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Symmetrization and Entanglement of Arbitrary States of Qubits

Asoudeh, M, Karimipour, V, Memarzadeh, L, Rezakhani, A T

Published Dec 25, 2002• 1 citation
OpenAlex

Abstract

Given two arbitrary pure states $ |\\phi>$ and $ |\\psi>$ of qubits or higher level states, we provide arguments in favor of states of the form $ \\frac{1}{\\sqrt{2}}(|\\psi> |\\phi> + i |\\phi> |\\psi>) $ instead of symmetric or anti-symmetric states, as natural candidates for optimally entangled states constructed from these states. We show that such states firstly have on the average a high value of concurrence, secondly can be constructed by a universal unitary operator independent of the input states. We also show that these states are the only ones which can be produced with perfect fidelity by any quantum operation designed for intertwining two pure states with a relative phase. A probabilistic method is proposed for producing any pre-determined relative phase into the combination of any two arbitrary states.

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Given two arbitrary pure states ∣phi> |\\phi>∣phi> and ∣psi> |\\psi>∣psi> of qubits or higher level states, we provide arguments in favor of states of the form frac1sqrt2(∣psi>∣phi>+i∣phi>∣psi>)\\frac{1}{\\sqrt{2}}(|\\psi> |\\phi> + i |\\phi> |\\psi>)frac1sqrt2(∣psi>∣phi>+i∣phi>∣psi>) instead of symmetric or anti-symmetric states, as natural candidates for optimally entangled states constructed from these states. We show that such states firstly have on the average a high value of concurrence, secondly can be constructed by a universal unitary operator independent of the input states. We also show that these states are the only ones which can be produced with perfect fidelity by any quantum operation designed for intertwining two pure states with a relative phase. A probabilistic method is proposed for producing any pre-determined relative phase into the combination of any two arbitrary states.
The nonlocal correlations between quantum states in an entangled system are essential to many quantum communications applications. A basic quantum operation, which permits the distribution of entanglement between two initially uncorrelated systems, is entanglement swapping. Here we present a rigorous formulation of entanglement swapping of any two partially mixed two-qubit states without limiting ourselves to any specific type of state or noise. Further, for two important classes of the input states, Bell diagonal and pure states, we describe how the concurrence of the final state is related to the concurrence of the initial states. First, we consider Bell diagonal states, and find bounds on the concurrence of the final state in terms of the concurrences of the initial states. These bounds are important for communications applications because polarization mode dispersion in fibers produces Bell diagonal states up to a local unitary rotation. Second, we show that swapping pure states occasionally results in a state of higher concurrence than either of the initial states. In addition, we find that two pure states are most likely to be capable of swapping to a state of increased concurrence when the two initial states have similar concurrences. Our analysis offers a completely general framework for investigating the behavior of any pair of two-qubit states when used for entanglement swapping.
We analyze various scenarios for entangling two initially unentangled qubits. In particular, we propose an optimal universal entangler that entangles a qubit in unknown state $|\ensuremath{\Psi}〉$ with a qubit in a reference (known) state ∣0〉.|0〉.∣0〉. That is, our entangler generates the output state that is as close as possible to the pure (symmetrized) state $(|\ensuremath{\Psi}〉|0〉+|0〉|\ensuremath{\Psi}〉).$ The most attractive feature of this entangling machine, is that the fidelity of its performance (i.e., the distance between the output and the ideally entangled---symmetrized state) does not depend on the input and takes the constant value $\mathcal{F}=(9+3\sqrt{2})/14\ensuremath{\simeq}0.946.$ We also analyze how to optimally generate from a single qubit initially prepared in an unknown state $|\ensuremath{\Psi}〉$ a two qubit entangled system, which is as close as possible to a Bell state $(|\ensuremath{\Psi}〉|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉+|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉|\ensuremath{\Psi}〉),$ where $〈\ensuremath{\Psi}|{\ensuremath{\Psi}}^{\ensuremath{\perp}}〉=0.$
