Intrinsic upper bound on two-qubit polarization entanglement predetermined by pump polarization correlations in parametric down-conversion

Abstract

We study how one-particle correlations transfer to manifest as two-particle correlations in the context of parametric down-conversion (PDC), a process in which a pump photon is annihilated to produce two entangled photons. We work in the polarization degree of freedom and show that for any two-qubit generation process that is both trace-preserving and entropy-nondecreasing the concurrence C(ρ) of the generated two-qubit state ρ follows an intrinsic upper bound with C(ρ) ≤ (1+P )/2, where P is the degree of polarization of the pump photon. We also find that for the class of two qubit states that is restricted to have only two non-zero diagonal elements such that the effective dimensionality of the two-qubit state is same as the dimensionality of the pump polarization state, the upper bound on concurrence is the degree of polarization itself, that is, C(ρ) ≤ P . Our work shows that the maximum manifestation of two-particle correlations as entanglement is dictated by one-particle correlations. The formalism developed in this work can be extended to include multiparticle systems and can thus have important implications towards deducing the upper bounds on multi-particle entanglement, for which no universally accepted measure exists. PACS numbers:The wave-particle duality, that is, the simultaneous existence of both particle and wave properties, is the most distinguishing feature of a quantum system. A quantum system is characterized in terms of physical observables such as energy, momentum, etc., as well as in terms of correlations, which, although, cannot be measured directly like the physical observables but the degree of which can be measured in terms of the contrast with which a system produces interference patterns [1][2][3]. In the context of quantum systems consisting of more than one particle, the wave-particle duality can manifest as entanglement [4]. Entanglement refers to intrinsic multi-particle correlations in a system and is quite often referred to as the quintessential feature of quantum systems [5]. There are many processes in which a quantum system gets annihilated to produce a new quantum system consisting of either equal or more number of particle. An example is the nonlinear optical process of parametric down-conversion (PDC), in which an input pump photon gets annihilated to produce two entangled photons called the signal and idler photons [6]. Another example is the four-wave mixing process, in which two input pump photons get annihilated to produce two new photons [7]. In such processes, it is known that the physical observables get transferred in a conserved manner [6,8]. For example, in parametric down-conversion, the energy of the pump photon remains equal to the sum of the energies of the down-converted signal and idler photons [6]. However, it is not very well understood as to how the intrinsic correlations in one quantum system get transferred to another quantum system.One of the main difficulties in addressing questions related to correlation transfer is the lack of a mathematical frame...

Transfer of temporal coherence in parametric down-conversion

“…Finally, we have shown that the concurrence of time-energy entangled two-qubit states is bounded by the degree of temporal coherence of the pump field. This result extends previously obtained results in the spatial [19] and polarization [31] degrees of freedom to the temporal degree of freedom and can thus have important implications for understanding how correlations of the pump field in general manifest as twoparticle entanglement. Our results can also be important for time-energy two-qubit based quantum communication applications.…”

“…We note that in situations in whichγ d (∆τ ′ ) ≈ 1 and κ 1 = κ 2 , the maximum achievable concurrence for a pulsed field can be unity in principle and for a continuous-wave field it can be unity as long as ∆τ is much smaller than the coherence time of the pump field. The above result is the temporal analog of the results obtained in the polarization [31] and spatial [19] degrees of freedom. However, unlike in the spatial degree of freedom, which does not involve any space-averaged detection scheme, the results derived in this article show that even for the time-averaged detection schemes, the temporal correlation properties of the pump do directly decide the upper limit on entanglement that a time-energy entangled two-qubit state can achieve.…”

“…There have been previous studies describing how correlations of the pump field in polarization and spatial degrees of freedom affect the entanglement of the generated two-qubit states. In the polarization degree of freedom it was shown [31] that the degree of polarization P of the pump photon puts an upper bound of (1 + P )/2 on the concurrence of the generated twoqubit state. In the spatial degree of freedom, effects of pump spatial coherence on the entanglement of the generated spatial two-qubit state have been worked out for two-qubit state that have only two non-zero diagonal elements, and for such states it has been shown that the concurrence is bounded by the degree of spatial coherence of the pump field [19].…”

Intrinsic degree of coherence of two-qubit states and measures of two-particle quantum correlations

“…One such example is the two-qubit state generator proposed in Ref. [40] using four PDC cyrstals. In Fig.…”

“…In recent years, transfer of the coherence properties of the pump photon to the signal and idler photons has been studied in various different contexts [40,45,46]. In this section, we investigate this transfer in terms of the intrinsic degree of coherence.…”