QIC Abstracts

 Vol.18 No.5&6, May 1, 2018

Research Articles:

Perfect sampling for quantum Gibbs states (pp0361-0388)
          
Daniel S. Franca
We show how to obtain perfect samples from a quantum Gibbs state on a quantum computer. To do so, we adapt one of the ``Coupling from the Past''-algorithms proposed by Propp and Wilson. The algorithm has a probabilistic run-time and produces perfect samples without any previous knowledge of the mixing time of a quantum Markov chain. To implement it, we assume we are able to perform the phase estimation algorithm for the underlying Hamiltonian and implement a quantum Markov chain such that the transition probabilities between eigenstates only depend on their energy. We provide some examples of quantum Markov chains that satisfy these conditions and analyze the expected run-time of the algorithm, which depends strongly on the degeneracy of the underlying Hamiltonian. For Hamiltonians with highly degenerate spectrum, it is efficient, as it is polylogarithmic in the dimension and linear in the mixing time. For non-degenerate spectra, its runtime is essentially the same as its classical counterpart, which is linear in the mixing time and quadratic in the dimension, up to a logarithmic factor in the dimension. We analyze the circuit depth necessary to implement it, which is proportional to the sum  of the depth necessary to implement one step of the quantum Markov chain and one phase estimation. This algorithm is stable under noise in the implementation of different steps. We also briefly discuss how to adapt different ``Coupling from the Past''-algorithms to the quantum setting.

Ent: A Multipartite entanglement measure, and parameterization of entangled states (pp0389-0442)
          
Samuel R. Hedemann
A multipartite entanglement measure called the ent is presented and shown to be an entanglement monotone, with the special property of automatic normalization.  Necessary and sufficient conditions are developed for constructing maximally entangled states in every multipartite system such that they are true-generalized X states (TGX) states, a generalization of the Bell states, and are extended to general nonTGX states as well.  These results are then used to prove the existence of maximally entangled basis (MEB) sets in all systems. A parameterization of general pure states of all ent values is given, and proposed as a multipartite Schmidt decomposition.  Finally, we develop an ent vector and ent array to handle more general definitions of multipartite entanglement, and the ent is extended to general mixed states, providing a general multipartite entanglement measure.

Candidates for universal measures of multipartite entanglement (pp0443-0471)
          
Samuel R. Hedemann
We propose and examine several candidates for universal multipartite entanglement measures.  The most promising candidate for applications needing entanglement in the full Hilbert space is the ent-concurrence, which detects all entanglement correlations while distinguishing between different types of distinctly multipartite entanglement, and simplifies to the concurrence for two-qubit mixed states. For applications where subsystems need internal entanglement, we develop the absolute ent-concurrence which detects the entanglement in the reduced states as well as the full state.

Composition of PPT maps (pp0472-0480)
          
Mathew Kennedy, Nicholas A. Manor, and Vern I. Paulsen
M. Christandl conjectured that the composition of any trace preserving PPT map with itself is entanglement breaking. We prove that Christandl's conjecture holds asymptotically by showing that the distance between the iterates of any unital or trace preserving PPT map and the set of entanglement breaking maps tends to zero. Finally, for every graph we define a one-parameter family of maps on matrices and determine the least value of the parameter such that the map is variously, positive, completely positive, PPT and entanglement breaking in terms of properties of the graph. Our estimates are sharp enough to conclude that Christandl's conjecture holds for these families.

Linear bosonic quantum channels defined by superpositions (pp0481-0496)
          
T.J. Volkoff
A minimal energy quantum superposition of two maximally distinguishable, isoenergetic single mode Gaussian states is used to construct the system-environment representation of a class of linear bosonic quantum channels acting on a single bosonic mode. The quantum channels are further defined by unitary dynamics of the system and environment corresponding to either a passive linear optical element $U_{\mathrm{BS}}$ or two-mode squeezing $U_{\mathrm{TM}}$.  The notion of nonclassicality distance is used to show that the initial environment superposition state becomes maximally nonclassical as the constraint energy is increased. When the system is initially prepared in a coherent state, application of the quantum channel defined by $U_{\mathrm{BS}}$ results in a nonclassical state for all values of the environment energy constraint. We also discuss the following properties of the quantum channels: 1) the maximal noise that a coherent system can tolerate, beyond which the linear bosonic attenuator channel defined by $U_{\mathrm{BS}}$ cannot impart nonclassical correlations to the system, 2) the noise added to a coherent system by the phase-preserving linear amplification channel defined by $U_{\mathrm{TM}}$, and 3) a generic lower bound for the trace norm contraction coefficient on the closed, convex hull of energy-constrained Gaussian states.

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