Local Quantum Operations Research Articles

Hiding bits in bell states.

We present a scheme for hiding bits in Bell states that is secure even when the sharers, Alice and Bob, are allowed to carry out local quantum operations and classical communication. We prove that the information that Alice and Bob can gain about a hidden bit is exponentially small in n, the number of qubits in each share, and can be made arbitrarily small for hiding multiple bits. We indicate an alternative efficient low-entanglement method for preparing the shared quantum states. We discuss how our scheme can be implemented using present-day quantum optics.

Local filtering operations on two qubits

We consider one single copy of a mixed state of two qubits and investigate how its entanglement changes under local quantum operations and classical communications (LQCC) of the type $\rho'\sim (A\otimes B)\rho(A\otimes B)^{\dagger}$. We consider a real matrix parameterization of the set of density matrices and show that these LQCC operations correspond to left and right multiplication by a Lorentz matrix, followed by normalization. A constructive way of bringing this matrix into a normal form is derived. This allows us to calculate explicitly the optimal local filterin operations for concentrating entanglement. Furthermore we give a complete characterization of the mixed states that can be purified arbitrary close to a Bell state. Finally we obtain a new way of calculating the entanglement of formation.

Four-party unlockable bound entangled state

I present a four-party unlockable bound entangled state, that is, a four-party quantum state which cannot be written in a separable form and from which no pure entanglement can be distilled by local quantum operations and classical communication among the parties, and yet when any two of the parties come together in the same laboratory they can perform a measurement which enables the other two parties to create a pure maximally entangled state between them without coming together. This unlocking ability can be viewed in two ways, either as a determination of which Bell state is shared in the mixture or as a kind of quantum teleportation with cancellation of Pauli operators.

Entanglement, information, and multiparticle quantum operations

Collective operations on a network of spatially-separated quantum systems can be carried out using local quantum (LQ) operations, classical communication (CC) and shared entanglement (SE). Such operations can also be used to communicate classical information and establish entanglement between distant parties. We show how these facts lead to measures of the inseparability of quantum operations, and argue that a maximally-inseparable operation on 2 qubits is the SWAP operation. The generalisation of our argument to N qubit operations leads to the conclusion that permutation operations are maximally-inseparable. For even N, we find the minimum SE and CC resources which are sufficient to perform an arbitrary collective operation. These minimum resources are 2(N-1) ebits and 4(N-1) bits, and these limits can be attained using a simple teleportation-based protocol. We also obtain lower bounds on the minimum resources for the odd case. For all $N{\geq}4$, we show that the SE/CC resources required to perform an arbitrary operation are strictly greater than those that any operation can establish/communicate.

Multipartite pure-state entanglement and the generalized Greenberger-Horne-Zeilinger states

We show that not all four-party pure states are Greenberger-Horne-Zeilinger (GHZ) reducible (i.e., can be generated reversibly from a combination of two-, three-, and four-party maximally entangled states by local quantum operations and classical communication asymptotically). We also present some properties of the relative entropy of entanglement for those three-party pure states that are GHZ reducible, and then we relate these properties to the additivity of the relative entropy of entanglement.

Exact and asymptotic measures of multipartite pure-state entanglement

In an effort to simplify the classification of pure entangled states of multi (m) -partite quantum systems, we study exactly and asymptotically (in n) reversible transformations among n'th tensor powers of such states (ie n copies of the state shared among the same m parties) under local quantum operations and classical communication (LOCC). With regard to exact transformations, we show that two states whose 1-party entropies agree are either locally-unitarily (LU) equivalent or else LOCC-incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger (GHZ) states are LOCC-incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations result in a simpler classification than exact transformations. We show that m-partite pure states having an m-way Schmidt decomposition are simply parameterizable, with the partial entropy across any nontrivial partition representing the number of standard ``Cat'' states (|0^m>+|1^m>) asymptotically interconvertible to the state in question. For general m-partite states, partial entropies across different partitions need not be equal, and since partial entropies are conserved by asymptotically reversible LOCC operations, a multicomponent entanglement measure is needed, with each scalar component representing a different kind of entanglement, not asymptotically interconvertible to the other kinds. In particular the m=4 Cat state is not isentropic to, and therefore not asymptotically interconvertible to, any combination of bipartite and tripartite states shared among the four parties. Thus, although the m=4 cat state can be prepared from bipartite EPR states, the preparation process is necessarily irreversible, and remains so even asymptotically.

