The space of quantum Hamiltonians has a natural partition in classes of operators that can be adiabatically deformed into each other. We consider parametric families of Hamiltonians acting on a bipartite quantum state space. When the different Hamiltonians in the family fall in the same adiabatic class, one can manipulate entanglement by moving through energy eigenstates corresponding to different values of the control parameters. We introduce an associated notion of adiabatic entangling power. This novel measure is analyzed for general $d\ifmmode\times\else\texttimes\fi{}d$ quantum systems, and specific two-qubit examples are studied
The space of quantum Hamiltonians has a natural partition in classes of operators that can be adiabatically deformed into each other. We consider parametric families of Hamiltonians acting on a bipartite quantum state space. When the different Hamiltonians in the family fall in the same adiabatic class, one can manipulate entanglement by moving through energy eigenstates corresponding to different values of the control parameters. We introduce an associated notion of adiabatic entangling power. This novel measure is analyzed for general d\ifmmode×\else\texttimes\fid quantum systems, and specific two-qubit examples are studied
The change of the entangling power of n fixed bipartite unitary gates, describing interactions, when interlaced with local unitary operators describing monopartite evolutions, is studied as a model of the entangling power of generic Hamiltonian dynamics. A generalization of the local unitary averaged entangling power for arbitrary subsystem dimensions is derived. This quantity shows an exponential saturation to the random matrix theory (RMT) average of the bipartite space, indicating thermalization of quantum gates that could otherwise be very non-generic and have arbitrarily small, but nonzero, entanglement. The rate of approach is determined by the entangling power of the fixed bipartite unitary, which is invariant with respect to local unitaries. The thermalization is also studied numerically via the spectrum of the reshuffled and partially transposed unitary matrices, which is shown to tend to the Girko circle law expected for random Ginibre matrices. As a prelude, the entangling power ep is analyzed along with the gate typicality gt for bipartite unitary gates acting on two qubits and some higher dimensional systems. We study the structure of the set representing all unitaries projected into the plane (ep,gt) and characterize its boundaries which contains distinguished gates including Fourier gate, CNOT and its generalizations, swap and its fractional powers. In this way, a family of gates with extreme properties is identified and analyzed. We remark on the use of these operators as building blocks for many-body quantum systems.
We consider physical Hamiltonians that can be represented by the multiparametric Gaussian ensembles, theoretically derive the state ensembles for its eigenstates and analyze the effect of varying system conditions on its bipartite entanglement entropy. Our approach leads to a single parametric based common mathematical formulation for the evolution of the entanglement statistics of different states of a given Hamiltonian or different Hamiltonians subjected to same symmetry constraints. The parameter turns out to be a single functional of the system parameters and thereby reveals a deep web of connection hidden underneath different quantum states.
The main part of this dissertation starts with a generalization of the entangling power, which quantifies the ability of unitary operators to generate entangled states by their action on the computational basis. The entangling power has been defined for an evenly split bipartite system, whose states live in a Hilbert space of the form H1⊗ H2 where dimH1= dimH2 . I generalize this so that we can consider an odd number of qubits. Entangling power is defined in terms of the linear entropy, a linearization of the von Neumann entropy, whose polynomial form allows one to derive simpler expressions for functions of the entropy. The linear entropy measure is lifted from the state space to the operator space to measure the entanglement of operators. In particular, we focus on the three qubit case, where dimH1 = 2 and dimH2 = 4, as a step to understanding entanglement in many qubit systems. This is the content of Chapter 3.
The goal of Chapter 4 is to explore the structure of entanglement producing operators in SU(2n), with the focus on SU(8). I first review the magic matrix formalism for two-qubit operators, a well known matrix that captures the entangling capability of operators in SU(4), using only 3 of the 15 parameters of the group. I then use a recursive definition of the Cartan decomposition of SU(2n) to construct a magic matrix for operators in SU(8). Because we view the entangling operation as between the first qubit and the other two as a single subsystem, the entanglement is invariant under the action of SU(2) ⊗ SU(4), but SU(8)/SU(2) ⊗ SU(4) is not a symmetric space, and we must use the decomposition in two stages to capture enough entangling operators to account for all of SU(8). The number of parameters appearing in the entangling power is then reduced from 63 to 8, keeping track of 7 entangling operators, one of which does not commute with the others. The next chapter then presents calculations of the entangling powers of operators that generate GHZ states from the computational basis, many of which are solutions to the Yang-Baxter equation and have strong connections to topological quantum computing. We find that the Kauffman-Lomonaco two-qubit gate that produces the Bell states from the computational basis has a maximal entangling power, while the GHZ gate on three qubits does not, adding another piece to the puzzle of entanglement.
We analyze the entangling capabilities of unitary transformations U acting on a bipartite ${(d}_{1}\ifmmode\times\else\texttimes\fi{}{d}_{2})$-dimensional quantum system. To this aim we introduce an entangling power measure e(U) given by the mean linear entropy produced acting with U on a given distribution of pure product states. This measure admits a natural interpretation in terms of quantum operations. For a uniform distribution explicit analytical results are obtained using group-theoretic arguments. The behavior of the features of e(U) as the subsystem dimensions d1 and d2 are varied is studied both analytically and numerically. The two-qubit case d1=d2=2 is argued to be peculiar.
