Loss-tolerant parity measurement for distant quantum bits

# Loss-tolerant parity measurement for distant quantum bits

Alain Sarlette  and Mazyar Mirrahimi QUANTIC project-team, INRIA Paris, France; and Department of Electronics and Information Systems, Ghent University, Belgium.QUANTIC project-team, INRIA Paris, France; and Department of Applied Physics, Yale University, USA
July 26, 2019
###### Abstract

We propose a scheme to measure the parity of two distant qubits, while ensuring that losses on the quantum channel between them does not destroy coherences within the parity subspaces. This capability enables deterministic preparation of highly entangled qubit states whose fidelity is not limited by the transmission loss. The key observation is that for a probe electromagnetic field in a particular quantum state, namely a superposition of two coherent states of opposite phases, the transmission loss stochastically applies a near-unitary back-action on the probe state. This leads to a parity measurement protocol where the main effect of the transmission losses is a decrease in the measurement strength. By repeating the non-destructive (weak) parity measurement, one achieves a high-fidelity entanglement in spite of a significant transmission loss.

The correlation of distant systems thanks to their entanglement is a proven hallmark of quantum physics [3, 2] and plays a fundamental role in envisioned quantum technology. Most fundamentally, quantum teleportation [4] shows how entanglement is a resource for effectively transmitting the unknown state of a quantum system between two locations, without physically transmitting quantum states. Towards the future quantum computer, such teleportation could transport information between the few-qubits processing units, and memory units which must be well isolated and hence should not be directly coupled to the rest of the system via physical interactions. This so-called modular architecture for quantum computing provides a viable solution to the major scaling problem for many-qubit quantum information processing [8, 19]. In quantum communication, similar ideas would allow quantum repeaters to purify information through local operations only, provided they can consume units of entanglement between the two communicating devices [6, 9]; such quantum repeaters are a necessary technology for exploiting accurate quantum communication, with associated e.g. cryptographic benefits, over long distances.

A major challenge towards enabling these applications is that generating entangled states between distant systems must rely, itself, on a quantum channel [1]. Microwave experiments have demonstrated how to deterministically entangle separate quantum subsystems via parity measurements [24], yet with a fidelity directly limited by the quality of the quantum channel: any losses on the probe field imply losses in entanglement. Channel losses can also be made to affect preparation success probability, instead of preparation fidelity. Indeed, many experiments in the optical domain have illustrated that by heralding the preparation on some rare photo-detection events, one can achieve significant entanglement despite propagation losses [7, 18, 22, 12, 5]; a similar scheme with microwaves has recently been implemented in [20]. The success rate of this probabilistic preparation is however usually very low. Furthermore, the preparation fidelity is still limited by imperfections such as the dark counts of the photodetector. The literature does not cover the possibility to highly entangle distant quantum bits deterministically by using a lossy quantum channel.

The present letter solves this question with an explicit proposal for the essentially equivalent [1] achievement of an eigenstate-preserving quantum non-demolition (QND) parity measurement between spatially separated qubits. Although such parity measurement is not feasible with a quantum channel subject to arbitrary errors [1], it becomes solvable if the channel features one dominant error source. Hence our key idea is to transmit over the quantum channel particularly engineered quantum states of light, i.e. “cat states”, for which the dominant photon loss errors almost reduce to photon-number parity flips [16]. With this we design the interaction between qubits and probe field such that (i) measuring the probe at the output performs a QND measurement of qubits parity and (ii) photon loss events on the transmitted probe field render the detection less decisive (weak measurement) but affect only minimally the parity eigenstates.

The abstract setting (Fig.1a) comprises two target qubits at different locations A, B and possibly embedded in auxiliary quantum machinery, e.g. a cavity in circuit quantum electrodynamics (QED) setups [30]. For each measurement, a source generates a controlled “probe” quantum state at A which then interacts with according to a unitary , is transmitted over a noisy quantum channel , before interacting with according to and finally hitting a detector at B. Those probe states play the role of parity meter. Since the quantum channel is the unequivocal bottleneck for remote entanglement in state-of-the-art technology [14], we focus on this issue and assume in this letter that all (reasonable) local actions (i.e. , generating , detection at ) are implemented perfectly.

