Sudden death of effective entanglement
Sudden death of entanglement is a well-known effect resulting from the finite volume of separable states. We study the case when the observer has a limited measurement capability and analyse the effective entanglement (i.e., entanglement minimized over the output data). We show that in the well defined system of two quantum dots monitored by single electron transistors, one may observe a sudden death of effective entanglement when real, physical entanglement is still alive. For certain measurement setups, this occurs even for initial states for which sudden death of physical entanglement is not possible at all. The principles of the analysis may be applied to other analogous scenarios, such as estimation of the parameters arising from quantum process tomography.
Sudden death of quantum entanglement Życzkowski et al. (2001); Rajagopal and Rendell (2001); Yu and Eberly (2004); Eberly and Yu (2007) is one of the phenomena related to the fact that in finite dimensional systems the set of nonentangled states is of finite volume Życzkowski et al. (1998). The phenomenon was explicitly identified in Ref. Życzkowski et al. (2001) (see also its implicit presence in independent analysis Rajagopal and Rendell (2001)). Full recognition of its importance and consequences was established with time Yu and Eberly (2004); Eberly and Yu (2007); Ficek and Tanaś (2006); Sainz and Björk (2008) (for the review see Ref. Yu and Eberly (2009)) and demonstrated experimentally Almeida et al. (2007).
In earlier analysis of entanglement evolution, it was assumed that the observer has the power to perform arbitrary measurements and can determine the state of the system completely. This is however not always true. In particular, there are natural systems, like quantum dots (QDs) which we will consider later, where limited measurement power is a natural and practical constraint (ie. single electron transistors (SETs) coupled to QD systems can be used to find a limited amount of information about the QD state Stace and Barrett (2004); Stace et al. (2004)) In all such cases of limited measurement capability, it is natural to consider the worst case scenario proposed in Ref. Horodecki et al. (1999): as real entanglement, one should consider the entanglement (i.e. chosen entanglement measure) minimized over the set of measurement data. This approach has found wide developments in terms of entanglement witnesses Eisert et al. (2007); Gühne et al. (2007); Audenaert and Plenio (2006); Gühne et al. (2008) with respect to experimental data Puentes et al. (2009); Schmid et al. (2008). The minimized entanglement will subsequently be called effective entanglement.
Here, we consider the system of a double QD (DQD) interacting with a phonon-bath which leads to an unavoidable partial pure dephasing effect typically on picosecond timescales Borri et al. (2001); Vagov et al. (2003, 2004). The time-evolution of physical entanglement in this system can be found in Ref. Roszak and Machnikowski (2006a). We take into account the limitations to the knowledge of the system state imposed by a realistic measurement setup (consisting of different configurations of SETs interacting with the double-dot system). This measurement scheme does not allow for state tomography and provides, in fact, a very limited set of observables. In each time-step we minimize the value of entanglement with respect to the data which can be measured and find the evolution of the effective entanglement with respect to the SET-defined observables. We show that such an evolution, besides leading to a quantitative reduction of entanglement (compared to physical entanglement), demonstrates qualitative changes such as the possibility of sudden death of effective entanglement in situations when physical entanglement lives for arbitrarily long times. Our approach may be extended to estimate the time evolution of quantities describing quantum processes (e. g., entangling power or fidelity of a quantum process).
Ii The model
The system under study consists of two parts, the measured and the measuring subsystems. The former consists of the DQD ensemble, in which superpositions of excitonic states undergo pure dephasing due to the interaction with the phonon modes of the surrounding crystal. The pure dephasing is only partial, as has been experimentally shown Borri et al. (2001) and later explained theoretically Vagov et al. (2003, 2004); experimental and theoretical data yield both qualitative and quantitative agreement Vagov et al. (2004). The and states of the qubit correspond to an empty QD and a QD with an exciton in its ground state, respectively (the two qubits are located in separate QDs). A number of SETs which determine the charge distribution in their vicinity constitute the latter. This allows for the measurement of a set of elements of the DQD density matrix (the precise elements measured are determined by the geometry of the measurement device with respect to the DQD), and the subsequent calculation of effective entanglement.
