results for au:Arvind in:quant-ph

- Oct 04 2017 quant-ph physics.optics arXiv:1710.01086v1Entanglement of states and Complete Positivity of maps are concepts that have achieved physical importance with the recent growth of quantum information science. They are however mathematically relevant whenever tensor products of complex linear (Hilbert) spaces are involved. We present such situations in classical scalar paraxial wave optics where these concepts play a role: propagation characteristics of coherent and partially coherent Gaussian beams; and the definition and separability of the family of Twisted Gaussian Schell Model (TGSM) beams. In the former, the evolution of the width of a projected one-dimensional beam is shown to be a signature of entanglement in a two-dimensional amplitude. In the latter, the partial transpose operation is seen to explain key properties of TGSM beams.
- Jul 24 2017 quant-ph arXiv:1707.06739v1We present the first NMR implementation of a scheme for selective and efficient quantum process tomography using local measurements (LMSEQPT) without ancilla. We generalize this scheme such that it can be implemented efficiently using only a set of local measurements involving product operators. The method allows us to estimate any element of the quantum process matrix to a desired precision, provided a set of quantum states can be prepared efficiently. As opposed to the standard implementation of selective and efficient quantum process tomography (SEQPT) which involves the preparation and complete characterization of a large number of quantum states, our local measurement-based technique requires much fewer experimental resources as it avoids complete characterization of the states and allows us to compute specific elements of the process matrix by a restrictive set of sub-system measurements. To demonstrate the efficacy of our scheme, we experimentally tomograph the processes corresponding to `no operation', a controlled-NOT (CNOT), and a controlled-Hadamard gate on a two-qubit NMR quantum information processor, with high fidelities.
- Jul 04 2017 quant-ph arXiv:1707.00289v1We have designed efficient quantum circuits for the three-qubit Toffoli (controlled-controlled NOT) and the Fredkin (controlled-SWAP) gate, optimized via genetic programming methods. The gates thus obtained were experimentally implemented on a three-qubit NMR quantum information processor, with a high fidelity. Toffoli and Fredkin gates in conjunction with the single-qubit Hadamard gates form a universal gate set for quantum computing, and are an essential component of several quantum algorithms. Genetic algorithms are stochastic search algorithms based on the logic of natural selection and biological genetics and have been widely used for quantum information processing applications. The numerically optimized rf pulse profiles of the three-qubit quantum gates achieve $> 99\%$ fidelity. The optimization was performed under the constraint that the experimentally implemented pulses are of short duration and can be implemented with high fidelity. Therefore the gate implementations do not suffer from the drawbacks of rf offset errors or debilitating effects of decoherence during gate action. We demonstrate the advantage of our pulse sequences by comparing our results with existing experimental schemes.
- May 10 2017 quant-ph arXiv:1705.03432v1We embarked upon the task of experimental protection of different classes of tripartite entangled states, namely the maximally entangled GHZ and W states and the ${\rm W \bar{W}}$ state, using dynamical decoupling. The states were created on a three-qubit NMR quantum information processor and allowed to evolve in the naturally noisy NMR environment. Tripartite entanglement was monitored at each time instant during state evolution, using negativity as an entanglement measure. It was found that the W state is most robust while the GHZ-type states are most fragile against the natural decoherence present in the NMR system. The ${\rm W \bar{W}}$ state which is in the GHZ-class, yet stores entanglement in a manner akin to the W state, surprisingly turned out to be more robust than the GHZ state. The experimental data were best modeled by considering the main noise channel to be an uncorrelated phase damping channel acting independently on each qubit, alongwith a generalized amplitude damping channel. Using dynamical decoupling, we were able to achieve a significant protection of entanglement for GHZ states. There was a marginal improvement in the state fidelity for the W state (which is already robust against natural system decoherence), while the ${\rm W \bar{W}}$ state showed a significant improvement in fidelity and protection against decoherence.
- Mar 20 2017 quant-ph arXiv:1703.06102v1We use the Majorana geometrical representation for a qutrit, where a pair of points on a unit sphere represents its quantum states. A canonical form for qutrit states is presented, where every qutrit state can be obtained from a one-parameter family of states via $SO(3)$ action. The notion of spin-1 magnetization which is invariant under $SO(3)$, is discussed and geometrically interpreted on the Majorana sphere. We study the dynamics of the pair of points representing a qutrit state under various useful quantum operations and connect them to different NMR operations. In our scheme, the rigid rotation of this pair of points under $SO(3)$ corresponds to non-selective pulses in NMR, while transition-selective NMR pulses are generated by the eight generators of $SU(3)$ and typically lead to the movement of only one point at a time on the Majorana sphere. Using the Majorana picture, we also describe the action of several quantum gates and the insights gained, are used to perform experiments on a spin-1 system (the NMR qutrit) oriented in a liquid crystalline environment. Finally, using the Gell Mann matrix picture we describe and experimentally implement a scheme for complete qutrit state tomography.