The QND measurement of a quantum observable discriminates possibly imperfectly between the eigenspaces of , but ensures that every eigenspace of remains unaffected for all possible detection results. In the case of an observable with degenerate eigenspaces, this only ensures that a state inside an eigenspace is sent to a state in the same eigenspace. Here, we define a slightly stronger Eigenstate-Preserving Quantum Non-Demolition (EP-QND) measurement, which stands for a QND measurement which does not affect any eigenstate of the quantum observable . In other words the EP-QND property ensures that the measurement acts as identity on each eigenspace. Consider the parity observable associated to two qubits in the canonical basis , with

 Q+=|00⟩⟨00|+|11⟩⟨11| , Q−=|01⟩⟨01|+|10⟩⟨10|.

In a projective measurement of , a detection result (resp.) would project the two qubits onto the even parity manifold (resp. the odd parity manifold ). A less decisive EP-QND measurement can result for instance from classical uncertainty in the detection, e.g. with probability an even (resp. odd) parity state gives detection result (resp. ). Then the probability to detect becomes with corresponding measurement back-action transforming initial state into [21]

 ~K+(|ψ⟩⟨ψ|)=ξQ+|ψ⟩⟨ψ|Q++(1−ξ)Q−|ψ⟩⟨ψ|Q−p+,

and similarly for detection result ,

 ~K−(|ψ⟩⟨ψ|) =ξQ−|ψ⟩⟨ψ|Q−+(1−ξ)Q+|ψ⟩⟨ψ|Q+p− p− =ξ⟨ψ|Q−|ψ⟩+(1−ξ)⟨ψ|Q+|ψ⟩.

Any even-parity (resp. odd-parity) state remains unchanged under such measurement back-action (whence the notation EP-QND) and predominantly gives detection result (resp. ), i.e. with probability . By repeating the EP-QND measurement sufficiently often, a precise conclusion about parity can be obtained without disturbing any initial state of definite parity.

We first sketch our concept with a probe qubit. Starting with we let (resp.) implement a CNOT gate on conditioned by (resp.), see Fig.1b. If the channel was perfect ( for on Fig.1b), then an initial state would remain unchanged, while an initial state would come out as just before detection of the probe. Thus the measurement operations correspond to with .

Now let the channel subject the probe to an unknown number of bit-flip operations ; this can be represented as a succession of CNOT gates, conditioning each bit-flip on an unknown hypothetical state of the environment. For even, the outcome is as for the perfect channel. For odd the final state of the probe is reversed, e.g. input yields output , but most importantly, the state of the target qubits remains unaffected. This essential property ensures that the expected evolution for unknown (equivalently, tracing over the unknown states of the environment) remains an EP-QND parity measurement, explicitly described by the operators with . One can easily adapt this to account for detection misses and errors. A broad distribution of values of implies low contrast for the measurement, pushing close to 1/2, but it does not impede its EP-QND character. Hence when sufficiently many measurements can be repeated within a relevant timescale, a conclusive result is obtained even for very close to (see details below).

This EP-QND property is not retained under general channel errors. Indeed if for instance includes a phase-flip , then the initial even-parity Bell state gets transformed by measurement back-action into ; thus this eigenstate of is not conserved, breaking the EP-QND character. Not knowing if was applied or not, the initial pure entangled state gets transformed into a statistical mixture of and , i.e. entanglement is lost. In fact it is impossible to ensure EP-QND measurement of parity when the channel disturbance can be arbitrary [1].

These abstract properties indicate a pathway towards loss-tolerant parity measurement: use a subspace of probe states on which the dominant decoherence channel acts as a unitary root of identity, e.g. a bit-flip. This fits a physical implementation where the probe is an electromagnetic field pulse whose logical states are materialized by so-called “cat states”, i.e. mesoscopic superpositions of coherent states. This encoding, proposed earlier as a resource towards quantum computing [16, 17], ensures that successive photon losses imply in good approximation, coherent logical bit flips. The rest of this letter describes such implementation in detail.