The Hamiltonian which governs the evolution of the excitonic states in the DQD is
where the two states of each QD are denoted by and , is the unit operator, are the transition energies in the two QDs, are exciton-phonon coupling constants, are creation and annihilation operators of the phonon modes, and are the corresponding energies (we put ). The explicit tensor notation refers to the DQD but is suppressed for the phonon reservoir components.
Exciton wave functions are modelled by anisotropic Gaussians with the extension in the plane for the electron/hole, and along for both particles. Then, the coupling constants for the deformation potential coupling between confined charges and longitudinal phonon modes have the form , where
is the normalisation volume of the bosonic reservoir, is the distance between the subsystems, are momentum components in the plane and along the axis, are deformation potential constants for electrons and holes, is the speed of longitudinal sound, and is the crystal density.
In our calculations we use parameters typical for two self-assembled GaAs/InGaAs QDs stacked on top of each other Roszak and Machnikowski (2006a); Roszak and Machnikowski (2009). The material parameters used are eV, eV, nm/ps, kg/m (corresponding to GaAs), and nm, nm, nm, nm (dot related parameters).
The Hamiltonian can be diagonalised exactly using the Weyl operator method Roszak and Machnikowski (2006b) and the evolution is calculated following Ref. Roszak and Machnikowski (2006a). The interaction with the phonon modes leads to partial pure dephasing Borri et al. (2001); Krummheuer et al. (2002); Vagov et al. (2004), leaving the state occupations unchanged; the explicit forms of the time dependence of the off-diagonal density matrix elements may be found in Ref. Roszak and Machnikowski (2006a).
The measuring subsystem is taken into account only in principle, in the sense that the information which can be gained about the state of the DQD ensemble is limited by realistic measurement capability. When measuring the state of a DQD by observing the current through a SET, the actual observable depends on its position with respect to the DQD. The number of the DQD density matrix elements which can be obtained is very limited; this restriction is a key point in this paper. There are a number of features inherent to this measurement scheme, e.g. finite measurement time and an uncertainty of the outcome, which are not taken into account for the sake of clarity.
Let us consider a SET located near the lower QD (configuration A). The interaction between the SET electron and the exciton in the dot shifts the energy levels in the SET when the QD is occupied, hence affecting the current. An appropriate choice of SET parameters allows for the maximisation of the difference in current flow and the measurement of the occupation of the lower dot Stace and Barrett (2004). The measurement projectors corresponding to this situation are and , so the measured quantity is
If such a SET is located near the upper QD, the measured quantity is
Configuration B involves a SET located symmetrically between the QDs and in close proximity to them in such a way that the energy level on the SET island is sensitive to the probability of finding an exciton in midpoint Stace et al. (2004). The corresponding projectors are and , where
This allows for the measurement of a linear combination of density matrix elements
If the SET is located further away from the DQD region (symmetrically), it is sensitive to the total number of excitons (configuration C). This allows for the measurement of
via projectors and , or via and depending on the SET parameters. Switching between the two modes can be accomplished by applying different voltage to the SET.
Iii Minimising entanglement from the SET data
Consider the general quantum state . There is a simple
Lemma .- For any convex entanglement measure (or entanglement parameter) which is invariant under complex conjugate one has , where the ”” symbol means real part of the quantum state.
Any reasonable entanglement measure should be invariant under the complex conjugate; all known entanglement measures like entanglement of formation, concurrence, lognegativity, relative entropy of entanglement and all distillable quantities fulfil this condition. Some of them are also convex i.e. entanglement of formation, concurrence and relative entropy of entanglement. In what follows we shall consider the concurrence Hill and Wootters (1997); Wootters (1998), which is both convex and invariant under the complex conjugate. The reason for this choice of entanglement measure is its mathematical simplicity and the fact that it is easily converted into Entanglement of Formation (which has a good physical interpretation). Since the SET measurement provides only real parts of off-diagonal density matrix elements, the first step of minimisation is to consider only the real parts of DQD states, .
