Theory of remote entanglement via quantum-limited phase-preserving amplification
We show that a quantum-limited phase-preserving amplifier can act as a which-path information eraser when followed by heterodyne detection. This ‘beam splitter with gain’ implements a continuous joint measurement on the signal sources. As an application, we propose heralded concurrent remote entanglement generation between two qubits coupled dispersively to separate cavities. Dissimilar qubit-cavity pairs can be made indistinguishable by simple engineering of the cavity driving fields providing further experimental flexibility and the prospect for scalability. Additionally, we find an analytic solution for the stochastic master equation—a quantum filter—yielding a thorough physical understanding of the nonlinear measurement process leading to an entangled state of the qubits. We determine the concurrence of the entangled states and analyze its dependence on losses and measurement inefficiencies.
Spatially separated objects can be entangled by the measurement backaction (1); (2); (3) of a joint measurement. In the process of concurrent remote entanglement generation, two remote stationary qubits are first entangled with separate flying qubits and which-path information is erased from them by interference effects (5); (4). A subsequent measurement of the flying qubits can then implement a joint measurement with a backaction that projects the stationary sources to an entangled state, demonstrated in various atomic and solid state systems (6); (7); (8); (9); (10); (11); (12) through coincidence detection of photons (13). In this concurrent scheme, entanglement generation occurs purely by measurement backaction; the entangled qubits exchange no information, not even unidirectionally. There need be no causal connection between the qubits. This represents an important conceptual difference from consecutive remote entangling configurations (14); (3) with unidirectional exchange of information due to both qubits seeing the same photon field.
Generating entanglement is a necessity for quantum communication, cryptography and computation (15); (16); (17). The concurrent configuration promotes scalability and modularity of a quantum network, allowing entangling operations between arbitrary nodes through routing independently generated parallel signals to a quantum eraser. Compared to consecutive configurations, high entanglement fidelity is harder to achieve since the single-qubit information is more exposed to losses. However, the concurrent method provides better on-off ratio for the effective entanglement since, with use of directional elements, no parasitic signal could in principle propagate from one qubit to the other.
We show that a quantum-limited phase-preserving amplifier can be used as a quantum eraser for the which-path information for concurrent microwave signals (Sec. II). When followed by detection of both quadratures of the amplified output signal, this novel ‘beam splitter with gain’ can implement continuous joint measurement on the remote signal sources. The quantum eraser configuration is general for systems involving continuous variables (18); (19). For concreteness, in Sec. III we propose and analyze remote entangling for superconducting qubits coupled to traveling continuous microwave signals (see also Ref. (20)). Analogously to the spatiotemporal mode shapes, the single-qubit information from dissimilar cavity-qubit pairs (unequal dispersive coupling or decay rates) is carried by the unequal temporal measurement amplitudes. However, because the measurement amplitudes depend on the cavity dynamics, they can be made indistinguishable through simple engineering of the cavity driving amplitudes, reinforcing the scalability and experimental flexibility of the entangling scheme. In Sec. IV we derive an analytic solution for the qubits’ stochastic measurement dynamics and analyze the fidelity of the resulting entanglement with realistic estimates before concluding in Sec. V.
Ii Phase-preserving amplifier as a which-path information eraser
Quantum-limited phase-preserving amplification can be implemented through non-degenerate parametric amplification. In this process two distinct incoming modes, denoted here as the signal and the idler at the frequencies and , are coupled to a strong pump mode by a nonlinear three-wave mixing element yielding amplified outgoing modes. We will first summarize the derivation of the amplifier input-output relations (21) before analyzing the erasure of the which-path information.