- Jan 31 2017 quant-ph arXiv:1701.08260v2We experimentally demonstrate the efficacy of a three-layer nested Uhrig dynamical decoupling (NUDD) sequence to preserve arbitrary quantum states in a two-dimensional subspace of the four-dimensional two-qubit Hilbert space, on an NMR quantum information processor. The effect of the state preservation is studied first on four known states, including two product states and two maximally entangled Bell states. Next, to evaluate the preservation capacity of the NUDD scheme, we apply it to eight randomly generated states in the subspace. Although, the preservation of different states varies, the scheme on the average performs very well. The complete tomographs of the states at different time points are used to compute fidelity. State fidelities using NUDD protection are compared with those obtained without using any protection. The nested pulse schemes are complex in nature and require careful experimental implementation.
- Jan 04 2017 quant-ph arXiv:1701.00606v2A bipartite quantum system in a mixed state can exhibit nonclassical correlations, which can go beyond quantum entanglement. While quantum discord is the standard measure of quantifying such general quantum correlations, the nonclassicality can be determined by simpler means via the measurement of witness operators. We experimentally construct a positive map to witness nonclassicality of two qubits in an NMR system. The map can be decomposed so that a single run of an experiment on an ensemble of spins suffices to detect the nonclassicality in the state, if present. We let the state evolve in time and use the map to detect nonclassicality as a function of time. To evaluate the efficacy of the witness operator as a means to detect nonclassicality, we measure quantum discord by performing full quantum state tomography at each time point and obtained a fairly good match between the two methods.
- Dec 09 2016 quant-ph arXiv:1612.02616v3The security of quantum key distribution protocols hinges upon features of physical systems that are uniquely quantum in nature. We explore the role of quantumness as qualified by quantum contextuality, in quantum key distribution schemes. We present a new quantum key distribution protocol whose security depends uponthe KCBS inequality and contextuality monogamy. We explicitly calculate the key rate, error rate introduced between Alice-Bob communication due to the presence of Eve, and the information-gain by Eve. This protocol provides a new framework for quantum key distribution which has conceptual and practical advantages over other quantum protocols.
- Nov 28 2016 quant-ph arXiv:1611.08071v2We examine a recent proposal to show the presence of nonlocal Pancharatnam type geometric phases in a quantum mechanical treatment of intensity interferometry measurements upon inclusion of polarizing elements in the setup. It is shown that a completely classical statistical treatment of such effects is adequate for practical purposes. Further we show that the phase angles that appear in the correlations, while at first sight appearing to resemble Pancharatnam phases in their mathematical structure, cannot actually be interpreted in that manner. We also describe a simpler Mach-Zehnder type setup where similar effects can be observed without use of the paraxial approximation.
- Oct 11 2016 quant-ph arXiv:1610.02472v1Random local measurements have recently been proposed to construct entanglement witnesses and thereby detect the presence of bipartite entanglement. We experimentally demonstrate the efficacy of one such scheme on a two-qubit NMR quantum information processor. We show that a set of three random local measurements suffices to detect the entanglement of a general two-qubit state. We experimentally generate states with different amounts of entanglement, and show that the scheme is able to clearly witness entanglement. We perform complete quantum state tomography for each state and compute state fidelity to validate our results. Further, we extend previous results and perform a simulation using random local measurements to optimally detect bipartite entanglement in a hybrid system of $2\otimes 3$ dimensionality.
- Oct 11 2016 quant-ph arXiv:1610.02755v1The discovery of the intriguing phenomenon that certain kinds of quantum correlations remain impervious to noise up to a specific point in time and then suddenly decay, has generated immense recent interest. We exploit dynamical decoupling sequences to prolong the persistence of time-invariant quantum discord in a system of two NMR qubits decohering in independent dephasing environments. We experimentally prepare two-qubit Bell-diagonal quantum states that interact with individual dephasing channels and demonstrate the effect of dynamical decoupling on the preservation of both quantum and classical correlations. We are able to freeze quantum discord over long time scales in the presence of noise, using dynamical decoupling. We use robust state-independent dynamical decoupling schemes for state preservation and demonstrate that these schemes are able to successfully preserve quantum discord.
- Jul 26 2016 quant-ph arXiv:1607.07396v1We study the dissipative dynamics of a wave packet passing through two different nonlinear media. The effect of dissipation on the known phenomenon of collapses and revivals of a wave packet as it evolves in a Kerr type nonlinear medium represented by the Hamiltonian $({a}^{\dag} a)^2$ is investigated. We find that partial revivals do take place when dissipation values are moderate. We consider the next order nonlinearity represented by the Hamiltonian $({a}^{\dag} a)^3$ where we observe the phenomena of super revivals. The effect of dissipation in this case has an additional feature of number dependence for the case of displaced number states. Overall our simulations show the robustness of the phenomena of collapses and revivals in a dissipative environment.
