Random quantum operations

Bruzda, Wojciech; Cappellini, Valerio; Sommers, Hans-Jürgen; Życzkowski, Karol

doi:10.1016/j.physleta.2008.11.043

Cited by 87 publications

(136 citation statements)

References 27 publications

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“…In particular, there exists a Jordan base in the Hilbert space (H ) in which the matrix of the map (1) has a block diagonal form (A1). Spectral properties of general quantum operations were studied in [12]. In the following we formulate several simple characteristic properties which are particularly useful for our subsequent considerations.…”

Section: Basic Properties Of Random Unitary Operationsmentioning

confidence: 99%

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Asymptotic evolution of random unitary operations

Novotný¹,

Alber

Jex³

2010

Open Physics

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Abstract:We analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations. Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic dynamics is described by a diagonalizable superoperator.We prove that this asymptotic dynamics takes place in a typically low dimensional attractor space which is independent of the probability distribution of the unitary operations applied. This vector space is spanned by all eigenvectors of the unitary operations involved which are associated with eigenvalues of unit modulus. Implications for possible asymptotic dynamics of iterated random unitary operations are presented and exemplified in an example involving random controlled-not operations acting on two qubits. PACS

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Section: Basic Properties Of Random Unitary Operationsmentioning

confidence: 99%

“…with the 1 × 1-matrices X (11) X (13) X (31) X (33) , with the 2 × 2 matrix X (22) , with the 1 × 2-matrices X (12) and X (32) , and with the 2×1 matrices X (21) and X (23) . Using Eqs.…”

Section: Asymptotic Dynamics Of a Twoqubit Cnot-systemmentioning

confidence: 99%

Asymptotic evolution of random unitary operations

Novotný¹,

Alber

Jex³

2010

Open Physics

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“…In this way, we define a model of random quantum channels by considering the image measure of the Haar distribution h dd ′ on the set of quantum channels. In the recent preprint [27], the authors study a similar model of random quantum channels, focusing on the spectral properties of the random matrix defining the channel.…”

Section: Non-random Repeated Interactions and A New Model Of Random Dmentioning

confidence: 99%

Random repeated quantum interactions and random invariant states

Nechita

Pellegrini

2010

Probab. Theory Relat. Fields

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Abstract. We consider a generalized model of repeated quantum interactions, where a system H is interacting in a random way with a sequence of independent quantum systems Kn, n 1. Two types of randomness are studied in detail. One is provided by considering Haar-distributed unitaries to describe each interaction between H and Kn. The other involves random quantum states describing each copy Kn. In the limit of a large number of interactions, we present convergence results for the asymptotic state of H. This is achieved by studying spectral properties of (random) quantum channels which guarantee the existence of unique invariant states. Finally this allows to introduce a new physically motivated ensemble of random density matrices called the asymptotic induced ensemble. IntroductionInitially introduced in [2] as a discrete approximation of Langevin dynamics, the model of repeated quantum interactions has found since many applications (quantum trajectories, stochastic control, etc.). In this work we generalize this model by allowing random interactions at each time step. Our main focus is the long-time behavior of the reduced dynamics.Our viewpoint is that of Quantum Open Systems, where a "small" system is in interaction with an inaccessible environment (or an auxiliary system). We are interested in the reduced dynamics of the small system, which is described by the action of quantum channels. When repeating such interactions, under some mild conditions on the spectrum of the quantum channel, we show that the successive states of the small system converge to the invariant density matrix of the channel.These considerations motivated us to consider random invariant states, and we introduce a new probability measure on the set of density matrices. There exists extensive literature [8,28,19,3] on what is a "typical" density matrix. There are two general categories of such probability measures on M 1,+ d (C): measures that come from metrics with statistical significance and the so-called "induced measures", where density matrices are obtained as partial traces of larger, random pure states. Our construction from Section 4 falls into the second category, since our model involves an open system in interaction with a chain of "auxiliary" systems.Next, we introduce two models of random quantum channels. In the first model, we allow for the states of the auxiliary system to be random. In the second one, the unitary matrices acting on the coupled system are assumed random, distributed along the Haar invariant probability on the unitary group, and independent between different interactions. Since the (random) state of the system fluctuates, almost sure convergence does not hold, and we state results in the ergodic sense.2000 Mathematics Subject Classification. Primary 15A52; Secondary 94A40, 60F15.

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“…where the Greek indices refer to the ancilla space and the Latin indices to the physical space (see [27,28] for related works on truncated random unitary operators). For homogeneous MPSs the index k is removed, implying that the unitary interaction is the same for all sites in the spin chain.…”

Section: Random Matrix Product Statesmentioning

confidence: 99%

Statistical properties of random matrix product states

et al. 2010

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We study the set of random matrix product states (RMPS) introduced in arXiv:0908.3877 as a tool to explore foundational aspects of quantum statistical mechanics. In the present work, we provide an accurate numerical and analytical investigation of the properties of RMPS. We calculate the average state of the ensemble in the non-homogeneous case, and numerically check the validity of this result. We also suggest using RMPS as a tool to approximate properties of general quantum random states. The numerical simulations presented here support the accuracy and efficiency of this approximation. These results suggest that any generalized canonical state can be approximated with high probability by the reduced density matrix of a random MPS, if the average MPS coincide with the associated microcanonical ensemble.Comment: 12 pages, 17 figures; published versio

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Random quantum operations

Cited by 87 publications

References 27 publications

Asymptotic evolution of random unitary operations

Asymptotic evolution of random unitary operations

Random repeated quantum interactions and random invariant states

Statistical properties of random matrix product states

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