ii.1 Input-output relations
We consider now a device with two ports and for each port we separate the incoming and outgoing modes. The device is operated in reflection but in the visualizations (see Figs. 1-2) we draw it in transmission for conceptual simplicity. For a reflective device, the signal input and output are related to the internal mode of the device through the coupling strength (22):
Because the internal mode is coupled to the external modes, it becomes damped at the rate and driven with :
Similar equations hold for the idler mode . To solve for the outputs as a function of the inputs in Eqs. (1)-(2), we need to know the internal dynamics. For that purpose and for concreteness, we take the quantum-limited phase-preserving amplifier to be realized with a Josephson parametric converter amplifier (JPC) (23); (24); (25), whose Hamiltonian is a three-wave mixer,
To operate the device as an amplifier, the pump mode is driven strongly at the frequency such that it reaches a steady state . The pump mode provides the energy for the amplification. By ignoring the remaining quantum fluctuations in and going into the frame rotating at the eigenfrequencies of the signal and the idler modes, the resulting dynamics is set by the Hamiltonian
The gain factor and the amplification bandwidth are
where the total coupling rate , the coupling asymmetry and the power gain
In the limit of a wide amplification bandwidth , i.e., when the input signals are slowly changing with respect to the time scale , the gain factor (6a) is a frequency independent constant to leading order (the zeroth order) in . This implies time-local input-output relations:
expressed in the frame rotating at the resonance frequencies; see Fig. 1(a). For simplicity we have ignored the phase between the idler and the signal ports. The next-to-leading order contribution is to ignore the second order terms but keep the first order terms in Eq. (6a), resulting in temporally non-local but causal responses with a delay kernel of the type in Eqs. (7).
ii.2 Erasure of the which-path information
We see from the input-output relations (7), that when the idler input is the vacuum, both quadratures of the signal input are amplified along with an added extra half a quantum of noise originating from the idler. When both input ports contain signals, they are coherently superposed in the outputs and the which-path information of the signals is erased in the frequency domain.
To further analyze the erasure of the which-path information, we specify that the signal output is measured by heterodyne detection (26); (27): equal sampling of quadratures and , see Fig. 1(a). For simplicity we scale the signal with and consider the high gain limit . The outcomes of a weak continuous measurement of infinitesimal duration are,
They consist of two parts: the part expected based on the prior knowledge of the system and the unexpected part (the ‘innovation’) . Alternatively, represent the quantum noise of the channels and are modeled by independent Wiener processes with variance (26); (27).
From Eq. (8), we see that the amplification-detection scheme is equivalent to a - beam-splitter followed by phase-sensitive amplifiers in both of the output arms implementing two single-quadrature (homodyne) measurements in the and directions, see Fig. 1. This interpretation also illustrates the nature of which-path erasure: observers of the output ports cannot know where the signals came from. In the high gain limit, the idler output is fully entangled with the signal output (28) containing no extra information. This can be understood as two-mode squeezing by the amplifier having effectively erased two of the four incoming quadratures.
The input-output relations (7) of the amplifier are expressed with the Hermitian conjugated input operators, and . But notice that they can be interpreted as complex conjugation in Eq. (8) when the amplifier is followed by the quadrature measurements. This is consistent with the physical picture of unidirectional information flow from the signal sources. The observer of the unidirectionally traveling signals can only make measurements whose measurement backaction to the system is expressed by the operators , that is, e.g., observations of discrete photon emissions or continuous leaking of the cavity field. The interpretation of the amplification-detection stage through beam-splitters and quadrature measurements gives practical means to handle components of cascaded quantum network with Bogoliubov transformations in their input-output relations (29); (30); (31).
Iii Concurrent generation of remote entanglement
We now study a phase-preserving amplifier as a which-path information eraser to concurrently generate remote entanglement. The considered configuration consists of two transmon qubits inside separate, remote superconducting cavities, see Fig. 2. The cavities are driven through weakly-coupled input ports and monitored through separate strongly-coupled transmission lines that form the signal and idler ports of a Josephson parametric converter amplifier. In the frame rotating at the cavity driving and the qubit frequencies , the Hamiltonian for a dispersively coupled qubit-cavity pair is
where denotes the dispersive coupling strength. When the cavities are driven at their resonance frequencies , they build up symmetric qubit-state dependent phase responses. In the ideal case these responses are identical and when they are amplified with a high-gain quantum-limited phase-preserving amplifier according to Eq. (7) there is no which-qubit information left in the outgoing modes. This allows pure joint measurements of the signal sources, here the transmon qubits, with measurement back-action that projects to an entangled subspace.