- Jul 14 2016 quant-ph arXiv:1607.03498v1We propose a non-contextual hidden variable model, consistent with all predictions of quantum mechanics (QM). A careful scrutiny of consistency requirements of any hidden variable model with quantum mechanics and the corresponding no-go theorems, leads us to the conclusion that the notion of contextuality is not a necessary feature of QM. The alternative view that emerges, hinges on identifying a new classical property which we call "multiplicativity". It turns out that by relaxing this condition of "multiplicativity" on hidden variable models, they can be made consistent with QM. The connection of non-multiplicativity to Bohm's model and to apparent contextuality is also discussed.
- Apr 19 2016 quant-ph arXiv:1604.04691v1Estimation of quantum states is one of the most important steps in any quantum information processing experiment. A naive reconstruction of the density matrix from experimental measurements can often give density matrices which are not positive, and hence not physically acceptable. How do we ensure that at all stages of reconstruction, we keep the density matrix positive and normalized? Recently a method has been suggested based on maximum likelihood estimation, wherein the density matrix is guaranteed to be positive definite. We experimentally implement this protocol and demonstrate its utility on an NMR quantum information processor. We discuss several examples where we undertake such an estimation and compare it with the standard method of state estimation.
- Jan 11 2016 quant-ph arXiv:1601.01936v2We present a scheme to estimate Gaussian states of one-dimensional continuous variable systems, based on weak (unsharp) quantum measurements. The estimation of a Gaussian state requires us to find position ($q$), momentum ($p$) and their second order moments. We measure $q$ weakly and follow it up with a projective measurement of $p$ on half of the ensemble, and on the other half we measure $p$ weakly followed by a projective measurement of $q$. In each case we use the state twice before discarding it. We compare our results with projective measurements and demonstrate that under certain conditions such weak measurement-based estimation schemes, where recycling of the states is possible, can outperform projective measurement-based state estimation schemes.
- Nov 03 2015 quant-ph arXiv:1511.00241v1We experimentally test quantum contextuality of a single qutrit using NMR. The contextuality inequalities based on nine observables developed by Kurzynski et. al. are first reformulated in terms of traceless observables which can be measured in an NMR experiment. These inequalities reveal the contextuality of almost all single-qutrit states. We demonstrate the violation of the inequality on four different initial states of a spin-1 deuterium nucleus oriented in a liquid crystal matrix, and follow the violation as the states evolve in time. We also describe and experimentally perform a single-shot test of contextuality for a subclass of qutrit states whose density matrix is diagonal in the energy basis.
- May 14 2015 quant-ph arXiv:1505.03266v2We explore the possibility of using "weak measurements" without "weak value" for quantum state estimation. Since for weak measurements the disturbance caused during each measurement is small, we can rescue the state, unlike for the case of projective measurements. We use this property of weak measurements and design schemes for quantum state estimation for qubits and for Gaussian states. We show, via numerical simulations, that under certain circumstances, our method can outperform the estimation by projective measurements both for qubits and for Gaussian states. It turns out that ensemble size plays an important role and the scheme based on recycling works better for small ensembles.
- Apr 21 2015 quant-ph arXiv:1504.04856v1We experimentally construct a novel three-qubit entangled W-superposition ($\rm W \bar{\rm W}$) state on an NMR quantum information processor. We give a measurement-based filtration protocol for the invertible local operation (ILO) that converts the $\rm W \bar{\rm W}$ state to the GHZ state, using a register of three ancilla qubits. Further we implement an experimental protocol to reconstruct full information about the three-party $\rm W \bar{\rm W}$ state using only two-party reduced density matrices. An intriguing fact unearthed recently is that the $\rm W \bar{\rm W}$ state which is equivalent to the GHZ state under ILO, is in fact reconstructible from its two-party reduced density matrices, unlike the GHZ state. We hence demonstrate that although the $\rm W \bar{\rm W}$ state is interconvertible with the GHZ state, it stores entanglement very differently.
- Apr 14 2015 quant-ph arXiv:1504.02991v1We introduce local filters as a means to detect the entanglement of bound entangled states which do not yield to detection by witnesses based on positive (P) maps which are not completely positive (CP). We demonstrate how such non-detectable bound entangled states can be locally filtered into detectable bound entangled states. Specifically, we show that a bound entangled state in the orthogonal complement of the unextendible product bases (UPB), can be locally filtered into a another bound entangled state that is detectable by the Choi map. We reinterpret these filters as local measurements on locally extended Hilbert spaces. We give explicit constructions of a measurement-based implementation of these filters for $2 \otimes 2$ and $3 \otimes 3$ systems. This provides us with a physical mechanism to implement such local filters.
- Mar 24 2015 quant-ph arXiv:1503.06624v1The quantum Fourier transform (QFT) is a key ingredient of several quantum algorithms and a qudit-specific implementation of the QFT is hence an important step toward the realization of qudit-based quantum computers. This work develops a circuit decomposition of the QFT for hybrid qudits based on generalized Hadamard and generalized controlled-phase gates, which can be implemented using selective rotations in NMR. We experimentally implement the hybrid qudit QFT on an NMR quantum emulator, which uses four qubits to emulate a single qutrit coupled to two qubits.
