On quantum operations as quantum states

We formalize Jamiolkowski’s correspondence between quantum states and quantum operations isometrically, and harness its consequences. This correspondence was already implicit in Choi’s proof of the operator sum representation of Completely Positive-preserving linear maps; we go further and show that all of the important theorems concerning quantum operations can be derived directly from those concerning quantum states. As we do so the discussion first provides an elegant and original review of the main features of quantum operations. Next (in the second half of the paper) we find more results stemming from our formulation of the correspondence. Thus, we provide a factorizability condition for quantum operations, and give two novel Schmidt-type decompositions of bipartite pure states. By translating the composition law of quantum operations, we define a group structure upon the set of totally entangled states. The question whether the correspondence is merely mathematical or can be given a physical interpretation is addressed throughout the text: we provide formulae which suggest quantum states inherently define a quantum operation between two of their subsystems, and which turn out to have applications in quantum cryptography.     

Random quantum states

This paper examines the statistical properties of random quantum states, for four different kinds of random state:(1) a pure state chosen at random with respect to the uniform measure on the unit sphere in a finite-dimensional Hilbert space;(2) a random pure state in a real space;(3) a pure state chosen at random except that a certain expectation value is fixed;(4) a random mixed state with fixed eigenvalues. For the first two of these, we give examples of simple states of a model system, the kicked top, which have the statistical properties of random states. Interestingly, examples of both kinds of randomness can be found in the same system. In studying the last two kinds of random state, we obtain new results concerning the application of information theory to quantum systems.     

Catalysis in nonlocal quantum operations

We show how entanglement can be used, without being consumed, to accomplish unitary operations that could not be performed without it. When applied to infinitesimal transformations, our method makes equivalent, in the sense of Hamiltonian simulation, a whole class of otherwise inequivalent two-qubit interactions. The new catalysis effect also implies the asymptotic equivalence of all such interactions.     

Von Neumann entropy-preserving quantum operations

For a given quantum state ρ and two quantum operations Φ and Ψ, the information encoded in the quantum state ρ is quantified by its von Neumann entropy S(ρ). By the famous Choi-Jamio?kowski isomorphism, the quantum operation Φ can be transformed into a bipartite state, the von Neumann entropy Smap(Φ) of the bipartite state describes the decoherence induced by Φ. In this Letter, we characterize not only the pairs (Φ,ρ) which satisfy S(Φ(ρ))=S(ρ), but also the pairs (Φ,Ψ) which satisfy Smap(Φ°Ψ)=Smap(Ψ).     

Random fields and quantum dynamics

We shall discuss quantum mechanical operators depending on the time or on the manifold in space. There is a similarity to the case of stochastic processes or random fields, where the innovation approach is one of the powerful tools to investigate their probabilistic structure. Having had some review of the innovation, similar attempt is made for some cases in quantum dynamics.     

Random quantum interference in microdevices

The conductance of small electronic devices at low temperatures is affected by random quantum interference of a suprisingly universal character that can represent large fractional effects. Would-be designers of small quantum-effect devices should be prepared either to “fix it” by obtaining unprecedented control over the microscopic details of the device structure, or “feature it” by figuring out ways to take advantage of these interference effects.     

Minimum construction of two-qubit quantum operations

Optimal construction of quantum operations is a fundamental problem in the realization of quantum computation. We here introduce a newly discovered quantum gate, B, that can implement any arbitrary two-qubit quantum operation with minimal number of both two- and single-qubit gates. We show this by giving an analytic circuit that implements a generic nonlocal two-qubit operation from just two applications of the B gate. Realization of the B gate is illustrated with an example of charge-coupled superconducting qubits for which the B gate is seen to be generated in shorter time than the CNOT gate.     

Sequential implementation of global quantum operations

We study the possibility for a global unitary applied on an arbitrary number of qubits to be decomposed in a sequential unitary procedure, where an ancillary system is allowed to interact only once with each qubit. We prove that sequential unitary decompositions are in general impossible for genuine entangling operations, even with an infinite-dimensional ancilla, being the controlled-NOT gate a paradigmatic example. Nevertheless, we find particular nontrivial operations in quantum information that can be performed in a sequential unitary manner, as is the case of quantum error correction and quantum cloning.     