The entangling power of a unitary operator acting on a bipartite Hilbert space measures the entanglement it generates from product states, averaged over the inputs. A finite-dimensional unitary has a spectral decomposition U=∑a=1neiθaPa, where eiθa are the eigenvalues, Pa the corresponding eigen-projectors, and n is the number of distinct eigenvalues. After removing an overall phase, the entangling power is a function on the (n−1)-torus of relative eigenphases at fixed spectral projectors. We prove that this function is stationary at all 2n−1 points on the torus where every relative phase is 0 or π, which we define as \textit{corners}. Up to an overall phase, U at each corner is a generalized reflection R=I−2Q satisfying R2=I, where Q is the sum of spectral projectors whose relative phase is π. At the corner the entangling power is expressed in terms of seven local-unitary invariants of Q. A unitary gate U can be realized as a corner of some projector family if and only if U2∝I, a condition satisfied by many Clifford and non-Clifford gates. We illustrate the theorem with two-qubit gates, SU(N) channel decompositions, and two-site spin chains, obtaining examples of minima, maxima, and saddle points. In addition, a corner that is a saddle point on the full phase torus can appear as a local maximum or minimum along different time-evolution trajectories.
The generation and exploitation of entangled states has become one of the most important technological challenges of the twenty-first century. In order to reliably generate multi-particle entangled states, one needs controllable two (or more) qubit interactions, and there have been many proposals for suitable systems. We have recently begun investigating adiabatic transfer protocols (Greentree et al. 2004), which are distinguished from conventional, nonadiabatic approaches by their robustness and relative insensitivity to noise. Focussing on the phosphorus in silicon paradigm we have shown how such transport leads to new quasi-2D architectures (Hollenberg et al. 2006), and surprising particle coupling schemes based on operator measurements (Devitt et al. 2005; Greentree et al. 2006), which in turn may lead to efficient mechanisms for syndrome identification in quantum error correction. Here we describe the realisation of these operator measurements more fully based on adiabatic passage of a control particle, highlighting the applicability of our protocols by showing how they can improve the scalability of other implementations. Introduction The pursuit of devices which can exploit quantum coherence is widely seen as one of the most important and challenging tasks facing twenty-first Century Technology. Defined by the use of coherent quantum systems (at their simplest: qubits, two-state systems) such devices enable a range of tasks, from quantum key distribution via single qubits (Bennett and Brassard 1984) or entangled states (Ekert 1991), to protocols for quantum teleportation (Bennett et al. 1993), efficient quantum simulation (Feynmann 1982; Abrams and Lloyd 1997) and factorisation (Shor 1994). At the heart of most such protocols is the exploitation of entanglement, where qubits that have been entangled, in some sense feel an interaction even if space-like separated. In fact, entanglement is a ubiquitous feature of quantum systems: the task for quantum information devices is to generate controlled and isolated entanglement, so that only the desired qubits are entangled with each other, and deleterious entanglement to the environment avoided. Broadly speaking, entanglement can be generated in one of two ways, either by direct interactions between qubits, or by collective measurement of two (or more) particles in particular bases that are, by their very nature, entangled. Such projective measurements are called operator measurements, and a flexible mechanism to realise them forms the basis of this paper. Operator measurements are intimately linked to the concept of quantum error correction (QEC) (Nielsen and Chuang 2000), and as QEC is an extremely important task for quantum computing, it is possible that our schemes may prove extremely useful in future quantum computers (QC). Operator measurements are also compatible with the concept of ‘repeat-until-success’ quantum computing (Lim et al. 2005; Barrett and Kok 2005; Lim et al. 2006) which has potential advantages for QC in realistic settings. Our method for generating operator measurements is based on protocols to transfer particles adiabatically around a quantum network. There are now several different protocols for such spatial adiabatic passage, which are all extensions of the original STIRAP (STImulated Raman Adiabatic Passage) (Vitanov et al. 2001) protocol. Spatial adiabatic passage protocols can be mediated via electromagnetic pulses (Brandes et al. 2001; Siewert et al. 2006), or direct control of wavefunction overlap (Greentree et al. 2004; Eckert et al. 2004; Petrosyan 2006). All such schemes share a common theme, namely that a particular eigenstate, which is in the null space of the Hamiltonian, can be smoothly evolved as a function of time. Such protocols are therefore extremely robust, requiring little regard for timing provided that the adiabaticity criterion (Vitanov et al. 2001) is respected. For concreteness, we will review the MRAP protocol (Multiple Recipient Adiabatic Passage) introduced by us earlier (Devitt et al. 2005; Greentree et al. 2006), and show explicitly how it can be used to mediate operator measurements. We will then comment on some of the implications of this protocol for QC designs, especially when concepts such as defect-tolerance are introduced. Multiple-Recipient Adiabatic Passage Consider a network, defined by four quantum dots that share a single electron, and two attached qubits, as depicted in Fig. 1(a). Employing quantum communication terminology, we suppose that Alice wishes to send a particle (with information encoded in the electronic spin states) to either Bob 1 or Bob 2, or to some arbitrary superposition of the Bobs, Australian Institute of Physics 17th National Congress 2006 – Brisbane, 3-8 December 2006 RiverPhys Paper No. 474 using the network. For clarity we refer to the particle transferred through the network as the bus particle. We label the quantum dots, A (Alice), 1 B (Bob 1), 2 B (Bob 2), C (Central). Also, the Bobs have qubits 1 Q and 2 Q which will be used later. We further assume that each participant has control of the coupling of their quantum dot to the central dot, and we label this coupling Ωα for α = A, B1, B2. With all dots maintained at constant energy, the Hamiltonian for this four dot system (ignoring the spin degree of freedom) is ( ) ( ) ( ) . . 2 2 1 1 c h C B t C B t A C t H B B A + Ω + Ω + Ω = where the couplings are allowed to vary as a function of time (for example by local control of surface gates to modify the barrier potential). The null space of the Hamiltonian above is a two-dimensional space, spanned by the vectors