- We prove that the projection on a completely entangled subspace S of maximum dimension in a multipartite quantum system obtained by Parthasarathy is not positive under partial transpose. We next show that several positive operators with range in S also have the same property. In this process we construct an orthonormal basis of S and provide a linking theorem to link the constructions of completely entangled subspaces due to Parthasarthy, Bhat and Johnston.
- Sep 09 2014 quant-ph arXiv:1409.2419v2We experimentally demonstrate the freezing of evolution of quantum states in one- and two-dimensional subspaces of two qubits, on an NMR quantum information processor. State evolution was frozen and leakage of the state from its subspace to an orthogonal subspace was successfully prevented using super-Zeno sequences, comprising of a set of inverting radio frequency (rf) pulses punctuated by pre-selected time intervals. We demonstrate the efficacy of the scheme by preserving different types of states, including separable and maximally entangled states in one- and two-dimensional subspaces of two qubits. The change in the experimental density matrices was tracked by carrying out full state tomography at several time points. We use the fidelity measure for the one-dimensional case and the leakage (fraction) into the orthogonal subspace for the two-dimensional case, as qualitative indicators to estimate the resemblance of the density matrix at a later time to the initially prepared density matrix. For the case of entangled states, we additionally compute an entanglement parameter to indicate the presence of entanglement in the state at different times. We experimentally demonstrate that the super-Zeno scheme is able to successfully confine state evolution to the one- or two-dimensional subspace being protected.
- Jul 24 2014 quant-ph arXiv:1407.6323v2Quantum Private Comparison (QPC) allows us to protect private information during its comparison. In the past various three-party quantum protocols have been proposed that claim to work well under noisy conditions. Here we tackle the problem of QPC under noise. We analyze the EPR-based protocol under depolarizing noise, bit flip and phase flip noise. We show how noise affects the robustness of the EPR-based protocol. We then present a straightforward protocol based on CSS codes to perform QPC which is robust against noise and secure under general attacks.
- Jul 15 2014 quant-ph arXiv:1407.3448v2We experimentally explore the state space of three qubits on an NMR quantum information processor. We construct a scheme to experimentally realize a canonical form for general three-qubit states up to single-qubit unitaries. This form involves a non-trivial combination of GHZ and W-type maximally entangled states of three qubits. The general circuit that we have constructed for the generic state reduces to those for GHZ and W states as special cases. The experimental construction of a generic state is carried out for a nontrivial set of parameters and the good fidelity of preparation is confirmed by complete state tomography. The GHZ and W-states are constructed as special cases of the general experimental scheme. Further, we experimentally demonstrate a curious fact about three-qubit states, where for almost all pure states, the two-qubit reduced states can be used to reconstruct the full three-qubit state. For the case of a generic state and for the W-state, we demonstrate this method of reconstruction by comparing it with the directly tomographed three-qubit state.
- Jun 20 2014 quant-ph arXiv:1406.5026v1We present the NMR implementation of a recently proposed quantum algorithm to find the parity of a permutation. In the usual qubit model of quantum computation, speedup requires the presence of entanglement and thus cannot be achieved by a single qubit. On the other hand, a qutrit is qualitatively more quantum than a qubit because of the existence of quantum contextuality and a single qutrit can be used for computing. We use the deuterium nucleus oriented in a liquid crystal as the experimental qutrit. This is the first experimental exploitation of a single qutrit to carry out a computational task.
- Feb 04 2014 quant-ph arXiv:1402.0447v3We explore the possibility of using "weak" measurements to carry out quantum state tomography. Given a certain fixed number of copies of identically prepared states of a qubit, we simulate state tomography using weak as well as projective measurements. Due to the collapse of the state after measurement, we cannot re-use the state after a projective measurement. If the coupling strength between the quantum system and the measurement device is made weaker, the disturbance caused to the state can be lowered. This then allows us to re-use the same member of the ensemble for further measurements and thus extract more information from the system. However, this happens at the expense of getting imprecise information from the first measurement. We simulate this scheme for a single qubit and show that under certain circumstances, it can outperform the projective measurement-based tomography scheme. This opens up the possibility of new ways of extracting information from quantum ensembles. We study the efficacy of this scheme for different coupling strengths.
- Radiative energy and momentum transfer due to fluctuations of electromagnetic fields arising due to temperature difference between objects is described in terms of the cross-spectral densities of the electromagnetic fields. We derive relations between thermal non-equilibrium contributions to energy and momentum transfer and surface integrals of tangential components of the dyadic Green's functions of the vector Helmholtz equation. The expressions derived here are applicable to objects of arbitrary shapes, dielectric functions, as well as magnetic permeabilities. For the case of radiative transfer, we derive expressions for the generalized transmissivity and generalized conductance that are shown to obey reciprocity and agree with theory of black body radiative transfer in the appropriate limit.