Optimal entanglement manipulation for an arbitrary state of two qubits

Starting from an arbitrary entangled state of two separated qubits, one can produce a unique state which has the maximal possible entanglement of formation by carrying out local quantum operations and classical communication. We show that this maximal entanglement of formation and the fidelity of the unique state can be expressed as functions of the initial state analytically. We also describe the most efficient procedure of the entanglement manipulation to produce the unique state.

Collective versus local measurements on two parallel or antiparallel spins

We give a complete analysis of covariant measurements on two spins. We consider the cases of two parallel and two antiparallel spins, and we consider both collective measurements on the two spins and measurements that require only local quantum operations and classical communication (LOCC). In all cases we obtain the optimal measurements for arbitrary fidelities. In particular, we show that if the aim is to determine as accurately as possible the direction in which the spins are pointing, it is best to carry out measurements on antiparallel spins (as already shown by Gisin and Popescu), second best to carry out measurements on parallel spins, and worst to be restricted to LOCC measurements. If the aim is to determine as accurately as possible a direction orthogonal to that in which the spins are pointing, it is best to carry out measurements on parallel spins, whereas measurements on antiparallel spins and LOCC measurements are both less suitable but equivalent.

Entanglement and collective quantum operations

We show how shared entanglement, together with classical communication and local quantum operations, can be used to perform an arbitrary collective quantum operation upon N spatially-separated qubits. A simple teleportation-based protocol for achieving this, which requires 2( N−1) ebits of shared, bipartite entanglement and 4( N−1) classical bits, is proposed. In terms of the total required entanglement, this protocol is shown to be optimal for even N in both the asymptotic limit and for `one-shot' applications.

Evidence for bound entangled states with negative partial transpose

We exhibit a two-parameter family of bipartite mixed states $\rho_{bc}$, in a $d\otimes d$ Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in $2\otimes 2$ can be distilled by local quantum operations and classical communication (LQ+CC). Evidence for this undistillability is provided by the result that, for certain states in this family, we cannot extract entanglement from any arbitrarily large number of copies of $\rho_{bc}$ using a projection on $2\otimes 2$. These states are canonical NPT states in the sense that any bipartite mixed state in any dimension with NPT can be reduced by LQ+CC operations to an NPT state of the $\rho_{bc}$ form. We show that the main question about the distillability of mixed states can be formulated as an open mathematical question about the properties of composed positive linear maps.

Schmidt number for density matrices

We introduce the notion of a Schmidt number of a bipartite density matrix, characterizing the minimum Schmidt rank of the pure states that are needed to construct the density matrix. We prove that Schmidt number is nonincreasing under local quantum operations and classical communication. We show that $k$-positive maps witness Schmidt number, in the same way that positive maps witness entanglement. We show that the family of states which is made from mixing the completely mixed state and a maximally entangled state have increasing Schmidt number depending on the amount of maximally entangled state that is mixed in. We show that Schmidt number {\it does not necessarily increase} when taking tensor copies of a density matrix $\rho$; we give an example of a density matrix for which the Schmidt numbers of $\rho$ and $\rho \otimes \rho$ are both 2.

Optimal Entanglement Enhancement for Mixed States

We consider the actions of protocols involving local quantum operations and classical communication (LQCC) on a single system consisting of two separated qubits. We give a complete description of the orbits of the space of states under LQCC and characterise the representatives with maximal entanglement of formation. We thus obtain a LQCC entanglement concentration protocol for a single given state (pure or mixed) of two qubits which is optimal in the sense that the protocol produces, with non-zero probability, a state of maximal possible entanglement of formation. This defines a new entanglement measure, the maximum extractable entanglement.