We look for local unitary operators W1⊗W2W_1 \otimes W_2W1​⊗W2​ which would rotate all equally entangled two-qubit pure states by the same but arbitrary amount. It is shown that all two-qubit maximally entangled states can be rotated through the same but arbitrary amount by local unitary operators. But there is no local unitary operator which can rotate all equally entangled non-maximally entangled states by the same amount, unless it is unity. We have found the optimal sets of equally entangled non-maximally entangled states which can be rotated by the same but arbitrary amount via local unitary operators W1⊗W2W_1 \otimes W_2W1​⊗W2​, where at most one these two operators can be identity. In particular, when W1=W2=(i/2)(σx+σy)W_1 = W_2 = (i/\sqrt{2})(σ_x + σ_y)W1​=W2​=(i/2​)(σx​+σy​), we get the local quantum NOT operation. Interestingly, when we apply the one-sided local depolarizing map, we can rotate all equally entangled two-qubit pure states through the same amount. We extend our result for the case of three-qubit maximally entangled state.
We show that using a fixed cloning machine, whether deterministic inexact or probabilistic exact, one can take an arbitrary blank state. Quantum information cannot be cloned. There cannot exist a machine which can produce two (or more) exact copies of an arbitrary state in a deterministic manner [1]. Replication is not allowed even when the input state is taken at random from a given set of two non-orthogonal states [2]. It has been further shown that probabilistic cloning is not possible if the input state is from a given linearly dependent set [3]. Henceforth we deal explicitly with qubits although all the results can be extended to higher dimensions. Although exact cloning is not possible, one can approximately clone an arbitrary input state [4, 5, 6]. In this scheme a fixed state is taken as the blank state, depending on which and the initial machine state, the cloning operation is constructed. We extend this operation such that any arbitrary (pure or mixed) state taken as the blank copy can do the job. The optimal universal 1 → 2 inexact qubit cloner of Brußet al.[5] takes an arbitrary qubit |ψ 〉 〈ψ | = 1 2 (I + − → s. − → σ) along with a fixed blank qubit |b 〉 and a machine state |M 〉 as input. An entangled state of the three qubits is produced as the output such that the reduced density matrices of the first two qubits are two similar approximate copies ρ = 1 2 (I + η− → s. − → σ) of |ψ 〉 〈ψ | with η = 2 3. The unitary operator realizing this process is defined on the combined Hilbert space of the input qubit, the blank qubit and machine by U ′ √ √
Entanglement in incoherent mixtures of pure states of two qubits is considered via the concurrence measure. A set of pure states is optimal if the concurrence for any mixture of them is the weighted sum of the concurrences of the generating states. When two or three pure real states are mixed, it is shown that 28.5% or 5.12% of the cases, respectively, are optimal. Conditions that are obeyed by the pure states generating such optimally entangled mixtures are derived. For four or more pure states, it is shown that there are no such sets of real states. The implication of these on the superposition of two or more dimerized states is discussed. A corollary of these results also show in how many cases rebit concurrence can be the same as that of qubit concurrence.
The action of a NOT gate on a classical bit results in a change of its value from a 0 to a 1 and vice versa. The action of the classical NOT gate is in principle perfect because with fidelity equal to unity it complements the value of a bit. The action of the quantum NOT gate in a computational basis |0⟩ and |1⟩ is very similar to the action of the classical NOT gate. However, a more general quantum mechanical operation which corresponds to a classical NOT gate would take a qubit in an arbitrary state |Ψ⟩ and produce a qubit in the state |Ψ⊥⟩ orthogonal to |Ψ⟩. This operation is anti-unitary and therefore, cannot be realized exactly. So how well we can do? We find a unitary transformation acting on an input qubit and some auxiliary qubits, which represent degrees of freedom of the quantum NOT gate itself, which approximately realizes the NOT operation on the state of the original qubit. We call this ‘device’ a universal-NOT gate because the size of the error it produces is independent of the input state. We show that an optimal U-NOT gate which has as its input N identical qubits and produces M outputs achieves a fidelity of F = (N + 1)/ (N + 2), which is equal to the fidelity of estimation of the input qubits. We also show that when a priori information about the state of the input qubit is available, the fidelity of a quantum NOT gate can be much better than the fidelity of estimation.