- Nov 15 2012 quant-ph arXiv:1211.3294v1In this paper we describe a new connection between UPB (unextendable product bases) and P (positive) maps which are not CP (completely positive). We show that inner automorphisms of the set of P maps which are not CP, produce extremal extensions of these maps that help in entanglement detection. By constructing such an extension of the well-known Choi map, we strengthen its power to unearth PPT (positive under partial transpose) entangled states. We further show that the class of maps generated from the Choi map via an inner automorphism naturally detects the entanglement of states in the orthogonal complement of certain UPB. This brings out a hitherto undiscovered connection between the Choi map and UPB. We also show that certain other recently considered extremal extensions are obtainable by such extensions of the Choi map.
- Jun 22 2011 quant-ph arXiv:1106.4279v1In this paper, we discuss extremal extensions of entanglement witnesses based on Choi's map. The constructions are based on a generalization of the Choi map due to Osaka, from which we construct entanglement witnesses. These extremal extensions are powerful in terms of their capacity to detect entanglement of positive under partial transpose (PPT) entangled states and lead to unearthing of entanglement of new PPT states. We also use the Cholesky-like decomposition to construct entangled states which are revealed by these extremal entanglement witnesses.
- We derive a first--principles method of determining the van der Waals or Casimir pressure in a dissipative and dispersive planar multilayered system by calculating the Maxwell stress tensor in a fictitious layer of vacuum, that is eventually made to vanish, introduced in the structure. This is illustrated by calculating the van der Waals pressure in a thin film with dissipative properties embedded between two semi--infinite media.
- Nov 07 2006 quant-ph arXiv:quant-ph/0611058v2We analyze the class of single qubit channels with the environment modeled by a one-qubit mixed state. The set of affine transformations for this class of channels is computed analytically, employing the canonical form for the two-qubit unitary operator. We demonstrate that, 3/8 of the generalized depolarizing channels can be simulated by the one-qubit mixed state environment by explicitly obtaining the shape of the volume occupied by this class of channels within the tetrahedron representing the generalized depolarizing channels. Further, as a special case, we show that the two-Pauli Channel cannot be simulated by a one-qubit mixed state environment.
- May 11 2006 quant-ph arXiv:quant-ph/0605092v2We describe a new implementation of the Bernstein-Vazirani algorithm which relies on the fact that the polarization states of classical light beams can be cloned. We explore the possibility of computing with waves and discuss a classical optical model capable of implementing any algorithm (on $n$ qubits) that does not involve entanglement. The Bernstein-Vazirani algorithm (with a suitably modified oracle), wherein a hidden $n$ bit vector is discovered by one oracle query as against $n$ oracle queries required classically, belongs to this category. In our scheme, the modified oracle is also capable of computing $f(x)$ for a given $x$, which is not possible with earlier versions used in recent NMR and optics implementations of the algorithm.
- Jan 11 2006 quant-ph arXiv:quant-ph/0601059v1The well known Poisson Summation Formula is analysed from the perspective of the coherent state systems associated with the Heisenberg--Weyl group. In particular, it is shown that the Poisson summation formula may be viewed abstractly as a relation between two sets of bases (Zak bases) arising as simultaneous eigenvectors of two commuting unitary operators in which geometric phase plays a key role. The Zak bases are shown to be interpretable as generalised coherent state systems of the Heisenberg--Weyl group and this, in turn, prompts analysis of the sampling theorem (an important and useful consequence of the Poisson Summation Formula) and its extension from a coherent state point of view leading to interesting results on properties of von Neumann and finer lattices based on standard and generalised coherent state systems.
- Jul 22 2003 quant-ph arXiv:quant-ph/0307153v2We discuss the ensemble version of the Deutsch-Jozsa (DJ) algorithm which attempts to provide a "scalable" implementation on an expectation-value NMR quantum computer. We show that this ensemble implementation of the DJ algorithm is at best as efficient as the classical random algorithm. As soon as any attempt is made to classify all possible functions with certainty, the implementation requires an exponentially large number of molecules. The discrepancies arise out of the interpretation of mixed state density matrices.
- May 05 2003 quant-ph arXiv:quant-ph/0305012v2We consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically generalised to accommodate new features which arise when the configuration space changes from $n$-dimensional Euclidean space ${\cal R}^n$ to a Lie group $G$. The notion of canonical momentum is carefully analysed, and the meanings of marginal probability distributions and their recovery from the Wigner distribution are clarified. For the case of compact $G$ an explicit definition of the Wigner distribution is proposed, possessing all the required properties. Geodesic curves in $G$ which help introduce a notion of the `mid point' of two group elements play a central role in the construction.
