantum systems is discussed.

quant-ph↗

arXiv2006

Sequential Generation of Matrix-Product States in Cavity QED

We study the sequential generation of entangled photonic and atomic multi-qubit states in the realm of cavity QED. We extend the work of C. Schoen et al. [Phys. Rev. Lett. 95, 110503 (2005)], where it was shown that all states generated in a sequential manner can be classified efficiently in terms of matrix-product states. In particular, we consider two scenarios: photonic multi-qubit states sequentially generated at the cavity output of a single-photon source and atomic multi-qubit states generated by their sequential interaction with the same cavity mode.

quant-ph↗

arXiv2006

Optimal squeezing and entanglement from noisy Gaussian operations

We investigate the creation of squeezing via operations subject to noise and losses and ask for the optimal use of such devices when supplemented by noiseless passive operations. Both single and repeated uses of the device are optimized analytically and it is proven that in the latter case the squeezing converges exponentially fast to its asymptotic optimum, which we determine explicitly. For the case of multiple iterations we show that the optimum can be achieved with fixed intermediate passive operations. Finally, we relate the results to the generation of entanglement and derive the maximal two-mode entanglement achievable within the considered scenario.

quant-ph↗

arXiv2006

Quantum computing

This article gives an elementary introduction to quantum computing. It is a draft for a book chapter of the "Handbook of Nature-Inspired and Innovative Computing", Eds. A. Zomaya, G.J. Milburn, J. Dongarra, D. Bader, R. Brent, M. Eshaghian-Wilner, F. Seredynski (Springer, Berlin Heidelberg New York, 2006).

quant-ph↗

arXiv2006

Criticality, the area law, and the computational power of PEPS

The projected entangled pair state (PEPS) representation of quantum states on two-dimensional lattices induces an entanglement based hierarchy in state space. We show that the lowest levels of this hierarchy exhibit an enormously rich structure including states with critical and topological properties as well as resonating valence bond states. We prove, in particular, that coherent versions of thermal states of any local 2D classical spin model correspond to such PEPS, which are in turn ground states of local 2D quantum Hamiltonians. This correspondence maps thermal onto quantum fluctuations, and it allows us to analytically construct critical quantum models exhibiting a strict area law scaling of the entanglement entropy in the face of power law decaying correlations. Moreover, it enables us to show that there exist PEPS within the same class as the cluster state, which can serve as computational resources for the solution of NP-hard problems.

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