Iv Sudden death of effective entanglement
Firstly, we will consider the situation, when a number of SETs provide all possible information that can be gained about the state of the DQD with this measurement technique. This requires a pair of SETs in configuration A located near the two QDs, giving and defined, respectively, by Eq. (2) and (3), and one in configuration C providing (Eq. (6)). Hence, all of the diagonal elements of the density matrix can be found,
A SET in configuration B provides (Eq. (5)) giving the real part of one of the off-diagonal density matrix elements
Since all of the diagonal density matrix elements are known, the set of initial maximally entangled states which cannot exhibit sudden death of effective entanglement is the same as the set of states with real off-diagonal density matrix elements which do not exhibit sudden death of physical entanglement Roszak and Machnikowski (2006a). The time evolution of effective and physical entanglement in these states under pure dephasing is the same and the concurrence is equal to . The situation is different for states where all diagonal density matrix elements are non-zero where the set of effectively entangled coherent states is substantially reduced. In Fig. 1 the effective entanglement of coherent states () as a function of and is plotted for two values of .
Secondly, let us consider the situation when only and have been measured (configuration A and B), so only linear combinations of some density matrix elements are known. The measurement outcome . The outcome , but it is easy to show that for non-zero effective entanglement . The minimisation of entanglement requires that the quantity be maximized, which occurs for (the unknown density matrix elements , , , and are defined in Eq. (7) and (8)); then Finding the maximal , which leads to set values for , , , and , and minimising over the five still unknown off-diagonal density matrix elements produces the effective entanglement for given measurement outcomes and . Effective entanglement as a function of and is plotted in Fig. 2.
It is interesting to consider here the time-evolution of effective entanglement under phonon-induced pure dephasing of an initially maximally entangled state (Eq. (4)). When measured it will yield and , so the effective concurrence (note that the state has , but and ). Phonon-induced evolution of the state does not change , but decreases with decreasing leading to sudden death of effective entanglement for sufficiently dephased states. This state does not exhibit sudden death of physical entanglement.
Thirdly, we will consider the simplest situation, where the measurement device is limited to configuration B and the measurement data yield only as defined in Eq. (5). The amount of information gained by the measurement is very limited. Minimising effective entanglement requires the maximisation of the same as in the previous measurement setup, yet now the maximum is easily found for . The dependence of effective entanglement on is plotted in the inset of Fig. 3.
The evolution of effective entanglement of the initial state in this setup under realistic phonon-induced pure dephasing is plotted in Fig. 3 for different temperatures. As is to be expected, effective disentanglement occurs faster than physical disentanglement. Furthermore, sudden death of entanglement appears for sufficiently high temperatures (e. g., when the dephasing is strong enough). For a limited range of temperatures, sudden birth of entanglement is also observed. The second phenomenon is due to the enhancement of coherence which occurs when wavepackets from the two QDs meet due to positive interference between them; this mechanism does not lead to the sudden birth of physical entanglement Roszak and Machnikowski (2006a).
V Conclusion and discussion
We have considered the evolution of entanglement in a DQD system under phonon-induced pure dephasing minimized over a set of attainable data (the considered measurement schemes are based on SETs). In all of the considered measurement setups a reduction of effective (minimized) entanglement compared to the physical one was observed. Furthermore, in the setups where the knowledge of QD occupations is limited, sudden death was found in situations when the physical entanglement lives for arbitrarily long times and its sudden death is not possible. For a limited range of temperatures (coupling strengths) even the sudden birth of entanglement occurs (due to a mechanism that does not cause sudden birth of physical entanglement). Hence, we have shown that the analysis of entanglement dynamics in systems with difficult measurement access can lead to qualitative differences in measured entanglement.
The present analysis may provide a starting point to analogous research in general dynamics including quantum process tomography (Chuang and Nielsen (1997); Riebe et al. (2006)). For instance, one may want to probe the entangling power Zanardi et al. (2000) of a sequence of quantum gates in time in a possibly cheap way, when only some of measurements are not costly. Then the principles of the above analysis may be applied with the help of the Choi-Jamiołkowski isomorphism Jamiołkowski (1972); Choi (1982). Fundamental parameters of the dynamics, including the fidelity of a quantum process, can be then estimated, i.e. by minimisation in time under a restricted set of data, especially when coarse-graining information about the dynamics is needed.
Acknowledgements.This work was supported by the UE IP SCALA project and by the Czech Science Foundation under Grant No. 202/07/J051. R. H. and P. H. acknowledge support from the Polish Ministry of Science under Grant No. NN202231937.
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