- Dec 10 2002 quant-ph arXiv:quant-ph/0212048v1We first give an $\O(2^{n/3})$ quantum algorithm for the 0-1 Knapsack problem with $n$ variables. More generally, for 0-1 Integer Linear Programs with $n$ variables and $d$ inequalities we give an $\O(2^{n/3}n^d)$ quantum algorithm. For $d =o(n/\log n)$ this running time is bounded by $\O(2^{n(1/3+\epsilon)})$ for every $\epsilon>0$ and in particular it is better than the $\O(2^{n/2})$ upper bound for general quantum search. To investigate whether better algorithms for these NP-hard problems are possible, we formulate a \emphsymmetric claw problem corresponding to 0-1 Knapsack and study its quantum query complexity. For the symmetric claw problem we establish a lower bound of $\O(2^{n/4})$ for its quantum query complexity. We have an $\O(2^{n/3})$ upper bound given by essentially the same quantum algorithm that works for Knapsack. Additionally, we consider CNF satisfiability of CNF formulas $F$ with no restrictions on clause size, but with the number of clauses in $F$ bounded by $cn$ for a constant $c$, where $n$ is the number of variables. We give a $2^{(1-\alpha)n/2}$ quantum algorithm for satisfiability in this case, where $\alpha$ is a constant depending on $c$.
- Oct 15 2002 quant-ph arXiv:quant-ph/0210097v1This paper is motivated by the computer-generated nonadditive ((5,6,2)) code described in an article by Rains, Hardin, Shor and Sloane. We describe a theory of non-stabilizer codes of which the nonadditive code of Rains et al is an example. Furthermore, we give a general strategy of constructing good nonstabilizer codes from good stabilizer codes and give some explicit constructions and asymptotically good nonstabilizer codes. In fact, we explicitly construct a family of distance 2 non-stabilizer codes over all finite fields of which the ((5,6,2)) is an special example. More interestingly, using our theory, we are also able to explicitly construct examples of non-stablizer quantum codes of distance 3. Like in the case of stabilizer codes, we can design fairly efficient encoding and decoding procedures.
- Jun 26 2002 quant-ph arXiv:quant-ph/0206174v1Using the Weyl commutation relations over a finite field we introduce a family of error-correcting quantum stabilizer codes based on a class of symmetric matrices over the finite field satisfying certain natural conditions. When the field is GF(2) the existence of a rich class of such symmetric matrices is demonstrated by a simple probabilistic argument depending on the Chernoff bound for i.i.d symmetric Bernoulli trials. If, in addition, these symmetric matrices are assumed to be circulant it is possible to obtain concrete examples by a computer program. The quantum codes thus obtained admit elegant encoding circuits.
- Jan 07 2002 quant-ph arXiv:quant-ph/0201009v1The U(2) invariant approach is delineated for the pair coherent states to explore their squeezing properties. This approach is useful for a complete analysis of the squeezing properties of these two-mode states. We use the maximally compact subgroup U(2) of Sp(4,R) to mix the modes, thus allowing us to search over all possible quadratures for squeezing. The variance matrix for the pair coherent states turns out to be analytically diagonalisable, giving us a handle over its least eigenvalue, through which we are able to pin down the squeezing properties of these states. In order to explicitly demonstrate the role played by U(2) transformations, we connect our results to the previous analysis of squeezing for the pair coherent states.
- This paper initiates a program which seeks to study the allowed spatial distributions of negative energy density in quantum field theory. Here we deal with free fields in Minkowski spacetime. Known restrictions on time integrals of the energy density along geodesics, the averaged weak energy condition and quantum inequalities are reviewed. These restrictions are then used to discuss some possible constraints on the allowable spatial distributions of negative energy. We show how some geometric configurations can either be ruled out or else constrained. We also construct some explicit examples of allowed distributions. Several issues related to the allowable spatial distributions are also discussed. These include spacetime averaged quantum inequalities in two-dimensional spacetime, the failure of generalizations of the averaged weak energy condition to piecewise geodesics, and the issue of when the local energy density is negative in the frame of all observers.
- Jul 03 2001 quant-ph arXiv:quant-ph/0107006v1We investigate the geometric phases and the Bargmann invariants associated with a multi-level quantum systems. In particular, we show that a full set of `gauge-invariant' objects for an $n$-level system consists of $n$ geometric phases and ${1/2}(n-1)(n-2)$ algebraically independent 4-vertex Bargmann invariants. In the process of establishing this result we develop a canonical form for U(n) matrices which is useful in its own right. We show that the recently discovered `off-diagonal' geometric phases [N. Manini and F. Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analysed in terms of the basic building blocks developed in this work. This result liberates the off-diagonal phases from the assumption of adiabaticity used in arriving at them.
- Dec 22 2000 quant-ph arXiv:quant-ph/0012116v1The existence of entangled quantum states gives extra power to quantum computers over their classical counterparts. Quantum entanglement shows up qualitatively at the level of two qubits. We show that if no entanglement is envolved then whatever one can do with qubits can also be done with classical optical systems. We demonstrate that the one- and the two-bit Deutsch-Jozsa algorithm does not require entanglement and can be mapped onto a classical optical scheme. It is only for three and more input bits that the DJ algorithm requires the implementation of entangling transformations and in these cases it is impossible to implement this algorithm classically.
