Perfect quantum state mirroring in a chain of N spins is defined as the condition in which the state |i〉 of the chain is swapped into the state |N — i〉 within a time evolution interval τ. Such a phenomenon is an interesting way of transfering entanglement. An expressions for the perfect mirroring of a single qubit contained in a spin chain were proposed in the past. We exploit such an expressions for calculating the evolution times in chains of both two and three spins. In the case of a chain of two qubits, we derive conditions under which the associated four Bell states diagonalize the Hamiltonian. It is found that for the two Bell states |Φ + 〉 and |Φ − 〉, perfect mirroring does not occur (i.e. entanglement is not preserved under swapping). On the other hand, perfect single qubit mirror effect (entanglement preservation) indeed occurs for the other two Bell states |Ψ + 〉 and |Ψ − 〉 which are mapped into |Φ + 〉 and |Φ − 〉 respectively. For the case of a chain of three qubits, the effects of a perfect single qubit mirroring on a set of four maximally entangled three qubit states ψ 1 , ψ 2 , χ 1 , and χ 2 are studied. Due to the fact that quantum mirroring preserves maximal entanglement, the states ψ 1 and ψ 2 are not altered. However, quantum mirroring changes the states χ 1 and χ 2 only if we apply perfect quantum state mirroring in the site a = 1 of the three qubits spin chain. The above constrains the preservation of maximal entanglement under qubit mirroring of such a state. Due to the fact that swapping has already been experimentally tested, a posible experimental implementations of single qubit mirroring is possible.
Perfect quantum state mirroring in a chain of N spins is defined as the condition in which the state |i〉 of the chain is swapped into the state |N — i〉 within a time evolution interval τ. Such a phenomenon is an interesting way of transfering entanglement. An expressions for the perfect mirroring of a single qubit contained in a spin chain were proposed in the past. We exploit such an expressions for calculating the evolution times in chains of both two and three spins. In the case of a chain of two qubits, we derive conditions under which the associated four Bell states diagonalize the Hamiltonian. It is found that for the two Bell states |Φ + 〉 and |Φ − 〉, perfect mirroring does not occur (i.e. entanglement is not preserved under swapping). On the other hand, perfect single qubit mirror effect (entanglement preservation) indeed occurs for the other two Bell states |Ψ + 〉 and |Ψ − 〉 which are mapped into |Φ + 〉 and |Φ − 〉 respectively. For the case of a chain of three qubits, the effects of a perfect single qubit mirroring on a set of four maximally entangled three qubit states ψ 1 , ψ 2 , χ 1 , and χ 2 are studied. Due to the fact that quantum mirroring preserves maximal entanglement, the states ψ 1 and ψ 2 are not altered. However, quantum mirroring changes the states χ 1 and χ 2 only if we apply perfect quantum state mirroring in the site a = 1 of the three qubits spin chain. The above constrains the preservation of maximal entanglement under qubit mirroring of such a state. Due to the fact that swapping has already been experimentally tested, a posible experimental implementations of single qubit mirroring is possible.
Quantum information transfer is an important part of quantum information processing. Several proposals for quantum information transfer along linear arrays of nearest-neighbor coupled qubits or spins were made recently. Perfect transfer was shown to exist in two models with specifically designed strongly inhomogeneous couplings. We show that perfect transfer occurs in an entire class of chains, including systems whose nearest-neighbor couplings vary only weakly along the chain. The key to these observations is the Jordan-Wigner mapping of spins to noninteracting lattice fermions that display perfectly periodic dynamics if the single-particle energy spectrum is appropriate. After a half-period of that dynamics, any state is transformed into its mirror image with respect to the center of the chain. The absence of fermion interactions preserves these features at an arbitrary temperature and allows for the transfer of nontrivially entangled states of several spins or qubits.
Quantum spin chains may be used to transfer quantum states between elements of a quantum information processing device. A scheme discovered recently [Phys. Rev. A 85, 022312 (2012)] was shown to have favorable transfer properties for single-qubit states even in the presence of built-in static disorder caused by manufacturing errors. We extend that scheme in a way suggested already in Bruderer et al. [Phys. Rev. A 85, 022312 (2012)] and study the transfer of the four Bell states which form a maximally entangled basis in the two-qubit Hilbert space. We show that perfect transfer of all four Bell states separately and of arbitrary linear combinations may be achieved for chains with hundreds of spins. For simplicity we restrict ourselves to systems without disorder.
