Quantum Jumps Are More Quantum Than Quantum Diffusion

It was recently argued [Phys. Rev. Lett 108, 220402 (2012)] that the stochastic dynamics of an open quantum system are not inherent to the system, but rather depend on the existence and nature of a distant detector. The proposed experimental tests involved homodyne detection, giving rise to quantum diffusion, and required total efficiencies of well over 50%. Here we prove that for no system is it possible to demonstrate detector-dependence using diffusive-type detection of efficiency less than 50%. However, this no-go theorem does not apply to quantum jumps, and we propose a test involving a qubit, using different jump-type detectors, with a threshold efficiency of only 37%.

tribution and terminology [14]). In particular, by correlating the continuous measurement record in (say) Alice's distant detector, in some interval [0, T ), with the result of various projective measurements performed directly on the atom by (say) Bob, at time T , a carefully constructed EPR-steering inequality may be tested. (Here T is randomly chosen by Bob.) If the inequality is violated, this proves that there can be no underlying OPDM for the atom, and thus the stochasticity in its evolution (jumps or diffusion) must emanate from the detectors.
In Ref. [9], the system considered was a strongly driven two-level atom, and an appropriate EPR-steering inequality was chosen. The best violation was found using the following pair of monitoring schemes, also known as unravellings: i) homodyne detection, giving rise to diffusion; and ii) photon counting, using spectral filtering and a weak local oscillator (LO) with adaptively controlled phase, giving rise to jumps. Assuming equal efficiencies for both detectors, the critical efficiency required was η c ≈ 0.58. Moreover, replacing the (extremely complicated) scheme (ii) with a different homodyne scheme raised this to η c ≈ 0.73.
In this paper, we show that the high efficiency required for tests involving homodyne detection is no accident: for any system, no matter the number L of outputs, and no matter the number M of different unravellings, if they are all diffusive and all efficiencies are below 0.5, it is impossible to demonstrate EPR-steering. Following the proof of this no-go theorem, we show that it does not apply to jump unravellings by exhibiting a qubit system, with L = 2 and M large, in which η c ≈ 0.37. Moreover, even restricting to M = 5, a decent (5%) violation is predicted for efficiency η d ≈ 0.455, whereas under the same conditions with diffusive unravellings the best we could obtain was η d ≈ 0.78. That is, the peculiarly quantum nature of open systems are far more easily manifest by quantum jumps than by quantum diffusion.

I. OPEN QUANTUM SYSTEMS AND DIFFUSIVE UNRAVELLINGS
We restrict to Markovian systems, since non-Markovian quantum systems do not, in general, allow arXiv:1312.1783v1 [quant-ph] 6 Dec 2013 for pure conditioned states even for 100% efficient nondisturbing detection [15]. Then the average, or unconditioned, evolution is described by a master equation (ME) (1) Hereĉ = (c 1 , · · · , c L ) is an arbitrary vector of operators (called Lindblad operators), and [16]. This equation results from tracing over that environment to which the system is coupled, but it is possible to monitor the environment and get further information about the system. This results in a conditioned state which is (in general) more pure, and which evolves stochastically according to the measurement record. Different ways of monitoring the environment give rise to different unravellings of the ME. For example, in quantum optics, a LO of arbitrary phase and amplitude may be added to the system's output signal prior to detection. For a weak LO (i.e. one comparable to the system's output field), individual photons may be counted, giving rise to quantum jumps in , but for a strong LO only a photocurrent is recorded, giving rise to quantum diffusion in [1,16].
The most general diffusive unravelling of Eq. (1) is described by the stochastic ME [16] where H[â]ρ ≡âρ + ρâ † − Tr[âρ + ρâ † ]ρ, and dZ(t) is a vector of c-number Wiener processes. Physically, these arise as noise in photocurrents, and have the correlations dZdZ † = Θdt and dZdZ = Υdt. Here Θ = diag(η 1 , · · · , η L ) is a real diagonal matrix, with 0 ≤ η l ≤ 1 being the efficiency with which output channel l is monitored. The complex symmetric matrix Υ, on the other hand, parametrizes all of the diffusive unravellings this allows. It is a subject only to the constraint that the unravelling matrix be Positive Semi-Definite (PSD), i.e. U (Θ, Υ) ≥ 0. Consider two different unravellings U and U 0 . If it is possible to write dZ 0 = dZ + dZ, where dZ is an unnormalized complex vector Wiener process uncorrelated with dZ, then clearly the first unravelling U can be realized by implementing the second, U 0 , and throwing out some of the information in record. This will be the case if the implied unravelling matrix for dZ,Ũ ≡ U 0 − U , is PSD, in which case we call U a coarse-graining of U 0 . Now say that Alice can implement a set {U m } M m=1 of different unravellings of the form of Eq. (3). A necessary condition for this set to be capable of demonstrating continuous EPR-steering, is that they not be coarsegrainings of a single unraveling [9]; that is, that there not exist a U 0 such that ∀m, U 0 − U m ≥ 0, since the stochastic evolution defined by U 0 would be an explicit OPDM compatible with all of the observed behaviour.
is non-negative for all l and m. To establish the result we need prove only that λ min (Ũ m ) ≥ 0 for all m. Using Weyl's inequality [17], It can be also proven, based on the properties of partitioned matrices [17], To reiterate: unless it is the case that for at least one output channel, and at least one unravelling, the monitoring efficiency is greater than 0.5, then there exists an unravelling U 0 = U (I, 0), which defines an OPDM which is consistent with all the observed conditional behaviour of the system, so that no detector-dependence can be proven. (It is interesting to note that this model, corresponding to the unravelling U (I, 0) is precisely that introduced, without a measurement interpretation, as quantum state diffusion in Ref. [8].) This is the first main result of our paper. The second is that this condition, η m l > 0.5, is not necessary for quantum jump unravellings, as we now show.

II. EVOLUTION VIA QUANTUM JUMPS
A more general class of unravellings (in that it contains quantum diffusion as a limiting case [16]) is that of quantum jumps, whereby the conditioned evolution of the system undergoes a discontinuous change upon certain events ("detector clicks"), and otherwise evolves smoothly [16]. There is not just one jump unravelling; for the general ME (1), for instance, each output channel can have a weak LO added to it prior to detection [16]. When a click is recorded in the lth output in interval [t, t + dt), the system state is updated via Here J [â] ≡â â † , and the jump operator isĉ l =ĉ l +µ l , where µ l is an arbitrary complex number. The norm of the unnormalized state˜ l is equal to the probability of this click. If no click is recorded the system evolves via [16] (t) →˜ 0 (t + dt) as required for the Eq. (1) to be obeyed on average, where again the norm is equal to the probability of their being no clicks in that infinitesimal interval [5].
In the case of efficient detection, the panoply of jumpy unravellings bestows an extraordinary power upon the experimenter: to confine the conditioned state of the system, in the long-time limit of an ergodic ME, to occupying only finitely many different states in Hilbert space [18]. Such a set of states, with the probabilities with which they are occupied, as in {(℘ k , |φ k )} K k=1 , is called a physically realizable ensemble (PRE) [19]. For the case of a qubit, a PRE of the minimum size (K = 2) always exists [18]. Moreover, there are as many K = 2 PREs as there are distinct (not necessarily linearly independent) real eigenvectors of the matrix A, which appears in the Bloch equation˙ r = A r + b equivalent to Eq. (1) for the qubit state represented by r = (σ x ,σ y ,σ z ) Consider now a particular qubit ME with L = 2 irreversible channels into the environment, as follows: whereσ ± = (σ x ± iσ y )/2 are raising and lowering operators respectively. That is, in the notation of Eq. (1), Note this ME is completely different from the single decoherence channel of Ref. [9]. Realization of this sort of ME has been recently investigated in the context of quantum computing, in the limit γ + = γ − , for which suitable unravellings allow universal computation to be performed [20]. The Bloch representation of Eq. (6) is˙ r = A r + b with where γ Σ = γ + + γ − and γ ∆ = γ + − γ − . This A has one real eigenvector in the z direction, giving rise to a K = 2 PRE, expressed in the Bloch representation {(℘ ± , r ± )} as E z ≡ {(γ ± /γ Σ , (0, 0, ±1) )}. This A also has infinitely many real eigenvectors in the x-y plane, which we can parametrize by the azimuthal angle ϕ, giving rise to the K = 2 PREs E ϕ ≡ {(1/2, (±C cos ϕ, ±C sin ϕ, z ss ) )} where C = 2 √ γ + γ − /γ Σ and z ss = γ ∆ /γ Σ . All of these ensembles average to give the steady-state Bloch vector r ss = (0, 0, z ss ) as shown in Fig. 1a. Following Ref. [18] we can determine the LO amplitudes to realize these PREs (assuming perfect detection). For E z it is trivial to see that no LO is required, aŝ c ± cause jumps between theσ z -eigenstates. For E ϕ we require an adaptive scheme with each µ ϕ l (t) taking two possible values, µ ϕ l ± = ± √ γ −l e liϕ /2. Here l, the label for the output channel, also takes the value ±, but that is independent of the ± defining the two values for the LO. The adaptivity required is that every time a detection in either channel occurs, the LO for both channels is swapped from their + values to the − values, or vice versa.

III. QUANTUM JUMPS ARE MORE LOSS-TOLERANT
For the unit-efficiency case, the z unravelling (i.e. that giving rise to ensemble E z ) is such that σ z 2 = 1. If this unravelling were the OPDM of the system then the complementary variableσ ϕ =σ − e iϕ +σ + e −iϕ (for any ϕ) would necessarily have zero mean, but we know that it has a nonzero conditional mean for the PRE E ϕ . Consider a finite set of n different ϕ values {ϕ j = (j/n)π}, so that, with the z unravelling, Alice has a total of M = n + 1 unravellings. Then the above, unit-efficiency, considerations suggest the following EPR-steering inequality [21]: (8) Here E z [•] means the ensemble average under the z unravelling, so that σ z appearing therein means Tr[ σ z ], where is the conditional state under that unravelling, and likewise for the ϕ j unravellings. The function f (n) is defined in Ref. [21] and asymptotes to 2/π as n → ∞.
In Eq. (8) we are not assuming unit efficiency; the unravellings are as defined above, but the long-time conditional states will not be pure (and will certainly not be just two in number for each unravelling). Our aim is to show that the no-go theorem for inefficient detection, applicable to diffusive unravellings, is not universal, by showing that Eq. (8) can be violated for η < 0.5. To do this we must evaluate the terms on the LHS for the n + 1 different unravellings, although we note that by the symmetry of the problem, E ϕ [| σ ϕ |] is independent of ϕ. The ensemble average E z can be done semi-analytically, while that for E ϕ requires stochastic simulation; see Appendix A. We plot these averages in Fig. 1, as well as S in Eq. (8), for varying R ≡ γ + /γ − , and varying η, and for n = 4 and n = ∞.
The critical threshold efficiency for jumps to violate Eq. (8), is η c ≈ 0.37, which is considerably below the limit of 0.5 necessary for diffusion according to our Theorem. This is the second main result of this paper. This η c is achieved in the limits R 1 and n → ∞, neither of which are convenient because the first implies that even when there is a violation it will always be very small (compared to the maximum possible violation of unity at η = R = 1), and the second because it requires infinitely many measurement settings. However, we show that a decent violation, of 0.05, is achievable with only n = 4, with an efficiency η d ≈ 0.455, which is still significantly below the 50% limit. This was for an optimized value of R, found numerically, of R o ≈ 0.16.

IV. COMPARISON WITH QUANTUM DIFFUSION
We now consider the same ME (6) and the same EPR-steering inequality (8), applied to diffusive unravellings (2). Here, withĉ = ( √ γ −σ− , √ γ +σ+ ) we have Θ = diag(η, η) and the optimal ϕ j unravelling is Υ = η diag(e −2iϕ , e 2iϕ ). For diffusive unravellings, there is no unravelling that is particularly useful for Alice to be able to predict Bob's value forσ z , so for the z unravelling we simply use an arbitrary ϕ unravelling (this is still better than using no unravelling i.e. replacing the E z [•] term by 1 − z 2 ss ). Since, as in the jump case, E ϕ [| σ ϕ |] is independent of ϕ, we only have to simulate one unravelling. This is described in Appnedix. A and the results are shown in Fig. 2. The critical efficiency is η c ≈ 0.59, greater than 0.5 as expected, for R 1. While this is less than the alldiffusive η c = 0.73 of Ref. [9], it is a long way above the quantum jump η c = 0.37 found above. Interestingly, analytical calculations (see Appnedix. A) show that for R 1, S ∼ = g(η) √ R, where g(η) changes sign at η c . Restricting to n = 4 and looking again for a decent (0.05) violation, we obtain η d = 78% at an optimal value of R o = 0.13.
In conclusion, for the experimental task of ruling out all objective pure-state dynamical models for an open quantum system we have: (i) proven it is impossible to achieve this by diffusive unravellings with efficiencies below 50%; and (ii) exhibited a set of quantum jump unravellings that would allow such a task, for a qubit, with an efficiency as low as η c = 37%. Moreover, even allowing for a decent margin of error and other experimental realities, a jump efficiency of only η d = 45.5% is required for our system, whereas the corresponding figure for diffusive unravellings is η d = 78%. That is, it is far easier to show that the stochasticity of quantum jumps arises in the distant detector (as opposed to being intrinsic to the system) than it is to show this for quantum diffusion, and in that sense the former are more quantum. For future work we believe that it will be possibly to prove even stronger no-go theorems for diffusive unravellings. But we also hope that better EPR-steering tests may allow experimentalists to get closer to the limits of such nogo theorems than our results here, so that the recently reported diffusive monitoring efficiency of 49% in superconducting qubit experiments [22] is encouraging.
This research was supported by the ARC Centre of Ex-cellence Grant No. CE110001027. We thank Jay Gam-betta for discussions.
Here ℘ appears as the probability for starting in state r at some time t jump , and p (t)w j (t) dt is the probability that, given this starting point, a jump occurs in the interval [t jump + t, t jump + t + dt) and puts the system into state r j . Averaging over the two possible initial states and all the possible times from one jump to the next, should give ℘ j (i.e. the same function as ℘ , since nothing distinguishes the first jump from the second in the long-time limit.) Solving Eq. (A.6) analytically gives ℘ + = ℘ − = 1/2. This very simple result cries out for an explanation, and here is the simplest one we can furnish. In the case of efficient detection, the system state is always either r + or r − , and alternates between them every time a jump occurs. Thus, after every jump it finds itself in either of them with the equal probability of ℘ ± = 1/2. We can model the case of imperfect detection, where both decoherence channels have the same efficiency η (as we have assumed) as randomly deleting a portion 1 − η of jumps from the full record for perfect efficiency. Since the remaining jumps are an unbiassed sample of the original set of jumps, on average the system state will be equally often in the two states (since the pure post-jump states in the two situations must agree).
The ensemble average we require can be obtained by calculating the below integral (A.7) This is directly comparable to Eq. (A.2). There the timeaverage was done by numerically simulating a typical trajectory of jumps. Here can calculate exactly the timeaverage by using the distribution over the initial state (immediately following a jump) and the time t until the next jump.

Simulating Averages for Diffusive Unravellings
The case we are interested in, where Υ = η diag(e −2iϕ , e 2iϕ ), corresponds to to homodyne detection of both channels, with phase ϕ. As noted in the main text, the ensemble averages are independent of ϕ so without loss of generality we can take ϕ = 0. Then the conditional state of the system evolves according to the following stochastic differential equation where dW ± are independent real Wiener processes. The state of qubit is confined to y = 0 plane such that at any instant of time it can be identified by a point x(t), 0, z(t) in the Bloch sphere. The evolution of this point is governed by the coupled stochastic differential equations We simulate these using the Milstein method [KloPla92]. Once the transients have decayed away (after several γ −1 Σ ) we record data for both x and z, to calculate E ϕ | σ ϕ | = E[|x|] and E ϕ 1 − σ z | 2 = E[ √ 1 − z 2 ] as time averages. From Eq. (A.11) one has a Gaussian distribution for x, which enables us to calculate the first term of the steering parameter, E[|x|]. For z, however, the above moments show that a Gaussian cannot be a good approximation for z (because it is bounded below by −1). However, we can consider a Taylor series expansion of √ 1 − z 2 about E[z]. This gives the analytic expression of steering parameter for small R as where h(η) comes from higher-order (beyond secondorder) moments of z, which are not negligible (they do not scale with R). Thus, whatever the form of h(η), this does not change the scaling with R: where g(η) = 8η/π − f (n) (4 − η 2 )/2 + h(η) . Ignoring h(η) and using f (∞) = 2/π [JonWis11], we predict a critical efficiency, where g(η c ) = 0, of η c ≈ 0.545. From the stochastic simulations with R = 0.01, we found (see main text) η c ≈ 0.59, showing that h(η) is non-negligible, as expected, but not very important.