- Jun 29 2000 quant-ph arXiv:quant-ph/0006123v1Quantum computing using two-dimensional NMR has recently been described using scalar coupling evolution technique [J. Chem. Phys.,109,10603 (1998)]. In the present paper, we describe two-dimensional NMR quantum computing with the help of selective pulses. A number of logic gates are implemented using two and three qubits with one extra observer spin. Some many-in-one gates (or Portmanteau gates) are implemented. Toffoli gate (or AND/NAND gate) and OR/NOR gates are implemented on three qubits. Deutsch-Jozsa quantum algorithm for one and two qubits, using one extra work qubit, has also been implemented using selective pulses after creating a coherent superposition state, in the two-dimensional methodology.
- Jun 23 2000 quant-ph arXiv:quant-ph/0006103v1We describe the experimental implementation of a recently proposed quantum algorithm involving quantum entanglement at the level of two qubits using NMR. The algorithm solves a generalisation of the Deutsch problem and distinguishes between even and odd functions using fewer function calls than is possible classically. The manipulation of entangled states of the two qubits is essential here, unlike the Deutsch-Jozsa algorithm and the Grover's search algorithm for two bits.
- Jun 15 2000 quant-ph arXiv:quant-ph/0006069v1The phenomenon of quantum entanglement is fundamental to the implementation of quantum computation, and requires at least two qubits for its demonstration. However, both Deutsch algorithm and Grover's search algorithm for two bits do not use entanglement. We develop a Deutsch-like problem, where we consider all possible binary functions for two bit inputs and distinguish their even or odd nature. The quantum algorithm to solve this problem requires entanglement at the level of two qubits. The final solution suggests that an NMR implementation of the problem would lead to interesting results.
- Feb 25 2000 quant-ph arXiv:quant-ph/0002070v1We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with an unitary irreducible representation of a (compact) Lie group. We show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining Clebsch-Gordan theory and the reciprocity theorems associated with induced unitary group representation. Applications to several examples involving $SU(2),$ $SU(3),$ and the Heisenberg-Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase-space description of quantum mechanics and quantum state reconstruction problems.
- Sep 23 1999 quant-ph arXiv:quant-ph/9909067v1A scheme to execute an n-bit Deutsch-Jozsa (D-J) algorithm using n qubits has been implemented for up to three qubits on an NMR quantum computer. For the one and two bit Deutsch problem, the qubits do not get entangled, hence the NMR implementation is achieved without using spin-spin interactions. It is for the three bit case, that the manipulation of entangled states becomes essential. The interactions through scalar J-couplings in NMR spin systems have been exploited to implement entangling transformations required for the three bit D-J algorithm.
- Jul 27 1999 quant-ph arXiv:quant-ph/9907080v1We develop the widest possible generalisation of the well-known connection between quantum mechanical Bargmann invariants and geometric phases. The key notion is that of null phase curves in quantum mechanical ray and Hilbert spaces. Examples of such curves are developed. Our generalisation is shown to be essential to properly understand geometric phase results in the cases of coherent states and of Gaussian states. Differential geometric aspects of null phase curves are also briefly explored.
- Jun 10 1999 quant-ph arXiv:quant-ph/9906027v2We demonstrate experimentally the usefulness of selective pulses in NMR to perform quantum computation. Three different techniques based on selective pulse excitations have been proposed to prepare a spin system in a pseudo-pure state. We describe the design of novel ``portmanteau'' gates using the selective manipulation of level populations. A selective pulse implementation of the Deutsch-Jozsa algorithm for a two-qubit and a three-qubit quantum computer is demonstrated.
- Jun 26 1998 quant-ph arXiv:quant-ph/9806085v1The connection between quantum optical nonclassicality and the violation of Bell's inequalities is explored. Bell type inequalities for the electromagnetic field are formulated for general states(arbitrary number or photons, pure or mixed) of quantised radiation and their violation is connected to other nonclassical properties of the field. Classical states are shown to obey these inequalities and for the family of centered Gaussian states the direct connection between violation of Bell-type inequalities and squeezing is established.
- Jun 16 1998 quant-ph arXiv:quant-ph/9806049v1The connection between quantum optical nonclassicality and the violation of Bell's inequalities is explored. Bell type inequalities for the electromagnetic field are formulated for general states of quantised radiation and their violation is connected to other nonclassical properties of the field. This is achieved by considering states with an arbitrary number of photons and carefully identifying the hermitian operators whose expectation values do not admit any local hidden variable description. We relate the violation of these multi-photon inequalities to properties of phase space distribution functions such as the diagonal coherent state distribution function and the Wigner function. Finally, the family of 4-mode states with Gaussian Wigner distributions is analysed, bringing out in this case the connection of violation of Bell type inequalities with the nonclassical property of squeezing.