iii.1 Stochastic master equation of the joint measurement process
To analyze the configuration and the entanglement generation in detail, it is modeled with a stochastic master equation (26); (27) (SME) for the density matrix of both qubit-cavity pairs. We derive it by using input-output theory for cascaded quantum systems (29); (30); (22) (essentially and ) and representing the amplification-detection stage as a beam-splitter followed by two quadrature measurements as in Eq. (8). See details of the derivation in Appendix A. The resulting Itō stochastic master equation for the monitored qubit-cavity pairs becomes
where . The first two rows describe the open quantum system dynamics. The dissipator terms model the coupling of the cavity fields to the transmission lines with loss rates . The dissipators for the qubits describe relaxation and pure dephasing with the rates and .
The last two rows describe the measurement backaction, which updates the best estimate of the quantum state based on the new information in the heterodyne measurement of the signal output. It is represented by innovation terms (27) that are linear in the measurement operators but nonlinear in the density matrix . The efficiencies , appearing in the innovation terms, describe the fraction of the information measured by the observer. The remaining fraction of the information is lost to the environments and averaging over it leads to dephasing. Here the efficiencies are and . They consist of the transmission coefficients and the measurement efficiency of the amplification-readout chain. With a finite amplification gain , there is an asymmetry between the idler and signal outputs in Eq. (7). When measuring only the signal output, the associated loss of information is represented with the inefficiency (in practice, dB (1); (25)).
iii.2 Stochastic master equation for the monitored qubits
Given that we consider driven and damped dispersively coupled qubit-cavity systems, the cavity states can be assumed as a superpositions of the qubit dependent coherent states (32) , see Fig. 2. The coordinates follow the classical equations of motion, , see Fig. 3(c)-(d). With modern superconducting technology, one can achieve typical values of s (33); (34) for the decay times of the qubit. In this regime, the probability for a relaxation event in either of the qubits can be assumed negligible small during the measurement time s. Then the qubits’ dynamics can be seen as frozen out, except for the measurement backaction. In this case, the cavity dynamics can be integrated out from the full SME (10) by using these time-dependent coherent states as an Ansatz (35); (32). Reduction of the full SME (10) is very helpful since we are primarily interested in the qubits and the full SME is rather inconvenient to deal with due to its large Hilbert space.
We approach here the integration of the cavity dynamics from a simple point of view. First, by exploiting the linearity of the innovation terms with respect to the measurement operators, we notice that the qubit-cavity pairs in Eq. (10) can be organized so that the two blocks look superficially decorrelated, except for the same stochastic ‘driving fields’ . In practice, Eq. (10) encapsulates a technical source of correlation through the terms and of the innovation operators. However, as these terms stem from the normalization condition, we can first uncorrelate the two blocks by sacrificing the normalization. Then, we make use of the single qubit-cavity results of Ref. (32) for integrating out the cavity individually on both qubit-cavity pairs, and finally add the normalization. We have verified these heuristic arguments with a numerical comparison and with an analytic calculation where we explicitly integrate out the cavity dynamics from the full SME (10) using a positive P representation (35); (36).
The resulting SME for the qubits’ density matrix is
where is the ac Stark effect induced by the photons in the cavities. The photons leaking out from the cavity carry information about the qubits’ population (35) causing measurement induced dephasing at the rates . The information is encoded into the distinguishability of the pointer states , see Fig. 2-3. Therefore we define the complex measurement amplitude of the operator as,
whose real and imaginary parts are related to the measurements in the and directions, respectively. The measurement rate is .