A single-step operation is proposed to produce multiqubit maximally entangled states in the NMR model. In the scheme, all qubits are initially in the ground state, and one single pulse of a multifrequency coherent magnetic radiation is applied to manipulate simultaneously the ``active states'' that satisfy the resonant conditions while all other ``inactive states'' remain unchanged. An effective Hamiltonian is derived in a generalized rotating frame, which allows us to predict the time evolution of ``active states'' generated by the magnetic pulse. The magnetic pulse parameters, such as frequencies, phases, amplitudes, and duration time, are obtained analytically to implement a Bell state of two qubits and a Greenberger-Horne-Zeilinger state of three qubits. The scheme has been generalized to create an N-qubit entangled state. Two rules are found to calculate the magnetic pulse parameters numerically, which are required to realize entangled states for even and odd qubits, respectively. The rules are successfully checked in the cases of $4\ensuremath{\leqslant}N\ensuremath{\leqslant}10$. The relation of lnt0 and lnN is found to be linear, and the duration time t0is approximately proportional to N.
Quantum states have high affinity for errors and hence error correction is of utmost importance to realise a quantum computer. Laflamme showed that 5 qubits are necessary to correct a single error on a qubit. In a Pauli error model, four different types of errors can occur on a qubit. Maximally entangled states are orthogonal to each other and hence can be uniquely distinguished by a measurement in the Bell basis. Thus a measurement in Bell basis and a unitary transformation is sufficient to correct error in Bell states. However, such a measurement is not possible for non-maximally entangled states. In this work we show that the 16 possible errors for a non-maximally entangled two qubit system map to only 8 distinct error states. Hence, it is possible to correct the error without perfect knowledge of the type of error. Furthermore, we show that the possible errors can be grouped in such a way that all 4 errors can occur on one qubit, whereas only bit flip error can occur on the second qubit. As a consequence, instead of 10, only 8 qubits are sufficient to correct a single error. We propose an 8-qubit error correcting code to correct a single error in a non-maximally entangled state. We further argue that for an n-qubit non-maximally entangled state of the form a∣0>n+b∣1>n, it is always possible to correct a single error with fewer than 5n qubits, in fact only 3n+2 qubits suffice.
Perfect (s,r)-state transfer
Much work has been done in the last two decades on the topic of quantum state transfer in a quantum spin network. One can model such a system of interacting qubits using an undirected graph, and studying vertex-to-vertex dynamics. This setup has recently been relaxed to allow for dynamics between linear combinations of two vertex states, i.e.\ from u=ea+seb to μ=eα+reβ, where r=s is either −1 (which corresponds to pair state transfer) or +1 (which corresponds to plus state transfer), or more recently r=s is taken to be any real number (which corresponds to s-pair state transfer).
Here, we broaden the investigation of s-pair state transfer to \textit{perfect (s,r)-state transfer}, which is perfect state transfer from u=ea+seb to μ=eα+reβ (up to some dilation) where r,s∈C. We identify infinite families of graphs with perfect (s,r)-state transfer and provide characterizations of cases when ∣r∣=∣s∣ and when ∣r∣=∣s∣, showing situations when the degree of entanglement between vertex states is preserved, and when it is not preserved. The latter is particularly important as it represents perfect state transfer from an entangled pair of qubits to another one where the degree of entanglement need not be the same\mdash in fact, it can be set up so as to ``boost'' (increase) entanglement. We provide an algorithm that finds the vector with two nonzero entries that maximizes the fidelity of transfer for a fixed time t starting from a given s-pair state u. Finally, we provide a sensitivity analysis, with respect to readout time errors, of perfect (s,r)-state transfer.
Spin chains have been proposed as a reliable and convenient way of transferring information and entanglement in a quantum computational context. Nonetheless, it has to be expected that any physical implementation of these systems will be subject to several perturbative factors which could potentially diminish the transfer quality. In this paper, we investigate a number of possible fabrication defects in the spin chains themselves as well as the effect of nonsynchronous or imperfect input operations, with a focus on the case of multiple excitation and qubit transfer. We consider both entangled and unentangled states and, in particular, the transfer of an entangled pair of adjacent spins at one end of a chain under the mirroring rule and also the creation of entanglement resulting from injection at both end spins.