Quantum Trajectories and Their Statistics for Remotely Entangled Quantum Bits

PHYSICAL REVIEW X 6, 041052 (2016)

Areeya Chantasri,1,2 Mollie E. Kimchi-Schwartz,3 Nicolas Roch,4 Irfan Siddiqi,3 and Andrew N. Jordan1,2,5 1Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA 2Center for Coherence and Quantum Optics, University of Rochester, Rochester, New York 14627, USA 3Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley, California 94720, USA 4Université Grenoble Alpes, Institut NEEL, F-38000 Grenoble, France and CNRS, Institut NEEL, F-38000 Grenoble, France 5Institute for Quantum Studies, Chapman University, 1 University Drive, Orange, California 92866, USA (Received 31 March 2016; revised manuscript received 6 October 2016; published 14 December 2016) We experimentally and theoretically investigate the quantum trajectories of jointly monitored transmon qubits embedded in spatially separated microwave cavities. Using nearly quantum-noise-limited superconducting amplifiers and an optimized setup to reduce signal loss between cavities, we can efficiently track measurement-induced entanglement generation as a continuous process for single realizations of the experiment. The quantum trajectories of transmon qubits naturally split into low and high entanglement classes. The distribution of concurrence is found at any given time, and we explore the dynamics of entanglement creation in the state space. The distribution exhibits a sharp cutoff in the high concurrence limit, defining a maximal concurrence boundary. The most-likely paths of the qubits’ trajectories are also investigated, resulting in three probable paths, gradually projecting the system to two even subspaces and an odd subspace, conforming to a “half-parity” measurement. We also investigate the most-likely time for the individual trajectories to reach their most entangled state, and we find that there are two solutions for the local maximum, corresponding to the low and high entanglement routes. The theoretical predictions show excellent agreement with the experimental entangled-qubit trajectory data.

DOI: 10.1103/PhysRevX.6.041052 Subject Areas: Quantum Physics, Quantum Information

I. INTRODUCTION The rapid development of quantum information science Measurement-induced entanglement of spatially sepa- in the superconducting domain [13] has seen an exponen- rated quantum systems is a startling prediction of quantum tial increase in qubit coherence time within the past decade, mechanics [1–7]. Recent experiments have demonstrated leading to many scientific advances [14]. This technologi- this effect via single-photon heralding [8–11], as well as via cal progress has led to a wide variety of advances in continuous measurement of photons interacting with qubits quantum physics, as observed and controlled in these [12]. The latter has the advantage of being able to systems, including greater than 99% fidelity in single-qubit investigate the physics of entanglement creation continu- quantum gates [15], multiqubit-entanglement-generating ’ ously, leading to new effects such as the sudden creation of gates [16], the violation of Bell s inequality [17], and entanglement after a finite measurement period, dubbed quantum-process tomography [18]. Recent developments entanglement genesis [6]. However, many questions are include the observation of quantum states of light in outstanding, such as the following: (a) What is the complete resonators [19], as well as nearly quantum-limited para- characterization of the dynamics of entanglement creation metric amplifiers [20,21]. as a continuous trajectory? (b) What is the statistical The improvement of coherent quantum hardware has distribution of the entanglement at any time during the brought with it a renewed focus on the physics of quantum process? (c) What is the most-likely way entanglement is measurement. Generalized measurements have been car- generated? In this work, we give a systematic answer to ried out in superconducting qubits [22], realizing probabi- these questions, as well as others, by analyzing experi- listic measurement reversal [23,24], weak values [25,26], mentally entangled quantum trajectories of jointly mea- and their connection with generalized Leggett-Garg sured transmon qubits and showing excellent agreement inequalities [27]. Continuous measurements [28] in super- with the theory developed here. conducting systems have only recently been realized, owing to the challenge associated with high-fidelity detection of microwave signals near the single-photon level. In particular, experimental achievements include continuous Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- feedback control [29,30] and the tracking of trajectories in bution of this work must maintain attribution to the author(s) and individual experiments in the plain measurement case the published article’s title, journal citation, and DOI. [31–34] as well as with a concurrent Rabi drive [35].

2160-3308=16=6(4)=041052(14) 041052-1 Published by the American Physical Society AREEYA CHANTASRI et al. PHYS. REV. X 6, 041052 (2016)

These experiments show detailed quantitative agreement distribution changes in time, from a separable state with with theory, indicating a good understanding of quantities zero concurrence, to projected states in either a separable such as the most-likely path of the quantum state between subspace or an entangled subspace. This can also be seen boundary conditions, predicted with an action principle of from the most-likely path analysis, showing the emergence an associated stochastic path integral [36–38]. of different most probable paths for each final state. Going beyond a single qubit opens the possibility of Moreover, we investigate the distribution of the time to measurement-induced entanglement, using a dissipative maximum concurrence, finding that the most probable time process as a tool to generate quantum correlations. For to maximum values of concurrence has a bimodal structure. quantum architectures building upon transitions at optical Studying the statistics of a large set of trajectories—rather frequencies [8–11], this feat has typically been achieved by than averaging over all of them to study the dynamics of the relying on the correlated detection of photons at the output ensemble—enables an unprecedented understanding of the of a beam splitter [1] to herald an entangled state. While dynamics of entanglement creation under measurement. powerful, this measurement protocol is binary and instan- The paper is organized as follows. In Sec. II, we describe taneous, and it allows no insight into the dynamical our transmon qubit measurement setup and the method of processes underlying the generation of the entangled state. reconstructing the joint trajectories. In Sec. III, we derive In solid-state systems, such as superconducting qubit the concurrence-readout relation and compute the distri- transitions in the microwave regime, there has been bution of concurrence as the measurement backaction tremendous interest in continuously generating bipartite proceeds. This distribution is then compared with the data [3–6,39] and multipartite [2,40,41] entangled states, using generated from the experiment. In Sec. IV, we turn to the weak measurements that slowly interact with the qubits, in most-likely path analysis, finding three possible paths that such a way that enables the resolution of the dynamical the joint system takes from a separable initial state to final aspects of the entangling backaction. Analog feedback entangled or separable states. The experimental most-likely control can naturally be applied in the weak continuous- paths are generated independently to compare with the measurement regimes [42–44], and digital feedback- theoretical prediction. We also discuss the distribution of generated entanglement has already been demonstrated the time for the two qubits to reach their maximum [45]. Joint measurement is uniquely useful as a means to entanglement. The conclusions are presented in Sec. V. generate entanglement between remote qubits [12,46–50], Additional details of the most-likely path calculations and a for which no local coupling exists and therefore no unitary discussion of the parity meter are presented in the means of generating entanglement are available. appendixes. The chief advantage of the continuous approach is in the efficiency of entanglement generation. In contrast to photon- II. TRAJECTORIES OF TRANSMON QUBITS counting schemes (in which entanglement-generation rates are heavily limited by photon losses), continuous measure- We consider a two-qubit system realized by super- ment enables an entanglement-generation rate limited only conducting transmon qubits embedded in spatially sepa- by the premeasurement initial state and the postselection rated microwave cavities in a setup optimized to reduce criterion, which are both experimental choices. However, losses between the cavities. These qubits are jointly continuous measurements may be highly sensitive to dis- measured via a dispersive readout in a bounce-bounce sipative losses and inefficiencies in amplification. geometry [see Fig. 1(a)] in which a microwave tone is Understanding and studying the dynamical processes under- sequentially reflected off of two copper cavities, each lying continuous measurement-induced entanglement is containing a transmon qubit, and subsequently amplified therefore critical for balancing these tradeoffs. and measured via homodyne detection. The cavities are In this work, we combine an efficient quantum amplifier directly joined by a circulator that enforces the unidirec- with a continuous half-parity measurement to conduct tional transfer of the coherent state. A single transmon detailed experimental and theoretical investigations of interacts with the light in its cavity via a dispersive the statistics of individual trajectories of qubit pairs as interaction of the form they undergo the entangling process. The half-parity χ σðjÞ † measurement can distinguish between the even-parity Hj ¼ j z a a; ð1Þ and odd-parity subspaces, similar to a full-parity measure- χ ment, but also between the two states j00i and j11i in the where j is the dispersive interaction strength of the light even-parity subspace, leaving the odd subspace entangled. interacting with the dipole moment of qubit j (cross-Kerr By peering into the ensemble, we can understand the nonlinearity between the qubit and the cavity), coupling to ðjÞ full spectrum of evolution paths as the two-qubit state its σz Pauli observable. The net effect of the bounce- gradually projects onto the entangled subspace or onto a bounce geometry is that the effective interaction of the trivial separable state. We explore the probability distribu- combined system is described by a Hamiltonian of the form tion of the qubits’ concurrence to understand how the H1 þ H2. After amplification and homodyne measurement,

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(a) (b)

FIG. 1. Experimental setup and transmon qubit trajectories. Panel (a) shows a simplified illustration of the experimental setup. Two transmon qubits in two remote cavities are linked via a single microwave circulator in a bounce-bounce geometry; the output of the circulator is routed to a high-efficiency amplification chain (not shown). The upper inset represents the distribution of the microwave field in the (X1, X2) quadrature plane at the output of the amplifier. The three possible outcomes correspond to three different subspaces: j00i, j11i, and the odd-parity subspace (spanned by the j01i and j10i states). Panel (b) shows the evolution in time of the measurement outcome probability distribution pðVtÞ. Panel (c) shows the agreement between the experimentally generated conditional tomography xiðVtÞ (symbols and shaded error bars) and the theoretical Bayesian reconstruction (dashed lines), for a single time t ¼ 0.48 μs. In panel (d), we use such reconstructions to predict and verify the trajectory of a single iteration of the experiment, showing both density matrix elements xi and concurrence (inset). this interaction gives a dispersive phase shift of ϕ for on the qubits; therefore, it can be used to track the evolution states j0i, j1i. The shift ϕ is given, in general, by of the system in time. By reducing the amplitude of the ϕ α − α α j ¼ arg½ j0i;j arg½ j1i;j, where i;j represents the intra- coherent state used to measure the system, we can engineer cavity coherent state conditioned on state i of qubit j. In the an entangling measurement with a characteristic measure- fully symmetric, weak measurement case (when all qubit ment time ranging from several hundred nanoseconds to χ ≪ κ and cavity parameters are identical and j¼1;2 ), this several microseconds: Critically, these time scales are phase shift is given by ϕj ≈ 2χj=κj, where κj is the cavity easily resolvable experimentally. The dynamics or the damping rate of cavity j. trajectory of the system state can be obtained via the full In the bounce-bounce geometry, there generally exists a master equations [46,47], using a two-cavity polaron probe frequency at which the dispersive shifts from both transformation to account for the cavity degree of freedom, transmon qubits are the same (ϕ1 ¼ ϕ2 ¼ φ), such that the giving the stochastic master equation for the qubit trajec- measurement tone can acquire a phase shift of either 2φ,0, tories. Alternatively, in a limit of large cavity decay rate 0, −2φ for states j00i, j01i, j10i, j11i. For small χ=κ, κ ≫ jχj, the qubit evolution can be continuously tracked via this results in a half-parity measurement on the two qubits, the quantum Bayesian approach [2,3], inferring the current where the measurement result can distinguish between states of the system from the measurement readouts and three subspaces: the j00i state, the j11i state, and the odd- how likely they are to occur. The two approaches both show parity subspace, but not within the odd-parity subspace, good agreement in tracking the qubit pair state [12]. spanned by the j01i and j10i states. Conditioning on the In this paper, we focus on the quantum Bayesian measurement results with a zero phase shift can lead to approach, as it is directly related to the probability creation of an entangled state within the odd-parity sub- distribution of the measurement readout and naturally leads space (superpositions of j01i and j10i). to the probability distribution of quantum trajectories. Let When the output of the second cavity [on the right in us denote pðVtjiÞ as a probability density function of a Fig. 1(a)] is directed to a nearly quantum-limited ampli- measurement readout Vt conditioned on the two-qubit fication chain, the instantaneous homodyne detection states i, where i ¼ 1, 2, 3, 4 represent the states j00i, signal can be correlated with the measurement backaction j01i, j10i, j11i, respectively. The quantum Bayesian update

041052-3 AREEYA CHANTASRI et al. PHYS. REV. X 6, 041052 (2016) for this type of double-qubit measurement provides a except ρ23. In an ideal half-parity measurement, the decay convenient way to calculate the joint state at time t,given of ρ23 would be limited only by the intrinsic lifetimes of the a known state at the initial time and the readout Vt, qubits; however, we must additionally account for exper- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi imental imperfections in the matching of δv2 and δv3 and −γ ρ ð0Þ pðV jiÞpðV jjÞe ijt for the loss of photons between the two cavities. These ρ t ij P t t ; ijð Þ¼ 4 ρ 0 ð2Þ effects are included in the (slightly time-dependent) k¼1 kkð ÞpðVtjkÞ dephasing rate γ23. where ρ denotes the ij element of the two-qubit density Since we expect most of the off-diagonal terms to damp ij quickly, we only consider five relevant density matrix matrix and γij is a decoherence rate associated with the elements: x1 ≡ ρ11, x2 ≡ ρ22, x3 ¼ ρ33, x4 ≡ ρ44, and x5 ≡ matrix element. The quantum Bayesian approach is equiv- jρ23j [52]. In order to compare the Bayesian prediction to alent to the positive-operator valued measure (POVM) the true density matrix, we perform conditional tomogra- formalism [51], provided we identify the measurement phy [12,53] to experimentally reconstruct the full exper- operators. In the case of no additional Rabi drives on either imental mapping V ↦ ρðV ;tÞ. In conditional tomography, qubit, the measurement operators are given by t t we collect an ensemble of trajectories that reach a particular pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vt at time t, and tomographically reconstruct that ensem- MV ¼ pðVtjoddÞðj01ih01jþj10ih10jÞ t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ble. This provides a density matrix conditioned on Vt.In Fig. 1(c), we show good agreement between the Bayesian þ pðVtj1Þj00ih00jþ pðVtj4Þj11ih11j; ð3Þ prediction and conditional tomography reconstruction of x ðV Þ for a single measurement time t ¼ 0.48 μs. We show where the distribution pðVtjoddÞ¼pðVtj2Þ¼pðVtj3Þ is i t the same for any state within the odd subspace. The examples of transmon qubit trajectories in Fig. 1(d) for an † initial state prepared in a product of xˆ states, i.e., the equal POVM elements, M MV , correctly integrate to the iden- Vt t superposition state ð1=2Þðj00iþj01iþj10iþj11iÞ, such tity operator in the two-qubit Hilbert space, understanding that the five relevant density matrix elements are that we integrate with a measure for V . Applying the state 0 0 0 0 0 t x ¼ x ¼ x ¼ x ¼ x ¼ 1=4. We show both the update rule, conditioned on an observation of the readout 1 2 3 4 5 † † Bayesian reconstruction and the tomographic verification, V , ρðtÞ¼M ρð0ÞM =Tr½M ρð0ÞM recovers Eq. (2) t Vt Vt Vt Vt which are in good agreement. This verifies that the γ 0 in the case of no extra decoherence Rij ¼ . Bayesian reconstruction can be used to faithfully translate t ~ 0 0 We define the readout Vt ≡ ðf=tÞ 0 Vðt Þdt − v0 as a a noisy measurement signal into the stochastic evolution of time average of a raw homodyne voltage signal rescaled a joint quantum state. with a weight factor f and an offset v0, where f is chosen so We conclude this section with a qualitative discussion σ2 1 4η that the variance V ¼ = mt is a function of a quantum of the mechanisms behind the entanglement dynamics, t which will be followed up with a quantitative analysis in efficiency of the homodyne measurement, ηm ≈ 0.22. The total probability distribution the next section. As discussed previously, the half-parity measurement can distinguish between the two even- X4 parity states but not within the odd-parity subspace. pðVtÞ¼ ρkkð0ÞpðVtjkÞ; ð4Þ We can think of this measurement as a continuous version k¼1 of an ideal projective measurement with three projectors Π 00 00 Π 01 01 10 10 Π 11 11 1 ¼j ih j; odd ¼j ih jþj ih j; 3 ¼j ih j, shown in Fig. 1(b), slowly resolves into the three peaks which partition unity in the two-qubit-state space. Given an expected for a half-parity measurement. arbitrary separable pure state, jψi¼ða1j0i1 þ b1j1i1Þ× The conditional readout distributions are well ap- ða2j0i2 þ b2j1i2Þ, textbook wave-function collapse upon proximated by Gaussian functions, giving pðVtjiÞ¼ measurement of the odd result gives the (un-normalized) 1=2 2 ðt=πsÞ expf−ðVt − δviÞ t=sg with the centering signals state jψ~ i¼a1b2j01iþb1a2j10i. Clearly, for certain val- δvi ¼ vi − v0 for i ¼ 1; …; 4, where s ¼ 1=2ηm. The ues of a1, b1, a2, b2, the postcollapse state will be measurement process cannot distinguish the two states entangled. In particular, the choice a1 b1 a2 b2 pffiffiffi ¼ ¼ ¼ ¼ in the odd-parity subspace; therefore, the readout distribu- 1= 2 previously mentioned will result in a maximally 01 10 tions corresponding to the states j i and j i are com- entangled state. Mathematically, entanglement is created δ ≈ δ ≈ 0 pletely (or nearly) overlapped, giving v2 v3 and because the odd projection operator cannot distinguish −δ ≈ δ ≈ δ v1 v4 v. The measurement strength is character- within the odd subspace. ized by an inverse of a characteristic measurement time Of the three possibilities between collapse to the two τ ≈ 1 δ 2η γ δ ≠ δ m = v m. The dephasing rates ij for vi vj are even states or to the odd subspace, quantum mechanics dominated by the effect of the distinguishability between only tells us the statistics of these events happening; it does −1 2 states i and j, γij ∼ ðηm − 1Þðδvi − δvjÞ =4s [2], resulting not predict which will occur in a given run. In addition, in in the strong suppression of all off-diagonal elements the continuous measurement case, we can only predict the

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0 1 … 5 statistics of the system by choosing one eventual path where xi for i ¼ ; ; are the matrix elements of the among the various choices. However, because the proba- initial qubit state, and γ ¼ γ23. We have used simplified ≡ bility distribution of the results is of the simple form (4),we notations Pi pðVtjiÞ for the probability distributions,P and may understand the gradual transition from separable to 4 0 N for a normalized factor given by N ¼ k¼1 xkð ÞPk. entangled as a given measurement result indicating which Substituting the probability distribution functions with basis state the system is collapsing into, masked by additive the Gaussian functions of the means δvi for i ¼ 1; …; 4,we quantum noise. Transitions from one branch to another can obtain a form of ct explicitly as a function of Vt and t, occur and may be understood as originating from the tails of the distributions overlapping. As time develops, those qffiffiffiffiffiffiffiffiffi tails decrease, so a larger (and less likely) quantum 2 0 α −β −γ 0 0 α −β c V ;t x eð 23Vt 23 Þt − x x eð 14Vt 14Þt ; 7 fluctuation is required in order to cause the quantum tð t Þ¼Mf 5 1 4 g ð Þ system to transition from one branch to another. This P increase of knowledge resulting in an altered quantum state 4 0 2α −2β M x e iVtt it may be understood as Bayesian inference, suitably gener- where the prefactor is given by ¼ i¼1 i α δ α α α alized to take quantum entanglement into account, Eq. (2). using a set of defined variables: i ¼ vi=s, ij ¼ i þ j, β δ 2 2 β β β 1 2η i ¼ vi = s, ij ¼ i þ j, and s ¼ = m. III. CONCURRENCE TRAJECTORIES AND The quantity ct in Eq. (7) would represent the actual ≥ 0 THEIR DISTRIBUTION concurrence of the qubit state at any time t,ifcðtÞ is satisfied. For our chosen initial state, a product of single- We now give a quantitative analysis of the statistical 0 0 0 0 0 qubit xˆ states, x1 ¼ x2 ¼ x3 ¼ x4 ¼ x5 ¼ 1=4, the quantity process of entanglement creation for continuous measure- ct is non-negative whenever the condition ðγ − α23Vt þ ment. As a measure of entanglement between two parties β23Þ < ðβ14 − α14VtÞ is true. From the experimental data such as the transmon qubits, concurrence is a convenient (e.g., for the setup with τm ¼ 0.60 μs), we have −δv1 ≈ δv4 choice and can be computed directly from the density and δv2 ≈ δv3 ≈ 0 (giving α14, α23, β23 ≈ 0) and matrix of the system [54]. The concurrence formula for β14 ∼ 3.2 MHz, while γ < 0.6 MHz. Therefore, the con- the half-parity setup is greatly simplified because of dition is always satisfied, and the second term in the the suppression of most matrix elements, resulting in an brackets of Eq. (7) decays faster than the first term, X-shape density matrix, of which the concurrence is resulting in a quantity that is always non-negative. calculated from [54,55] Consequently, the quantity in Eq. (7) gives the concur- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rence-readout relationship CðV ;tÞ¼c ðV ;tÞ, and the C 2 0 − t t t ðtÞ¼ max f ;x5ðtÞ x1ðtÞx4ðtÞg: ð5Þ concurrence at any time t can be determined directly from the time-averaged measurement readout Vt. The value of concurrence ranges from 0 for a separable The concurrence formula in Eq. (7) can be simplified state to 1 for the Bell states. The concurrence trajectory of further by considering a perfectly symmetric half-parity an exemplar trajectory is shown in the inset to Fig. 1(d).In measurement, δv2 ¼ δv3 ¼ 0 and −δv1 ¼ δv4 ¼ δv. Given this section, we use the simplified formula Eq. (5) to show the initial state, a product of two qubit xˆ states, the that the concurrence of the two-qubit state can be deter- concurrence is then given by mined directly from the measurement readout, which then leads to the derivation of the concurrence probability 2 distribution as a function of the measurement readout e−γt − e−δv t=s C ˆ ðV ;tÞ¼ 2 ; ð8Þ and measuring time. ps;x t −δv t=s 1 þ coshð2Vtδvt=sÞe A. Concurrence-readout relationship where the subscript “ps, xˆ” indicates the perfectly Let us consider the concurrence formula in Eq. (5), symmetric half-parity measurement given the specific where its value is determined by the second term in the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi initial state. brackets, which we denote ct ≡ 2fx5ðtÞ − x1ðtÞx4ðtÞg.If c is a non-negative quantity (i.e., c ≥ 0), then the t t B. Probability density function for concurrence is simply given by CðtÞ¼ct. At the end of this section, we show that this is always the case for our concurrence trajectories chosen initial qubit state and parameter regimes, but it is not From the direct relationship between the measurement true in general [6]. From the Bayesian update in Eq. (2),we readout and the concurrence of the qubit state, the prob- calculate the quantity ability density function of the concurrence can be derived qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi from the probability distribution of the time-averaged 2 n pffiffiffiffiffiffiffiffiffiffiffi o 0 −γt 0 0 signal V . The distribution of the time-averaged readout c ¼ x P2P3e − x x P1P4 ; ð6Þ t t N 5 1 4 is given by

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X4 0 (a) pðVtÞ¼ xi pðVtjiÞ: ð9Þ i¼1

The variance of the distribution σ2 ¼ s=2t narrows as time Vt increases, leading to the collapse of the joint qubit state into three categories: j00i state, j11i state, and some superposition state of j01i and j10i after a few characteristic measurement times τm. Knowing the probability density function of the time- averaged signal, we follow the transformation of random variables Vt → C using the concurrence-readout relationship, Eq. (7) [or Eq. (8) for a perfectly symmetric case]. The concurrence CðVt;tÞ is not a monotonic function in Vt; instead, it has a bell-like shape, as shown in the inset of (b) Fig. 2(a). We write the cumulative distribution function of the concurrence FC;tðcÞ¼pC;tðC ≤ cÞ¼pðVt ≤ V−Þþ 1 − ≤ f pðVt VþÞg, where pC;tðcÞ is a probability density function for the concurrence, and Vþ, V− are two solutions that arise from solving Eq. (7) [or Eq. (8)], CðVt;tÞ¼c. The concurrence distribution is then obtained by taking a derivative of the cumulative distribution, (c) ∂ ∂ V− Vþ pC ðcÞ¼pðV−Þ þ pðV Þ ; ð10Þ ;t ∂c þ ∂c noting that V−ðc; tÞ and Vþðc; tÞ are functions of the concurrence c and time t. The full solution of pC;tðcÞ is quite lengthy and is not shown. We show in Fig. 2(a) the plots of concurrence probability distributions (10) for different values of time t, and in Figs. 2(b) and 2(c) the density plot comparing the theory FIG. 2. Concurrence distribution for the qubits under the half- and the transmon experiment. At an early time, the parity measurement. In panel (a) we plot the concurrence distribution of concurrence is narrowly peaked near its probability density function, Eq. (10), for different values of maximum, which increases over time, whereas at later time. The values of time for the presented curves are chosen so as to see their unique features as they develop. The grey dotted curve times, a second peak emerges near the zero concurrence, joining the high-concurrence peaks shows the concurrence upper showing a bimodal distribution. In Fig. 2(b), the theoretical bound, Eq. (11). The inset shows an example of how the histogram for the concurrence is obtained by integrating the concurrence (at time t ¼ 1 μs) varies as a function of the readout theory curves, Eq. (10), for the probability over small Vt. Panels (b) and (c) are the histograms of the concurrence at any intervals δc ≈ 0.015. This integration makes a fair com- time points from t ¼ 0 to T ¼ 1.6 μs (with a step size 0.01 μs), parison with the histogram of the experimental data in comparing theory and experimental data. For the theory plot, we Fig. 2(c), calculated with a bin size of 0.015. We note that a coarse-grain the distribution pC;tðcÞ in Eq. (10) by integrating it short delay in the experimental entanglement creation is a with a pixel size δc ∼ 0.015, which is the bin size of the result of the cavity ring-up time, which will be discussed experimental histogram. For presentation purposes, a histogram more in the next section. at any time t is normalized by its maximum element. We stress that the concurrence distribution has a sharp upper bound [shown as a grey dotted curve in Fig. 2(a)], the concurrence, plotted as a function of the measured which the concurrence cannot exceed. In order to under- signal Vt, is bounded from above for any fixed time t by C stand why the probability distribution for the concurrence some amount we call max, and consequently, any value of has a sharp upper cutoff at any time, we recall that the concurrence higher than that maximum (whose value will density matrix of the two-qubit system, conditioned on the change as the time increases) cannot be realized. Therefore, time-integrated readout Vt, is entirely specified by that the probability distribution of concurrence has a sharp C (random) outcome, together with the initial state, the upper cutoff given by maxðtÞ. Physically, this indicates that dephasing rate, and other parameters of the problem, there is an upper limit on how fast entanglement can be Eq. (2). Consequently, the concurrence is controlled by created by the continuous measurement in this situation, Vt, as in Eq. (8). As can be seen from the inset of Fig. 2(a), even for rare events of the measurement process.

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For the perfectly symmetric case in Eq. (8), we find an pforffiffiffiffiffiffiffiffiffiffiffi an instantaneous readout is given by pðvtjiÞ¼ analytic solution for the upper bound of the concurrence, δt=πs − v − δv 2δt=s i 1 0 expf ð t iÞ g for ¼ ,2,3,4. knowing that coshðxÞ has its minimum at x ¼ and To derive the most-likely path for these two-qubit C ˆ V V 0 ps;xð tÞ has its maximum at t ¼ ; consequently, the trajectories, we use the Bayesian update equation for the concurrence upper bound is given by two-qubit state Eq. (2), adapted to a state update every time

2 step δt, and then we maximize the joint probability density −γt − −δv t=s C e e Eq. (12), constraining the state update equations. Following max;ps;xˆ ðtÞ¼ 2 : ð11Þ 1 þ e−δv t=s the most-likely path analysis for quantum states under continuous measurement [36,37] and introducing Lagrange The behavior of this bound is a result of two competing multipliers fp1;p2;p3;p4;p5g for the constraints, we rates—the extra dephasing rate γ and a measurement rate 2 obtain differential equations for an optimal path in the δv =s. Equation (11) increases from zero for small times qubit-state space, and decays for a long time after reaching its maximum 4 concurrence, as seen in Fig. 2. The maximum possible x X concurrence for this qubit half-parity measurement and the ∂ x ¼þ i x f2v ðδv − δv Þ − δv2 þ δv2g; ð13aÞ t i s k t i k i k time this happens can be obtained from this relation. More k¼1 about the time to reach maximum concurrence will be δ 2 δ 3 discussed in Sec. IV C. x5 ð v2 þ v3Þ ∂ x5 ¼ −γx5 þ v ðδv2 þ δv3Þ − t s t 2 IV. MOST-LIKELY PATH ANALYSIS X4 x δv2 − 2v δv ; 13b In addition to the distribution analysis in the previous þ kð k t kÞ ð Þ 1 section, where we treated the concurrence at each point in k¼ time independent from any other times, we now incorporate where i ¼ 1; …; 4 for the first line and the variables xk are the notion of connected trajectories, finding a probability time-dependent functions. We note that vt in these equa- density function for quantum trajectories and their most- tions behaves as a “smooth” optimal readout, which is a likely paths. These most-likely paths describe routes with function of the qubit density matrices and their Lagrange the highest probability density, taking into account that all multipliers, determining an optimal path from an initial of the points in the trajectory ensemble are connected via state to its final state [36]. An optimal path is a solution of quantum-state update rules, e.g., in Eq. (2).Aswehave ten differential equations: five for the qubit-state variables, seen previously, the concurrence distribution exhibits a Eq. (13), another five for Lagrange multipliers, and the transition from a single-peak distribution to double-peak optimal readout as a function of both sets of variables (see distributions, one at the upper bound concurrence and Appendix A). These equations describe most-likely paths another near zero concurrence. Here, with the notion of for a measurement with any values of δv1;…;4 and can also trajectories, we also see that the two-qubit state starting in be generalized to include the effect of external drives on the its initial state gradually collapses to three subspaces: the 00 11 two-qubit state. However, in the absence of an external j i subspace, the j i subspace, and the odd subspace, as drive, these can be simplified, as we will see in the next described by three most-likely routes. section. Let us consider the five nontrivial elements of the two- qubit density matrix fx1;x2;x3;x4;x5g and treat each element as an independent variable. Each state trajectory A. Most-likely paths for joint measurement is represented by a time series of these five elements, which of transmon qubits corresponds to one realization of the measurement readout In this work, where the transmon qubits only evolve with … the influence of the measurement backaction, the optimal fvtg¼fR v0;vδt;v2δt; ;vtg, where we define vt ¼ tþδt ~ 0 0 ðf=δtÞ Vðt Þdt − v0 as an instantaneous readout at readout is found to be constant in time (see Appendix A). t We can therefore bypass solving the full set of differential time t with an integration time δt. Since the readout vt’s are assumed to be Markovian and only depend on the qubit equations [49] and only compute the qubit evolution in states right before the measurement, a joint probability Eq. (13) with constant vt, looking for measurement results density function for the readout realization is given by with maximum likelihood density. To estimate the likelihood density of a two-qubit trajectory, we evaluate a Yt X4 logarithm of the probability density function Eq. (12) δ PðfvtgÞ ¼ xk;t0 pðvt0 jkÞ ; ð12Þ approximated to first order in t [36] to obtain t0¼0 k¼1 Z 4 t 1 X 0 P ≈ S − 0 − δ 2 a product of probability distributions of vt0 from t ¼ 0 to log ðfvtgÞ 0 dt ðvt0 vkÞ xk;t0 ; ð14Þ 0 s t0 ¼ t with a time step δt. The probability density function k¼1

041052-7 AREEYA CHANTASRI et al. PHYS. REV. X 6, 041052 (2016)

Xt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where S0 represents a state-independent part of the joint δt P D ¼ Tr ðρ ðt0Þ − ρ ðt0ÞÞ†ðρ ðt0Þ − ρ ðt0ÞÞ ; probability density function . We show in the insets of a;b 2T a b a b Figs. 3(a)–3(c) examples of the approximated log- t0¼0 likelihood density as functions of optimal readout vt, for ð15Þ three different measurement strengths. In the case of τ ≈ 2 10 μ m . s, there is only one maximum likelihood value for all possible pairs. The goal is to pick the first few ≈ δ located at vt v2;3, which gives the most-likely path with trajectories with minimum total distance to other trajecto- high concurrence, shown as a solid curve in Fig. 3(a); ries in the ensemble and average them to get an estimate of whereas, in the stronger measurement cases, Figs. 3(b) the experimental most-likely paths. For this particular set of and (c), the approximated log-likelihood density has three data, we choose about 102 highly likely trajectories to get a ≈ δ local maxima: the middle ones, vt v2;3, corresponding to smooth estimate of the most-likely paths. However, for the most-likely paths collapsed to entangled states (high-con- case in which there exist multiple (e.g., three) most-likely ≈ δ currence branches); and the sided peaks, vt v1 and paths, we first divide the ensemble into subensembles ≈ δ vt v4, corresponding to two branches of most-likely according to their trace distance and then apply the 00 11 paths collapsed to j i and j i states (low-concurrence minimum total distance procedure to the trajectories in branches), respectively. We note that this technique of each subensemble separately. In Figs. 3(a)–3(c), we show finding multiple most-likely paths is not under final-state the concurrence of the experimental most-likely paths as constraints as in Refs. [35,36]. data points. We choose to compare the theoretical most-likely paths Ideally, the theoretical most-likely paths predicted from with high-trajectory density paths extracted from the the log-likelihood density, Eq. (14), for each set of experimental data, using the trace distance method. We measurement strengths using the initial state fx0; …;x0g¼ 104 1 5 collect an ensemble of transmon trajectories and then f1=4; …; 1=4g would be good enough to compare with the compute average trace distances between any two trajec- experimental data. However, in the experiment, after the ρ ρ tories (e.g., a and b), initial state has been prepared, the cavities take some time

(a) (b) (c)

(d) (e) (f)

FIG. 3. The most-likely paths from the theoretical prediction and the experimental data. The first, second, and third columns are from three different data sets, measuring the same qubits and cavities with three different measurement-readout powers (indicated by the characteristic measurement time): τm ¼ 2.10 μs, 0.60 μs, and 0.36 μs, respectively. Panels (a)–(c) show the concurrence of the multiple most-likely paths: the path with high concurrence (solid purple line), and the two paths with low concurrence projecting onto j00i (dashed orange line) and j11i (dotted green line) subspaces. The vertical lines (labeled as A1;2, B, C1;2, D) represent the times at which the concurrence is maximum for each of these paths. The insets show the log-likelihood density as functions of optimal measurement readouts, and the grey-scale histograms in the background show experimental concurrence histograms similar to the one in Fig. 2(c). Panels (d)–(f) present the evolution of the density matrix elements of the high-concurrence most-likely paths: The solid curves indicate theoretical solutions, whereas the data points indicate experimental most-likely paths postselected with the most entangled state at the final time T ¼ 1.6 μs. Examples of the postselected trajectories are shown as fluctuating curves.

041052-8 QUANTUM TRAJECTORIES AND THEIR STATISTICS FOR … PHYS. REV. X 6, 041052 (2016) to reach their steady-state condition, making the parameters surprising because the distribution of concurrence shows δv1; …; δv4 unstable during the first ∼0.13 μs time. sharp peaks along the concurrence bound. We note that the Therefore, we need to let the initial qubit state evolve solutions for the low-concurrence branch (for vt ¼ δv and and then find new “initial” states at time t ¼ 0.13 μs for the vt ¼ −δv) can be found numerically. theoretical most-likely path calculation. We use the exper- In the above analysis, for the half-parity case, a simple imental most-likely states at t ¼ 0.13 μs (one for each analytic solution for the most-likely path for arbitrary branch) as the initial states in calculating the log-likelihood values of vt was not forthcoming. However, if we further density functions (the insets), which then leads to an simplify the problem by considering a parity meter [6], then excellent prediction of the most-likely paths and their we can solve the equations of motion, Eqs. (13) and their concurrence for the rest of the evolution. conjugate equations, exactly. A parity meter has the same We also note that the two unequal low-concurrence detector outputs for the even- and odd-parity subspaces, but branches in Figs. 3(b) and 3(c) happen because the the detector can distinguish between the subspaces. More population of the states drifts more toward the ground detail of the calculation is presented in Appendix C. state j00i during the transient time, as a result of the qubit relaxation during the measurement. Moreover, in C. Distribution of time to maximum concurrence Figs. 3(d)–3(f), we present the evolution of matrix elements of the high-concurrence most-likely paths, showing a good In the process of the entanglement generation, there are agreement between the theoretical most-likely paths and interesting quantities to investigate, such as the maximum experimental most-likely paths postselected with the most concurrence each individual trajectory can reach and the entangled state at the final time T ¼ 1.6 μs. time it takes to reach the highest value. We previously showed that qubit trajectories branch out to high- and low- concurrence subspaces. Therefore, one would expect that B. Most-likely paths for perfectly there are two most-likely times for the qubit trajectories to symmetric half-parity measurement reach their maximum concurrences (or their most entangled We are also interested in finding analytic solutions for states). the most-likely paths for the symmetric case: δv2 ¼δv3 ¼0 In Fig. 4, we show the normalized histograms of time for and −δv1 ¼ δv4 ¼ δv. The differential equations (13) for transmon qubit trajectories to reach their maximum con- the qubit state simplify to currence. The histograms for the τm ¼ 0.36 μs and 0.60 μs measurement cases explicitly show double peaks, which 2 ∂txp ¼ −bxp − axm þ axpxm þ bxp; ð16aÞ agree with the branching of concurrence and the most- likely qubit paths in Figs. 3(b) and 3(c). The times at which 2 ∂txm ¼ − bxm − axp þ bxmxp þ axm; ð16bÞ these peaks are located can be theoretically predicted from

∂tx2;3 ¼þx2;3ðaxm þ bxpÞ; ð16cÞ

∂tx5 ¼ − γx5 þ x5ðaxm þ bxpÞ; ð16dÞ where we have defined new variables xp ≡ x1 þ x4 and xm ≡ x1 − x4, and constant parameters a ¼ vtδv=s and b ¼ δv2=4s. The first two equations can be solved independently from the rest. For the case vt ¼ δv2;3 ¼ 0, which corresponds to the most-likely odd-parity result, we obtain an analytic solution for the high-concurrence branch of the most-likely path,

∝ 0 −δ 2 4 x1;4ðtÞ x1;4 expð v t= sÞ; ð17aÞ ∝ 0 x2;3ðtÞ x2;3; ð17bÞ FIG. 4. Histograms of the time to maximum concurrence for 0 individual trajectories for three measurement strengths indicated x5ðtÞ ∝ x5 expð−γtÞ; ð17cÞ by the values of τm. For the two cases with strong readout powers (shown in red and green histograms), there exists two peaks where the proportionality factor is the inverse of a normal- 0 0 0 0 2 corresponding to two most-likely times to reach their most ized factor N ¼ð1 − x1 − x4Þþðx1 þ x4Þ expð−δv t=4sÞ. entangled states. The theoretical predictions of these times are The concurrence of this path exactly coincides with the shown as vertical dashed lines labeled as A1;2, B, C1;2, D [same as upper concurrence bound derived in Eq. (11), which is not in Figs. 3(a)–3(c)].

041052-9 AREEYA CHANTASRI et al. PHYS. REV. X 6, 041052 (2016) the time to maximum concurrence of the solutions of the full-parity measurement, and we have made some predic- most-likely paths. As shown by the vertical dashed lines in tions about that case in this work. Figs. 3(b), 3(c), and 4, A1;2 and B are the two most-likely times to reach maximum concurrence (for low- and high- ACKNOWLEDGMENTS concurrence branches, respectively) for τm ¼ 0.60 μs, and C1;2 and D are the same but for the case with τm ¼ 0.36 μs. This work was supported by US Army Research Office The agreement between the theoretical prediction of the Grants No. W911NF-15-1-0496 and No. W911NF-13-1- peaks and the peaks of the histograms are as good as the 0402, by National Science Foundation Grant No. DMR- agreement of the theory-experiment most-likely paths in 1506081, by John Templeton Foundation Grant ID 58558, Figs. 3(b) and 3(c). We note that for the weak measurement and by the Development and Promotion of Science and regime, τm ¼ 2.10 μs, the bifurcation has not occurred yet Technology Talents Project Thailand. M. E. K. S. acknowl- during the measurement time T ¼ 1.6 μs. One would edges support from the Fannie and John Hertz Foundation. expect to see a branching effect when the total measurement time is long enough. APPENDIX A: OPTIMIZED PATHS (MOST- LIKELY PATHS) WITH PRESELECTED OR V. CONCLUSION POSTSELECTED STATES We have investigated the process of entanglement gen- We follow the outline in Refs. [36,37] for the stochastic eration between two spatially separated superconducting path-integral formalism and the action principle for con- transmon qubits and their statistical properties. The entan- tinuous quantum measurement. A joint probability density glement of the two qubits is created as a result of the half- function of quantum states and measurement readouts, parity dispersive measurement via the microwave pulses from time t0 ¼ 0 to t0 ¼ t, is given by a path integral sequentially interacting with both qubits. The strength of the joint measurement is arbitrary, and we have studied Z three different values of the measurement strength. We Pðfxtg; fvtgjx0Þ¼N DpðtÞ exp S; ðA1Þ examined the concurrence of individual trajectories and theoretically calculated its distribution from the quantum Z 4 5 t 1 X X Bayesian approach, gradually projecting the two-qubit 0 2 S ¼ − dt ðvt0 − δviÞ xi þ pjð∂t0 xj − F jÞ ; 0 s states to entangled states with high concurrence and to i¼1 j¼1 separable states with zero concurrence. A2 The most-likely path analysis was also carried out, ð Þ predicting the most probable paths for the qubit trajectories. where fx g¼fx 0 g denotes a set of quantum-state matrix We found that in the two-qubit-state space, there are three t j;t 1 2 … 5 0 likely paths conforming to the three branches projecting to elements at all times (with j ¼ ; ; ; , and t runs from 0 0 0 Dp the j00i subspace, the j11i subspace, and the odd (j01i, time t ¼ to t ¼ t), and ðtÞ denotes the path-integral j10i) subspaces; the first two correspond to the lower measure for the conjugate variables (or Lagrange multi- branches of the concurrence bifurcation, and the last pliers as mentioned in the main text). The action of the S corresponds to the high-concurrence branch. These theo- path integral [37] is given in terms of the two-qubit retical most-likely paths show good agreement with the density matrix variables xj and their conjugates pj for 1 2 … 5 0 experimental most-likely ones extracted from the transmon j ¼ ; ; ; , which are implicitly functions of time t . S trajectory data (for three independent data sets) based on The first sum in the time integral of is the logarithm of the the trace distance between trajectories in two-qubit-state joint probability density function of the measurement P δ space. Moreover, we have presented the distributions of the readout ðfvtgÞ, Eq. (12), truncated to first order in t F time to the maximum concurrence for individual trajecto- (taking a time continuum limit), and the functional j is the ries. The most-likely path analysis was shown to be useful right-hand side of Eq. (13). in predicting the peaks of these time distributions. The joint probability density function in a path-integral We conclude that the accurate tracking of quantum representation, Eq. (A2)Q , is written with a measure Dx D ≡ t trajectories of a jointly measurement qubit system is ðtÞ vðtÞ limδt→0 t0¼0 dx1;t0 dx2;t0 dx5;t0 dvt0 , con- possible and that the physics of the entanglement creation sidering the density matrix elements and the measurement statistics is well described by a quantum trajectory theo- readout as random variables. The probability density retical approach. The theoretical most-likely paths to function multiplied by the measure gives a probability that entanglement and concurrence distribution match the is vanishingly small but invariant under changes of mea- experiment excellently. This work shows the way to use sure. However, as in usual probability theory, one can ask this process as a control mechanism to entangle remote about “peaks” or the most-likely values of an associated systems for quantum-information-processing purposes. probability density function. The most-likely paths for the In future work, similar questions can be posed about the quantum trajectories starting from an initial state and

041052-10 QUANTUM TRAJECTORIES AND THEIR STATISTICS FOR … PHYS. REV. X 6, 041052 (2016) ending at a final state after some time t can be found by space only. This similar definition of the optimal path optimizing the action of the stochastic path integral (A2). is also used in classical stochastic processes, for example, by By extremizing the action Eq. (A2) over all variables xi, the variation of the Martin-Siggia-Rose action formalism pi, vt, we get a set of ten ordinary differential equations [56,57]. (ODEs) for the optimized path, and one equation for optimal measurement readout. The set of ODEs includes APPENDIX B: TRANSFORMATION OF THE five differential equations of the two-qubit variables xi,as MOST-LIKELY PATH UNDER CHANGE shown in Eq. (13), and five equations for the conjugate OF MEASURE variables p , i We further investigate the transformations of the optimal X4 paths under change of measure. In probability theory, given ∂tpi ¼ xjAij þ x5p5Bi þ Ci for i ¼ 1; 2; 3; 4; a probability density function of a continuous random j¼1 variable V, one can consider a change of variable V → ζ ¼ gðVÞ, where ζ is a new random variable. The ðA3aÞ total probability contained in the differential region should 4 X be invariant, i.e., pVðVÞdV ¼ pζðζÞdζ, so the new prob- ∂ γ tp5 ¼þ 5p5 þ p5 xjBj; ðA3bÞ ability density function can be reformulated as j¼1 −1 −1 pζðζÞ¼pVðg ðζÞÞj½d=ðdζÞg ðζÞj, assuming that g is where monotonic. If there exists an optimal value Vopt, obtained by solving d= dV p V 0, and there similarly ½ ð Þ Vð ÞjVopt ¼ Aij ¼ −ðpi − pjÞð2vt − δvi − δvjÞðδvi − δvjÞ; exists an optimal value in the ζ variable, one can show that ζ does not need to be the same as g V . Thus, a 1 2 2 2 opt ð optÞ B ¼ fv ð4δv − 2δv2 − 2δv3Þ − ð2δv − δv − δv Þg; i 2s t i i 2 3 question regarding an optimal (most-likely) value is de- 1 pendent on the random variables one is interested in; that is, C ¼ ðv − δv Þ2: it is measure dependent. We also note that for a special case i 2s t i when gðVÞ is a linear function, the optimal values in the ζ The optimal readout is given as a function of the two-qubit two variables coincide, opt ¼ glinearðVoptÞ. variables and the conjugate variables, We stress that, even though the joint optimization of the quantum states and the measurement readouts (A1) can X4 X4 give a different solution under a change of the readout v p x x δv − δv x δv t ¼ f i i jð i jÞþ i ig variables, we can avoid this change of measure by 1 1 i¼ j¼ integrating out all the readout variables and only consid- X4 ering the optimal path in the quantum-state space, as p5x5 δ δ − 2 δ þ 2 ð v2 þ v3 xi viÞ; ðA4Þ discussed at the end of the previous appendix. For the 1 i¼ rescaled shifted homodyne signal as a readout variable, the connecting the ODEs for the qubit variables and the joint optimization happens to coincide with the quantum- conjugate variables. We note that by taking a time deriva- state optimization after integrating out the readout variables. This way we have thus proven that our quantum-state tive of the function in Eq. (A4) and substituting both ∂txi optimal path is independent of the readout measure, while and ∂tpj with the ODEs above, we find that ∂tvt ¼ 0, therefore implying that an optimal readout is a constant the optimal readout itself vt is a measure-dependent in time. quantity in general. From the joint probability density function (A1), we can consider integrating out the measurement-readout variables APPENDIX C: ANALYTIC SOLUTION IN THE CASE OF A PARITY METER to get a probability density function just inR the quantum- P x x D P x state variables, i.e., ðf tgj 0Þ¼ vðtÞ ðf tg; For the full-parity measurement case, we assume fvtgjx0Þ. Since the action (A2) is bilinear in vt, its δv1 ¼ δv4 ¼ dv, while δv2 ¼ δv3 ¼ 0. With this special Gaussian-functional integral gives exactly the same result case, we find the equations of motion (up to a constant normalization factor) as extremizing the action over the readout and substituting the optimal ∂tx1 ¼þλx1xo; ðC1aÞ solution (A4) to the action. By extremizing the rest of the action over the quantum-state variables and their ∂tx2 ¼ −λx2xe; ðC1bÞ conjugates, we get the same set of ten differential equations [Eqs. (13) and (A3)] with the substitution of the ∂tx3 ¼ −λx3xe; ðC1cÞ optimal readout (A4). The solutions to these equations can ∂ λ now be interpreted as an optimal path in the quantum-state tx4 ¼þ x4xo; ðC1dÞ

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where the value of λ can be fixed once we have specified the ∂tx5 ¼ − γx5 − λx5xe; ðC1eÞ 0 0 initial and final values of the density matrix, x1;x2; …; 0 2 x5;x1;f;x2;f; …;x5;f. where we define λ ¼ 2v δv=s − δv =s, and x ¼ x1 þ x4 ¼ t e Finally, we may find the concurrence of the most-likely ρ00;00 þ ρ11;11 is the probability of being in the even-parity pffiffiffiffiffiffiffiffiffi path by calculating C ¼ 2 maxf0;x5 − x1x4g. We find the subspace, while x ¼ x2 þ x3 ¼ ρ01 01 þ ρ10 10 is the prob- o ; ; result ability of being in the odd-parity subspace. ∂ ∂ pffiffiffiffiffiffiffiffiffi Taking the sum of tx1 and tx4, we can derive an 0 −ðγþλÞt 0 0 x5je j − x1x4 equation for xo alone since xe þ xo ¼ 1, C 2 0 ðtÞ¼ max ; 0 0 −λt : ðC6Þ j1 − ðx2 þ x3Þð1 − e Þj

x_o ¼ −x_e ¼ −λð1 − xoÞxo: ðC2Þ In order to cross the entanglement border [6] from an λ Integrating this equation gives the solution unentangled state, must take on a negative value to enhance the first term over the second, taking the system x0e−λt into the odd-parity subspace. x ðtÞ¼ o ; ðC3Þ The entanglement border will be crossed when the o 1 − 0 1 − −λt xoð e Þ concurrence is zero or when the condition 0 0 where xo is the initial condition for x2 þ x3. Similarly, we γ λ 1 pffiffiffiffiffiffiffiffiffiffiffiffix5 find for the even probability, þ ¼ð =tÞ ln 0 0 ðC7Þ jx1x4j x0 e is satisfied. xeðtÞ¼ 0 −λt ; ðC4Þ 1 − xoð1 − e Þ

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