The outcomes of a weak continuous measurement of infinitesimal duration of both and quadratures (heterodyne) of the amplified signal, from Eq. (8), can be expressed as
where the expectation values, for example , are taken for the instantaneous qubits’ density matrix . Additionally, the terms involving are not informationally meaningful since they only deterministically offset the signals, and will be ignored in what follows. Importantly, SME (11) and the measurement outcomes (13) show that heterodyne measurement of the output of a quantum-limited phase-preserving amplifier implements a pair of two-qubit measurements corresponding to the operators
The complex phase and the magnitude of the measurement amplitudes are tunable in situ by the cavity driving. Thus, the measurement operators can be changed continuously from a simultaneous separate readout of and into a joint entangling readout .
iii.3 Balanced driving for perfect erasure of the which-path information from dissimilar sources
To utilize the readout for remote heralded entangling, we drive the cavities at resonance implying . For the most efficient entangling readout, we would like make both measurement amplitudes equal throughout the measurement—including cavity transients and unequal cavity-qubit parameters. In an entangling readout, one does not want to gather any single-qubit information. Remarkably, the matching of the measurement amplitudes can be achieved by simple engineering of the drive amplitudes. This result improves the flexibility and scalability of the concurrent remote entangling scheme. We get the balanced driving amplitude,
as a result of solving the input of the cavity 2 for a given output , visualized in Fig. 3. Cavity responses in the direction are left unmatched because they carry no information about the qubits’ population, see Figs. 2-3. However, the noise needs to be recorded because it encodes the stochastic relative phase shift between the qubits due to the unequal photon shot noise in each cavity (1).
As the erasure of the which-qubit information in the amplifier output is made perfect with the balanced driving, the subsequent heterodyne measurement realizes a measurement of the joint operator . For the initial state , where , the measurement backaction projects the system into a heralded entangled state with the success probability .
Iv The quantum filter
In the quantum filtering, one assumes that the measurement records and are known from the initialization up to a time and then one would like to know the best estimate of the state of the open quantum system conditioned on this particular measurement record and the initial condition. The stochastic master equation gives the incremental update from to given the new information in the measurement records and knowledge of the system . Naturally, a way to obtain the quantum filter is to solve the SME numerically for each measurement trajectory individually. However, this may generally be a computationally expensive and slow task. Thus, an analytic quantum filter would be much more appealing.
We now consider SME (11) with the balanced real measurement amplitudes ,
where we have explicitly written the backaction of the -measurement in the form of stochastic phase rotation (the second term). The measurement currents are
In the following we take into account only the increased dephasing rate of the qubits by the relaxation processes, , but ignore its effect on evolution of the qubits’ population. This is justified by long typical times, s, with respect to typical measurement time s.
Based on this Itō calculus, we have found the analytic solution for SME (11) that expresses the two-qubit state conditioned on an actual stochastic measurement record and the initial state . The time-dependent full solutions for the most important two-qubit Bloch coordinates are (the rest are shown in Appendix B)
The ac Stark effect induced rotation angles are denoted with and . We have used the notation of and defined the combined measurement efficiency that has an important role in the analysis of the entanglement fidelity.
We define as the apparent total information content recorded by the observer up to time . The stochastic measurement records and are weighted integrals of the measurement outcomes (17),
The measurement amplitude is the correct relative weighting function—a matched filter—between the different time instances of the measurement. The analytic solutions of Eqs. (19) and (30) have been verified by comparing them to the numerical solution of the stochastic master equation with perfect overlap within the accuracy of the numerical methods.
This reduction is due to an accumulated temporal mismatch between the acts that cause the qubit dephasing and the measurement backaction, at the rates and , respectively. Physically, each qubit is entangled with cavity photons and the entanglement is not removed by the measurement backaction until the photons have leaked out. At the end of a measurement , when the cavities have been brought back to the vacuum and all the available information has been recorded, the purity revives. Interestingly, during cavity transients the dephasing rate can have negative values, which implies revival of qubit coherence. This non-Markovian dynamics originates from coupling the qubits to Markovian reservoirs indirectly through the cavities. The solutions (19)-(20) generalize and go beyond the previous results (32); (37); (1); (3); (39); (38) by deriving the quantum filter for a concurrent entangling two-qubit readout and verifying the purity reduction directly from stochastic calculus.