- Sep 16 1997 quant-ph arXiv:quant-ph/9709030v1We exploit results on the classical Stieltjes moment problem to obtain completely explicit necessary and sufficient conditions for the photon number distribution p(n) of a radiation field mode to be classical. These conditions are given in two forms - respectively local and global in the individual photon number probabilities. Central to the first approach is the recognition of the important fact that the quantities n!p(n) are moments of a quasiprobability distribution, notwithstanding the fact that p(n)'s can by themselves be considered as a probability distribution over the nonnegative integers. This leads to local classicality conditions involving p(n)'s for only a small number of values of n. This local approach enables us to present detailed quantitative statements on the connection between nonclassicality and oscillations in the photon number distribution. The second approach is in terms of the traditional factorial moments of p(n). Equivalence of the two approaches is established.
- Aug 26 1997 quant-ph arXiv:quant-ph/9708038v1We exploit results from the classical Stieltjes moment problem to bring out the totality of all the information regarding phase insensitive nonclassicality of a state as captured by the photon number distribution p_n. Central to our approach is the realization that n !p_n constitutes the sequence of moments of a (quasi) probability distribution, notwithstanding the fact that p_n can by itself be regarded as a probability distribution. This leads to classicality restrictions on p_n that are local in n involving p_n's for only a small number of consecutive n's, enabling a critical examination of the conjecture that oscillation in p_n is a signature of nonclassicality.
- Dec 09 1996 quant-ph arXiv:quant-ph/9612024v1We present an operator approach to the description of photon polarization, based on Wigner's concept of elementary relativistic systems. The theory of unitary representations of the Poincare group, and of parity, are exploited to construct spinlike operators acting on the polarization states of a photon at each fixed energy momentum. The nontrivial topological features of these representations relevant for massless particles, and the departures from the treatment of massive finite spin representations, are highlighted and addressed.
- Nov 12 1996 quant-ph arXiv:quant-ph/9611014v2A recently introduced hierarchy of states of a single mode quantised radiation field is examined for the case of centered Guassian Wigner distributions. It is found that the onset of squeezing among such states signals the transition to the strongly nonclassical regime. Interesting consequences for the photon number distribution, and explicit representations for them, are presented.
- May 30 1996 quant-ph arXiv:quant-ph/9605042v1We describe a generalisation of the well known Pancharatnam geometric phase formula for two level systems, to evolution of a three-level system along a geodesic triangle in state space. This is achieved by using a recently developed generalisation of the Poincare sphere method, to represent pure states of a three-level quantum system in a convenient geometrical manner. The construction depends on the properties of the group $SU(3)\/$ and its generators in the defining representation, and uses geometrical objects and operations in an eight dimensional real Euclidean space. Implications for an n-level system are also discussed.
- Dec 18 1995 quant-ph arXiv:quant-ph/9512020v2A new operator based condition for distinguishing classical from non-classical states of quantised radiation is developed. It exploits the fact that the normal ordering rule of correspondence to go from classical to quantum dynamical variables does not in general maintain positivity. It is shown that the approach naturally leads to distinguishing several layers of increasing nonclassicality, with more layers as the number of modes increases. A generalisation of the notion of subpoissonian statistics for two-mode radiation fields is achieved by analysing completely all correlations and fluctuations in quadratic combinations of mode annihilation and creation operators conserving the total photon number. This generalisation is nontrivial and intrinsically two-mode as it goes beyond all possible single mode projections of the two-mode field. The nonclassicality of pair coherent states, squeezed vacuum and squeezed thermal states is analysed and contrasted with one another, comparing the generalised subpoissonian statistics with extant signatures of nonclassical behaviour.
- Sep 06 1995 quant-ph arXiv:quant-ph/9509001v2The non-classical property of subpoissonian photon statistics is extended from one to two-mode electromagnetic fields, incorporating the physically motivated property of invariance under passive unitary transformations. Applications to squeezed coherent states, squeezed thermal states, and superposition of coherent states are given. Dependences of extent of non-classical behaviour on the independent squeezing parameters are graphically displayed.
- Sep 06 1995 quant-ph arXiv:quant-ph/9509002v3text of abstract (We present a utilitarian review of the family of matrix groups $Sp(2n,\Re)$, in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the so-called unitary metaplectic representation of $Sp(2n,\Re)$. Global decomposition theorems, interesting subgroups and their generators are described. Turning to $n$-mode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U(n)-invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under $Sp(2n,\Re)$ action are delineated.)
- Jan 03 1995 quant-ph arXiv:quant-ph/9412011v1A general analysis of squeezing transformations for two mode systems is given based on the four dimensional real symplectic group $Sp(4,\Re)\/$. Within the framework of the unitary metaplectic representation of this group, a distinction between compact photon number conserving and noncompact photon number nonconserving squeezing transformations is made. We exploit the $Sp(4,\Re)-SO(3,2)\/$ local isomorphism and the $U(2)\/$ invariant squeezing criterion to divide the set of all squeezing transformations into a two parameter family of distinct equivalence classes with representative elements chosen for each class. Familiar two mode squeezing transformations in the literature are recognized in our framework and seen to form a set of measure zero. Examples of squeezed coherent and thermal states are worked out. The need to extend the heterodyne detection scheme to encompass all of $U(2)\/$ is emphasized, and known experimental situations where all $U(2)\/$ elements can be reproduced are briefly described.