This physical implementation of loss-tolerant parity measurement is sketched in Figure 1c. We denote

 ∣∣C±β⟩=(|β⟩±|−β⟩)/N±β,N±β=√2±2 e−2|β|2,

the superpositions between two coherent states and , ; the normalization constants rapidly approach as the coherent amplitude becomes large. The probe field is initially prepared in the state and interacts with two qubit-cavity systems in a cascaded manner. Between the two setups, it is exposed to losses that are modeled by the mixing with the vacuum state of an ancillary mode. This is represented by a unitary operator modeling a beam-splitter Hamiltonian with transmittance and reflectance . The unitary operators apply:

 UA|0⟩A∣∣C±α⟩p=|0⟩A∣∣C±α⟩p,UA|1⟩A∣∣C±α⟩p=|1⟩A∣∣C∓α⟩p, (1) UB|0⟩B∣∣C±√ηα⟩p=|0⟩B∣∣C±√ηα⟩p,UB|1⟩B∣∣C±√ηα⟩p=|1⟩B∣∣C∓√ηα⟩p,

Finally, after interaction with the second qubit, a measurement projects the probe’s state onto or . Identifying and respectively with the logical and , we recover the above abstract scheme where implement CNOT gates.

We now analyze the performance of this scheme. Imagine a virtual detector for the ancillary field modeling the losses, also projecting it to one of the two states or , but with unread detection result. The measurement outcomes of the two detectors (one real and one virtual) are associated to four Kraus operators , modeling the back-action of the measurements on the target qubits: e.g. their state , after measuring even parities for both detectors, should be modified to . Following the simple computations of the supplementary material, these Kraus operators are

 M+,+ M+,− M−,+ M−,− (2)

Discarding the inaccessible outcome of the virtual detector, the back-action induced by the measurement of the probe field follows the partial Kraus maps:

 K+(ρ) =M+,+ρM†+,++M+,−ρM†+,−Tr(M+,+ρM†+,++M+,−ρM†+,−), (3) K−(ρ) =M−,+ρM†−,++M−,−ρM†−,−Tr(M−,+ρM†−,++M−,−ρM†−,−).

In the lossless case (), and vanish as , and the coefficients in front of and in and in front of and in are identical, equal to 1. This corresponds to a projective parity measurement, as described above by with .

The effect of transmission losses () is twofold. First, it reduces the measurement strength. Indeed, when the probe field is detected in a given parity (e.g. ), the qubits could be projected to the opposite parity manifold (e.g. by the Kraus operator ). However, each measurement does increase the conditional probability of finding the qubits in the same parity manifold as the one indicated by the probe detections, because . A projective parity measurement under perfect transmission () is thus replaced by a less decisive measurement for , where at each shot we gain partial information on the parity. By repeating the measurement the state gets projected onto a well-defined parity subspace. An initial state of definite parity will always keep this parity (e.g. ).

The second, more harmful effect of the transmission loss is a slight perturbation of the EP-QND property, by introducing slow mixing within each given parity manifold. This is due to the coherent states and not being perfectly orthogonal, so that . This effectively induces a dephasing inside parity manifolds, e.g.  implies that drives the even-parity states towards , while would drive them towards .

The simulations of Fig. 2a illustrate the competition between parity measurement and undesired dephasing while varying , the average number of photons in the probe field. Initializing both qubits in the state , an EP-QND parity measurement should project the joint state towards one of the two Bell states or . This is the dominant tendency on Fig. 2(a), while the transmission loss induces a slow dephasing mixing the target Bell states with the undesired ones and . By increasing , this undesired dephasing gets suppressed significantly, at the expense of a slower convergence, i.e. weaker parity measurement. As soon as , one can achieve arbitrarily high fidelity in this way.

This near EP-QND parity measurement can be used to stabilize a particular Bell state through a simple feedback protocol. For instance, to stabilize , (i) apply a -pulse around the -axis on the first qubit whenever the measurements estimate a probability higher than to be in the odd parity manifold, (ii) after that, apply a -pulse on both qubits around the -axis irrespectively of the detection result. The measurement back-action favors convergence towards the dominant parity, the pulse correcting the parity whenever the state is converging towards the wrong one. This pushes the state towards the span of , without favoring the target . The two -pulses then leave untouched and send the undesired onto , such that the next parity measurement stochastically moves the corresponding population as well towards the target Bell state. The simulations of Fig. 2b illustrate the performance of this protocol, having fixed and varying Feedback stabilization of entanglement is further discussed in the supplementary material.

Convergence rates can be calculated analytically for both the parity measurement and the spurious dephasing, yielding respectively [1]:

 rparity =12log(1−e−4|α|21−e−4(1−η)|α|2), rdephasing =12log(1−e−4|α|21−e−4η|α|2).

This allows to estimate the measurement performance as a function of and . Consider again the evolution depicted on Fig.2a. The fidelity to the closest Bell state is 1/2 times the sum of two terms: dominant parity population, which converges from to at roughly a rate , and dominant phase population, which decreases from to at a rate . The two terms contribute equally to the error after a number of measurements that satisfies . The corresponding estimate of Bell state fidelity is . By solving numerically the above transcendental equation, Figure 3 illustrates these estimates of our parity measurement’s performance. For a transmission efficiency as low as we can get as high as with less than 400 measurement runs, taking . Increasing to one can achieve the same with only 17 measurements, taking .

All the required operations for this proposal have been individually implemented within the framework of quantum superconducting circuits. The strong dispersive coupling of a transmon qubit to a high-Q cavity mode [26], provides the universal controllability of the state of the quantum harmonic oscillator modeling the cavity mode [15, 13]. This controllability has been experimentally illustrated with circuit QED setups [29, 11]. Such a coupling enables to prepare the probe field in a cat state and to perform the CNOT gates and of (1) between the qubits and and associated intra-cavity fields. Recent experiments realizing a variable coupling between cavity modes and a transmission line [32, 31, 10], provide the possibility of catching the propagating microwave field, performing the required gate between the qubit and the cavity field, and finally releasing back the cavity photons. Finally, while the measurement of photon-number parity has also been realized in a similar setup [27], one can further simplify the protocol by letting map the intra-cavity field to coherent states instead of the cat states . This would allow to replace a photon-number parity measurement by a simple homodyne detection of the released field using a parametric amplifier [28]. A single measurement duration in such an experimental realization depends mainly on the time required to perform the operations and . As explained in [15, 13] this gate time is roughly the inverse of the dispersive coupling strength. Recent experiments where this coupling strength is more than three orders of magnitude larger than both the qubit and the cavity decay rates [29, 23, 11] indicate that high entanglement fidelities should be achievable whenever a transmission efficiency of more than is achieved.

We have shown that it is possible to perform a near EP-QND measurement of the parity of two distant qubits despite an important loss through the transmission channel between them. By preparing the probe field in a quantum superposition of two coherent states with opposite phases, we avoid any back-action of the probe losses on the target qubits’ state inside any parity subspace. Indeed, such losses mainly decrease the measurement strength but barely affect its EP-QND character. Therefore, even with an inefficient transmission channel, by repeating the measurement many times one can efficiently project the joint qubits state onto a definite parity subspace. This could be used not only to deterministically and efficiently prepare an entangled state of two distant qubits, but combined with a quantum feedback strategy it could even protect such entangled state against the local decay channels of the qubits [1]. The operations required to perform this loss-tolerant parity measurement are within the reach of state of the art experiments with quantum superconducting circuits. Their implementation will lead to an important step forward for implementing quantum teleportation protocols in a loss-tolerant way, and more particularly towards the modular architecture solution for large scale quantum information processors [8].

The authors thank the Agence Nationale de la Recherche for financial support under grant ANR-14-CE26-0018.

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## Supplementary Material

### Constraining the channel loss operators is necessary for QND parity measurement

We prove that it is impossible to design the measurement setup of Fig.1a of the main text in order to obtain an EP-QND parity measurement under arbitrary channel noise, i.e. when the in Fig.1b of the main text can be arbitrary.

• Note that any unconditioned local unitaries on the qubits along the measurement chain, can be merged into and . Indeed, even if physically we apply such to qubit after , this commutes with since it acts as identity on both and . We will thus assume without loss of generality a setting with just two arbitrary unitaries and , which make the arbitrary probe interact respectively with and .

• In order to keep invariant the even-parity states and , we have to take such that and for some . Indeed consider a contrario that e.g. an initial was mapped by into a state involving ; then, since no further action is applied on qubit , at the end of the nominal measurement chain there would unavoidably be a nonzero probability to end up in a state involving this , which would contradict the EP-QND objective for an initial state .

• If we chose for some , then we could rewrite

 UA|qA⟩|ψp⟩=(Z|qA⟩)|ϕ0⟩p

with acting only on qubit and defined by , . This clarifies that the probe state would carry no information about the state of qubit , hence we can get no parity information by later measuring the probe system at .

• On the other hand if we have , then it is always possible to write a noise channel which keeps in place but, conditionally on the unknown state of some environment variable, moves . I.e. referring to Fig.1b of the main text,

 E1|ϕ0⟩p=|ϕ0⟩pandE1|ϕ1⟩p=|ϕ2⟩p≠|ϕ1⟩p.
• Consider now what happens to the initial even-parity state . After has been applied, the qubits are entangled with the probe in the state . In order to have an EP-QND measurement, the remaining action has to disentangle from the probe before detection. In the case where is not applied (no channel loss), we thus need

 UB|0⟩B|ϕ0⟩p=|0⟩B|ϕb⟩p\scriptsize AND UB|1⟩B|ϕ1⟩p=|1⟩B|ϕb⟩p (4)

for some . In the case where in contrast is applied by the lossy channel, we need

 UB|0⟩B|ϕ0⟩p=|0⟩B|ϕa⟩p\scriptsize AND UB|1⟩B|ϕ2⟩p=|1⟩B|ϕa⟩p (5)

for some . For the experiment we must select a unique , without knowing the environment state i.e. without knowing whether was applied or not. The left parts of (4),(5) thus impose

 |ϕb⟩=|ϕa⟩.

But then the second parts of (4),(5) lead to a contradiction, as they require the unitary to map two different initial states onto the same final state.

This shows the impossibility for the measurement to be EP-QND with respect to the “conditional ” channel noise. We have taken care to keep our construction fully general, such that we can conclude: whatever our design for , , , we will obtain a measurement setup either which gains no parity information, or for which some particular noise actions can destroy the EP-QND character and perturb definite-parity states.

For example, in our proposed construction with CNOT gates:

• For S1 we have , ; we are obviously in the situation of S3a, not of S2.

• Regarding S3a, the phase-flip channel for instance would keep at but move to . Then to save the situation would have to satisfy in particular which is not possible.

This example shows that an EP-QND parity measurement is not possible if the transmitted probe state can be subject to both bit-flip and phase-flip errors.

### Channel with constrained loss operators: generalization of loss-tolerant measurement

In the main text we discuss the case, motivated by realistic experimental conditions, where the transmission channel only subjects the probe to an unknown number of bit-flips. This is not the only favorable situation.

Consider indeed a situation where the transmission channel subjects the probe to an unknown number of unitary operations of the same type , where Identity for some integer . Now let and apply conditioned -gates on the probe, where

 V=(UC)N/2.

Then all the operations , , commute with each other, and an initial probe state gets mapped just before detection onto

 (UC)N/2⋅(UC)N/2⋅UnC|ψ0⟩p=UnC|ψ0⟩p

if the target qubits have even parity, or onto

 (UC)N/2⋅UnC|ψ0⟩p

if they have odd parity, while the target qubits remain unaffected. Thus not knowing makes the final probe state uncertain, as with the bit-flip, implying that detection results will not allow to perfectly discriminate the parity. But the probe ends up in the same (unknown) state for all even parity states of the target qubit; and it ends up in another same state for all odd parity states of the target qubit. This ensures preservation of the EP-QND property under channel losses.

### Kraus operators computation

We here summarize the calculations that lead to the Kraus operators of Eq.(2) in the main text. We start with an initial joint state of the two qubits, the probe field, and the ancillary mode modeling the transmission loss, given by

 |ψ0⟩=(c00|00⟩A,B+c11|11⟩A,B+c01|01⟩A,B+c10|10⟩A,B)∣∣C+α⟩p|0⟩env.