The solution (19)-(20) is an analytic quantum filter without need for stochastic numerical solutions that would be limited by time step approximations (40); (3). The quantum filter can be interpreted in two ways. First, given an actual measurement record it draws the stochastic quantum trajectories of the two-qubit state, see Fig. 4(a)-(b). In another interpretation, the quantum filter gives the two-qubit state as function of measurement outcomes representing the effect of the measurement backaction, visualized in Fig. 4(c). The measurement outcomes are normally distributed with zero mean and variance . For a strong measurement, the distribution of the measurement outcomes is a mixture of three normal distributions, each with variance , centered at . This gives a definition of the measurement strength as the distinguishability of the parity subspaces.
To examine the fidelity of the entanglement, we calculate the concurrence (42) from the quantum filter solutions [see Fig. 4(b)]. In the limit of strong measurement , the state has collapsed with high probability either to an entangled state with or to a product state with . In this limit, the concurrence can be accurately approximated from the simplified expression (43): . Expressing this with the quantum filter solutions results in
where is the sum of qubits’ dephasing rates.
Let us now consider the case of where the measurement has ended such that . Curiously, there exists an important bound for the total measurement efficiency since the numerator of Eq. (22) needs to be positive for an entangled state with concurrence . Even for ideal qubits (), there is a threshold for forming a entangled state. For symmetric transmission this corresponds to . In the presence of decoherence the bound naturally becomes stricter:
Above the threshold, the purification by the measurement backaction dominates over the measurement induced and qubits’ natural dephasing. In addition one can see from Eq. (22) that given a fixed measurement time and non-ideal efficiencies , there exists an optimal measurement strength that maximizes the concurrence. Physically this can be understood as follows: with the optimal , the purification by the measurement backaction is in balance with the measurement induced and qubits’ natural dephasing.
V Discussion and conclusions
Compared to corresponding photon-counting based concurrent remote entangling schemes (12); (13) whose entanglement fidelity is more robust to losses and inefficiencies, the proposed continuous variable scheme achieves very high generation rate of entangled qubit pairs (s). Experimental values ( s, s, ) reachable in near-future superconducting circuit experiments result in concurrence . The current experimental capabilities are such that the cavity-qubit asymmetries can be reduced through our pulse engineering scheme to the point that they will not be a limiting factor—rather it is the overall efficiency of transmission and measurement, where future technical improvements will lead to considerably better concurrencies.
In conclusion, we considered a readout chain of a quantum-limited phase-preserving amplifier followed by heterodyne detection and developed a physically intuitive description compatible with the theory of cascaded quantum systems. Based on a stochastic master equation approach, we theoretically demonstrated that the amplifier can be utilized as an eraser for the which-qubit information, even from dissimilar sources, and an element in a promising protocol for concurrent entanglement generation between remote superconducting qubits. This protocol is feasible with existing technologies and can be expected to demonstrate formation of entangled remote qubits, primitive constituents of quantum communication and distributed quantum computation.
Acknowledgements.We are very grateful for R. T. Brierley and Shyam Shankar for many useful discussions. We acknowledge support from ARO W911NF-14-1-0011, W911NF-14-1-0563, NSF DMR-1301798 and the Yale Center for Research Computing.
Appendix A Cascaded quantum systems with a quantum-limited phase-preserving amplifier and heterodyne detection
To rigorously derive the stochastic master equation of a cascaded quantum system (29), an effective method is to construct the unidirectional quantum network by using the input-output triplets of the network elements (30). The -triplet contains the scattering matrix for the input-output ports of the element, the vector that specifies the coupling to input-output ports and the internal Hamiltonian . For compiling a network, one needs to know the rules for the cascade (series) and concatenation (parallel) products. Let us consider two systems and , then the cascade and concatenation products are, respectively,
In the cascade product, the outputs of the system are connected to inputs of the system , and the cascaded Hamiltonian has a corresponding driving term .
The unidirectional quantum network of the concurrent remote entanglement setup of Figs. 1 and 2 is shown in Fig. 5. The -triplets of the individual elements are: