We provide a physical prescription based on interferometry for introducing the total phase of a mixed state undergoing unitary evolution, which has been an elusive concept in the past. We define the parallel transport condition that provides a connection form for obtaining the geometric phase for mixed states. The expression for the geometric phase for mixed state reduces to well known formulas in the pure state case when a system undergoes noncyclic and unitary quantum evolution.

We present a simple device based on the controlled-SWAP gate that performs quantum state tomography. It can also be used to determine maximum and minimum eigenvalues, expectation values of arbitrary observables, purity estimation as well as characterizing quantum channels. The advantage of this scheme is that the architecture is fixed and the task performed is determined by the input data.PACS numbers: 03.67. Hk, 03.67.Lx One of the the key issues in quantum information is, given an unknown quantum system, what can we learn about it. In particular, we are concerned not only with the resources needed (number of identical unknown physical systems), but also with the complexity of quantum operations required (number of different devices, networks, etc.), in order to obtain certain information about a quantum state, characterized by its density matrix ̺. There are many interesting parameters of ̺ we can determine, such as its maximum and minimum eigenvalues, its purity or even ̺ itself (state tomography [1]), but we also can use ̺ to determine expectation values of arbitrary observables or to characterize unknown quantum channels. However, this usually involves building separate devices for each task, or even building different devices for different measurements within the same task.In this paper we present a simple, universal device, whose architecture is fixed but whose behaviour is determined by the choice of input data [2] (see also [3] for a quantum optical realization of a similar idea). In fact, with suitable input, we can directly measure all the properties mentioned before.Consider a typical interferometric set-up for a single qubit: Hadamard gate, phase shift ϕ, Hadamard gate, followed by a measurement in the computational basis. Here and in the following, we borrow terminology from quantum information science and describe quantum interferometry in terms of quantum logic gates [4]. We modify the interferometer by inserting a controlled-U operation between the Hadamard gates, with its control on the qubit and with U acting on a quantum system described by some unknown density operator ρ. We do not assume anything about the form of ρ, it can, for example, describe several entangled or separable subsystems. This set-up is shown in Fig. 1. The action of the controlled-U on ρ modifies the interference pattern by the factor,where v is the new visibility and α is the shift of the interference fringes, also known as the Pancharatnam phase [5]. Thus, the observed visibility gives a straightforward way of estimating the average value of unitary operators U in state ρ and has a variety of interesting applications. For example, it can be used to measure some entanglement witnesses W , as long as they are unitary operators and the corresponding controlled-W operations are easy to implement [6]. Here, we focus on the applications related to quantum state state tomography. Clearly the interferometer in Fig. 1 can be used to estimate any d × d

We describe in detail a general strategy for implementing a conditional geometric phase between two spins. Combined with single-spin operations, this simple operation is a universal gate for quantum computation, in that any unitary transformation can be implemented with arbitrary precision using only single-spin operations and conditional phase shifts. Thus quantum geometrical phases can form the basis of any quantum computation. Moreover, as the induced conditional phase depends only on the geometry of the paths executed by the spins it is resilient to certain types of errors and offers the potential of a naturally fault-tolerant way of performing quantum computation.

We give a simple way of characterising the average fidelity between a unitary and a general operation on a single qubit which only involves calculating the fidelities for a few pure input states

We introduce a model of quantum computation intermediate between the gate-based and measurement-based models. A quantum register is manipulated remotely with the help of a single ancilla that "drives" the evolution of the register. The fully controlled ancilla qubit is coupled to the computational register only via a fixed unitary two-qubit interaction and then measured in suitable bases, driving both single- and two-qubit operations on the register. Arbitrary single- qubit operations directly on register qubits are not needed. We characterize all interactions E that induce a unitary, stepwise deterministic measurement back-action on the register sufficient to implement any quantum channel. Our scheme offers experimental advantages for computation, state preparation, and generalized measurements, since no tunable control of the register is required

Examples of geometric phases abound in many areas of physics. They offer both fundamental insights into many physical phenomena and lead to interesting practical implementations. One of them, as indicated recently, might be an inherently fault-tolerant quantum computation. This, however, requires to deal with geometric phases in the presence of noise and interactions between different physical subsystems. Despite the wealth of literature on the subject of geometric phases very little is known about this very important case. Here we report the first experimental study of geometric phases for mixed quantum states. We show how different they are from the well understood, noiseless, pure-state case.PACS numbers: 03.65. Bz, 42.50.Dv, A quantum system can retain a memory of its motion when it undergoes a cyclic evolution, e.g its quantum state may acquire a geometric phase factor in addition to the dynamical one [1,2]. For pure quantum states this effect is well understood and it has been demonstrated in a wide variety of physical systems [3]. Its potential application to perform the fault-tolerant quantum computation has been the subject of more recent investigations [4,5,6]. In contrast, relatively little is known about geometric phases, and more generally, about quantum holonomies of mixed or entangled quantum states. Here we report an NMR experiment which constitutes the first experimental study of quantum holonomies for mixed quantum states. We observed and measured the geometric phase of a mixed state of a spin half nuclei. Our experimental data are in a good agreement with the recent theoretical predictions by Sjöqvist et al [7].The geometric phase of pure states is an intriguing property of quantum systems undergoing parallel cyclic evolutions. The parallel transport of a particular vector |Ψ implies no change in phase when |Ψ(t) evolves into |Ψ(t + dt) , for some infinitesimal change of the parameter t. Although locally there is no phase change, the system may acquire a non-trivial phase after completing a closed loop parameterized by t. The origin of this phase can be traced to an underlying curvature of the parameter space, depending only on the geometry of the path and is resilient to certain dynamical perturbations of the evolution, e.g. it is independent of the speed of the evolution. Therefore, it is a potential method for performing intrinsically fault-tolerant quantum logic gates, a very desirable feature for practical implementations of quantum computation. However, quantum systems that interact with other systems, be it components in a quantum computer or otherwise, become entangled and cannot be described by a state vector |Ψ . In this context the notion of parallel transport and geometric phases must be extended to mixed quantum states.Mathematically, Uhlmann was the first to address the issue of a mixed state holonomy [8]. In his approach a system in a mixed state is embedded, as a subsystem, in a larger system that is in a pure state. Given a mixed state of the subsystem there are infinitely man...

We generalize the notion of relative phase to completely positive maps with known unitary representation, based on interferometry. Parallel transport conditions that define the geometric phase for such maps are introduced. The interference effect is embodied in a set of interference patterns defined by flipping the environment state in one of the two paths. We show for the qubit that this structure gives rise to interesting additional information about the geometry of the evolution defined by the CP map.Berry's [1] discovery of a geometric phase accompanying cyclic adiabatic evolution has triggered an immense interest in holonomy effects in quantum mechanics and has led to many generalizations. The restriction of adiabaticity was removed by Aharonov and Anandan [2], who pointed out that the geometric phase is due to the curvature of the projective Hilbert space. It was extended to noncyclic evolution by Samuel and Bhandari The geometric phase has shown to be useful in the context of quantum computing as a tool to achieve faulttolerance [8]. For practical implementations of geometric quantum computing, it is important to understand the behavior of the geometric phase in the presence of decoherence. For this, we generalize in this Letter the idea in [7] to completely positive (CP) maps, i.e. we define the relative (Pancharatnam) phase and introduce a notion of parallel transport with concomitant geometric phase for such maps. These generalized concepts reduces to that of [7] in the case of unitary evolutions.Let us first consider a Mach-Zehnder interferometer with a variable relative U (1) phase χ in one of the interferometer beams (the reference beam) and assume that the interfering particles carry an additional internal degree of freedom, such as spin or polarization, in a pure state |k . The other beam (the target beam) is exposed to the unitary operator U i , yielding the output interference pattern I ∝ 1 + ν cos(χ − α), which is completely determined by the complex quantityThus, by varying χ, the relative phase α and visibility ν can be distinguished experimentally. We note that α is a shift in the maximum of the interference pattern, a fact that motivated Pancharatnam [5] to define it as the relative phase between the internal states |k and U i |k of the two beams. Pancharatnam's analysis was generalized in [7] to mixed states undergoing unitary evolution as follows. Assume that the incoming particle is in a mixed internal state ρ = N k=1 w k |k k|, where N is the dimension of the internal Hilbert space. Each pure component |k of this mixture contributes an interference profile given by k|U i |k = ν k e iα k weighted by the its probability w kNoting that Tr(U i ρ) = w k k|U i |k , this can also be written asThe key result is that the interference fringes, produced by varying the phase χ, is shifted by α = arg Tr(U i ρ) and that this shift reduces to Pancharatnam's original prescription for pure states. These two facts are the central properties for α being a natural generalization of Pancharatnam's relative ...

We describe in detail a general strategy for implementing a conditional geometric phase between two spins. Combined with single-spin operations, this simple operation is a universal gate for quantum computation, in that any unitary transformation can be implemented with arbitrary precision using only single-spin operations and conditional phase shifts. Thus quantum geometrical phases can form the basis of any quantum computation. Moreover, as the induced conditional phase depends only on the geometry of the paths executed by the spins it is resilient to certain types of errors and offers the potential of a naturally fault-tolerant way of performing quantum computation.