Following the definition of the unitary operators , and in Eq.(1) of the main text, just before performing the photon-number parity measurements of the probe and ancillary fields, this joint state has evolved to:

 |ψ⟩

Here, we have used the fact that

 |±β⟩=N+β2∣∣C+β⟩±N−β2∣∣C−β⟩,

for and . Detecting the probe and the ancillary fields both in even parity leads to applying the following projection operator as the measurement back-action

 Π+,+ = IdA,B⊗Πevenp⊗Πeven% env = IdA,B⊗(∞∑k=0|2k⟩⟨2k|p)⊗(∞∑k=0|2k⟩⟨2k|% env)

with Id the identity map. Since has even parity and has odd parity, the projected wave function can be simply read off the above expression,

 Π+,+|ψ⟩=N+√1-ηα2 ⎛⎝c00N+√ηαN+α|00⟩+c11N−√ηαN−α|11⟩⎞⎠A,B∣∣C+√ηα⟩p∣∣C+√1−ηα⟩env.

In a similar way, we have the following projected wave functions for other measurement outcomes

 Π+,−|ψ⟩=N−√1-ηα2 ⎛⎝c01N−√ηαN+α|01⟩+c10N+√ηαN−α|10⟩⎞⎠A,B∣∣C+√ηα⟩p∣∣C−√1−ηα⟩env
 Π−,+|ψ⟩=N+√1-ηα2 ⎛⎝c01N+√ηαN+α|01⟩+c10N−√ηαN−α|10⟩⎞⎠A,B∣∣C−√ηα⟩p∣∣C+√1−ηα⟩env
 Π−,−|ψ⟩=N−√1-ηα2 ⎛⎝c00N−√ηαN+α|00⟩+c11N+√ηαN−α|11⟩⎞⎠A,B∣∣C−√ηα⟩p∣∣C−√1−ηα⟩env.

This corresponds to the Kraus operators as defined by Eq.(2) in the main text.

### Convergence rates

The rate at which the parity measurement acquires information can be expressed analytically as a function of and . Define the Lyapunov function

 Vparity(ρ) = √⟨00|ρ|00⟩⟨10|ρ|10⟩ = wB02√1−(PB0)2+wB12√1−(PB1)2,

where denotes the population with qubit in state and measures the parity, conditioned on qubit being in state , for . One can show that

 ⟨Vparity(ρk+1)⟩=√1−e−4(1−η)|α|2√1−e−4|α|2⟨Vparity(ρk)⟩, (7)

where is the joint qubits state after the ’th measurement and denotes the ensemble average of over measurement realizations. Indeed, we have

 E(Vparity(ρk+1) | ρk)=Vparity(K+(ρk))P(+ | ρk)+Vparity(K−(ρk))P(− | ρk),

where

 P(+ | ρk) =Tr(M+,+ρkM†+,++M+,−ρkM†+,−) P(− | ρk) =Tr(M−,+ρkM†−,++M−,−ρkM†−,−)

are respectively the conditional probabilities of achieving a positive/negative outcome at ’th measurement. Therefore, we have

 E(Vparity(ρk+1) | ρk)= √⟨00|M+,+ρkM†+,+|00⟩⟨10|M+,−ρkM†+,−|10⟩+ √⟨11|M+,+ρkM†+,+|11⟩⟨01|M+,−ρkM†+,−|01⟩+ √⟨00|M−,−ρkM†−,−|00⟩⟨10|M−,+ρkM†−,+|10⟩+ √⟨11|M−,−ρkM†−,−|11⟩⟨01|M−,+ρkM†−,+|01⟩ =(N+√1−ηαN−√1−ηαN+αN−α)Vparity(ρk).

Taking the expectation value of both sides we find the result of (7). Thus exponentially decays to zero at a rate

 rparity=12log(1−e−4|α|21−e−4(1−η)|α|2).

In the case of a fully EP-QND measurement, i.e. assuming and in the Kraus operators, the two terms in the Lyapunov function (Convergence rates) would decay at the same rate and represents precisely the parity measurement strength. Indeed, in this case we would obtain the same result with the alternative Lyapunov function where .

One can also analytically calculate the dephasing rate induced by the transmission loss. To this aim we define the coherence function

 C(ρ)=∣∣⟨00|ρ|11⟩∣∣+∣∣⟨01|ρ|10⟩∣∣.

Similar calculations as above lead to

 ⟨C(ρk+1)⟩=√1−e−4η|α|2√1−e−4|α|2⟨C(ρk)⟩.

Therefore the coherence function exponentially decays to zero at a rate

 rdephasing=12log(1−e−4|α|21−e−4η|α|2).

Figure 4 provides a comparison of the simulations to the above analytical results. As can be seen, the performance of the parity measurement protocol can be well-explained using the above two rates. The slight mismatch between the raising rate of the blue curve (fidelity to Bell states) and the theoretical can be explained by the nonlinear relation between the fidelity and the Lyapunov function . This nonlinearity makes it impossible to translate the rate into a precise exponential rate for the raising dynamics of the average fidelity.

As explained through the main text, using these rates, it is possible to estimate the maximum achievable fidelity as a function of and . Indeed, this maximum fidelity is achieved, approximately, when the contribution of the first effect (parity projection) becomes equivalent to that of the second one (dephasing inside a parity eigenspace). This leads to the transcendental equation . In the limit of (equivalent to ), one can approximately replace by , and therefore the solution to the transcendental equation is well estimated by

 Tmeas=1rparityW0(rparityrdephasing),

where is the Lambert W-function. The simulations of Fig. 3 in the main text illustrate this result.

### Feedback stabilization of an entangled state

As outlined in the main text, the EP-QND parity measurement allows to stabilize a highly entangled state through a simple feedback mechanism. We here provide some details about this scheme.

We first note that the feedback decision requires to know the respective parity populations. This knowledge is typically obtained by a quantum filter, i.e. a computer estimating the state by simulating the evolution Eq.(3) of the main text associated to the respective detection results. Such filters are known to be stable [25].

In the present case, the quantum filter can be simplified significantly, namely by discarding all off-diagonal components in the Bell state basis. Indeed, first note that the measurement does not depend on coherences among subspaces of different parity. Moreover, the measurement itself completely destroys, in a single iteration, any coherences between subspaces of different parity. The pulses “export” coherences possibly present between e.g.  and into coherences between and , i.e. between different parity subspaces, which are thus destroyed at the next measurement; while they “import” into e.g. the even parity eigenspace, the coherences previously present between and , which are none since those two Bell states belong to different parity eigenspaces and their coherences were thus destroyed by the last measurement. Thus after one initial weak measurement step at most, no relevant coherences among Bell states will survive. This allows to update just the populations on the four Bell states, as in a classical filter for a partially observed Markov chain. This filter, updating the populations on the four Bell states, is the only computation required for the feedback: the action itself just requires a binary decision, to switch towards the predominantly populated parity.

To analyze the feedback more explicitly, we note that instead of applying the pulses to rotate the state in the Schrödinger picture, we can reformulate the dynamics by applying the pulses to the measurement scheme, in an equivalent Heisenberg picture. The corresponding measurements then alternate between the parity measurement in -basis, and a parity measurement in -basis. The corresponding state is subject to -pulses in the respective or basis. A pulse in basis (resp.  basis) does not change the populations of -parity subspaces (resp. of -parity subspaces), and is applied to increase the population in (resp. ). The stochastic convergence of the measured system towards a definite parity in both and coordinates ensures that, in absence of other effects, the population in would increase in expectation until reaching 100%. The dephasing effect limits the actual fidelity for a given . Taking larger allows to increase the fidelity, but slows down the convergence, such that a tradeoff value of must be selected with respect to other decoherence effects acting on the system.

We can quantify the performance of this feedback scheme using the characteristic convergence rates computed in the previous section. We here keep the viewpoint of the previous paragraph, of performing the measurement alternatively in -basis and in -basis. Then a similar continuous-time model for the populations of Bell state writes:

 ddtpBe+ = rparity(