Classes of operations in quantum theory

Recent work of Davies and Lewis has shown how partially ordered vector spaces provide a setting in which the operational approach to statistical physical systems may be studied. In this paper, certain physically relevant classes of operations are identified in the abstract framework, some of their properties are derived and applications to the Von Neumann algebra model for quantum theory are discussed.     

Probability-fidelity tradeoffs for targeted quantum operations

We present probability-fidelity tradeoffs for a varying quantum operation with fixed input-output states and for a varying inversion of a fixed quantum operation.     

How to decompose arbitrary continuous-variable quantum operations

We present a general, systematic, and efficient method for decomposing any given exponential operator of bosonic mode operators, describing an arbitrary multimode Hamiltonian evolution, into a set of universal unitary gates. Although our approach is mainly oriented towards continuous-variable quantum computation, it may be used more generally whenever quantum states are to be transformed deterministically, e.g., in quantum control, discrete-variable quantum computation, or Hamiltonian simulation. We illustrate our scheme by presenting decompositions for various nonlinear Hamiltonians including quartic Kerr interactions. Finally, we conclude with two potential experiments utilizing offline-prepared optical cubic states and homodyne detections, in which quantum information is processed optically or in an atomic memory using quadratic light-atom interactions.     

Controlled unknown quantum operations on hybrid systems

Any unknown unitary operations conditioned on a control system can be deterministically performed if ancillary subspaces are available for the target systems [Zhou X Q, et al. 2011 Nat. Commun. 2 413]. In this paper, we show that previous optical schemes may be extended to general hybrid systems if unknown operations are provided by optical instruments. Moreover, a probabilistic scheme is proposed when the unknown operation may be performed on the subspaces of ancillary high-dimensional systems. Furthermore, the unknown operations conditioned on the multi-control system may be reduced to the case with a control system using additional linear circuit complexity. The new schemes may be more flexible for different systems or hybrid systems.     

Random delayed-choice quantum eraser via two-photon imaging

We report on a delayed-choice quantum eraser experiment based on a two-photon imaging scheme using entangled photon pairs. After the detection of a photon which passed through a double-slit, a random delayed choice is made to erase or not erase the which-path information by the measurement of its distant entangled twin; the particle-like and wave-like behavior of the photon are then recorded simultaneously and respectively by only one set of joint detection devices. The present eraser takes advantage of two-photon imaging. The complete which-path information of a photon is transferred to its distant entangled twin through a “ghost" image. The choice is made on the Fourier transform plane of the ghost image between reading “complete information" or “partial information" of the double-path.     

Random sets,q-distributions and quantum groups

We have shown that the non-extensivity of classical set theory is related to unitary quantum groups. Using this non-extensivity property, we define a q-distribution, a binomial q-distribution and a Poisson q-distribution.     

Random quantum correlations and density operator distributions

Randomly correlated ensembles of two quantum systems are investigated, including average entanglement entropies and probability distributions of Schmidt decomposition coefficients. Maximal correlation is guaranteed in the limit as one system becomes infinite dimensional. The reduced density operator distributions are compared with distributions induced via the Bures and Hilbert-Schmidt metrics.     

Random quantum magnets with broad disorder distribution

We study the critical behavior of Ising quantum magnets with broadly distributed random couplings (J), such that P(ln J) ∼ | ln J|^{-1 - α}, α > 1, for large | ln J| (Lévy flight statistics). For sufficiently broad distributions, α < , the critical behavior is controlled by a line of fixed points, where the critical exponents vary with the Lévy index, α. In one dimension, with = 2, we obtained several exact results through a mapping to surviving Riemann walks. In two dimensions the varying critical exponents have been calculated by a numerical implementation of the Ma-Dasgupta-Hu renormalization group method leading to ≈ 4.5. Thus in the region 2 < α < , where the central limit theorem holds for | ln J| the broadness of the distribution is relevant for the 2d quantum Ising model. Received 6 December 2000 and Received in final form 22 January 2001