Robust multipartite quantum correlations without complex encodings

# Robust multipartite quantum correlations without complex encodings

## Abstract

One of the main challenges for the manipulation and storage of multipartite entanglement is its fragility under noise. We present a simple recipe for the systematic enhancement of the resistance of multipartite entanglement against any local noise with a privileged direction in the Bloch sphere. For the case of exact local dephasing along any given basis, and for all noise strengths, our prescription grants full robustness: Even states with exponentially decaying entanglement are mapped to states whose entanglement is constant. In contrast to previous techniques resorting to complex logical-qubit encodings, such enhancement is attained simply by performing local-unitary rotations before the noise acts. The scheme is therefore highly experimentally friendly, as it brings no overhead of extra physical qubits to encode logical ones. In addition, we show that, apart from entanglement, the resiliences of the relative entropy of quantumness and the usefulness as resources for practical tasks such as metrology and nonlocality-based protocols are equivalently enhanced.

###### pacs:
03.67.-a, 03.67.Mn, 42.50.-p

Introduction.— Multipartite quantum correlations in composite systems subject to local noise are in general extremely fragile, typically decaying more quickly as the number of particles increases. For instance, one of the most important genuinely multipartite entangled states is the Greenberger-Horne-Zeilinger (GHZ) state GHZ (). It represents coherent superpositions of the kind of the celebrated Schrödinger’s cat state cat (). Unfortunately, under the action of local noise, its entanglement decays exponentially fast, with a decay-rate that grows proportionally to  expdecay (); expdecayrobustdecay (); aolita (). Thus, the system is very rapidly taken into, or close to, a separable mixture. This exponential fragility entails serious drawbacks for the practical applicability of GHZ states as resources for quantum information processing in realistic scenarios. For example, for any noise strength, the quantum gain provided by GHZ entanglement in parameter estimation metrology () or distributed-computing scenarios CCP (); CCPBrukner (); chaves () vanishes almost instantaneously already for a moderate system size.

A possible way to enhance the robustness of quantum correlations is to encode logical qubits into error-correction codewords, consisting of entangled states of many physical qubits expdecayrobustdecay (); dur (). For instance, for small noise strengths , the decay rate of logical GHZ entanglement under local white noise can be decreased exponentially with the number of physical qubits in the codeword when the codewords are themselves GHZ states dur (). This is remarkable because the enhancement is achieved passively, i.e. without any active error correction. However, there is a price to pay in experimental overhead: For the logical state to achieve full entanglement robustness – in the sense that its logical entanglement becomes independent on –, each logical qubit requires a number of genuine-multipartite entangled physical qubits that scales logarithmically with the number of logical qubits.

With the maximally mixed state as its only steady state, local white noise (local depolarization) is the most detrimental type of local noise. Nevertheless, in many realistic situations the noise can be assumed, up to good approximation, to possess privileged directions in the Bloch sphere, including pure states as steady states. This is the case, for instance, in many experiments with atomic or ionic qubits, where the dominant source of noise is dephasing from magnetic-field and laser-intensity fluctuations, and from spontaneous emissions during Raman couplings ion_review (). Another example is provided by birefringent polarization-mantaining optical fibers pol_mantaining_fibers (), where mechanical stress and temperature induce index-refraction fluctuations that dephase polarization qubits. In the former case, the privileged noise direction is that of the quantization axis defined by the magnetic field, while in the latter, that of the linear polarizations along the ordinary and extraordinary axes of the fiber.

In this work we study the action on graph states of local noisy channels with an approximately well-defined privileged basis. Graph states constitute a family of genuine multi-qubit entangled states with remarkable applications graph_review (). Relevant examples thereof are the previously mentioned GHZ state, or the cluster state, which allows for measurement-based quantum computation cluster (). We introduce an experimentally friendly recipe, consisting of local-unitary rotations before the noise acts, to enhance the resistance of graph-state quantum correlations. Remarkably, and despite its simplicity, for exact dephasing this prescription supplies the states with full robustness: It gives an -independent lower bound for the decay of graph-state entanglement. In particular, the exponentially fragile entanglement of GHZ states is enhanced to decay only linearly with , for all . In addition, the bound holds for quantum correlations other than entanglement Kavan () and is robust against mixedness in the initial states. Finally, for GHZ states, we show that the local-unitary protection resists small noise deviations from exact dephasing, and that the enhancement applies also to the usefulness for physical tasks such as metrology  metrology () and distributed-computing CCP (); CCPBrukner () protocols.

Enhancement of the robustness of graph-state quantum correlations.— We consider local completely positive trace-preserving channels , defined on any state as

 E(ϱ) ≐ (1−p2)ϱ+p2(αXXϱX (1) + αYYϱY+αZZϱZ),

where , , and are, respectively, the first, second, and third Pauli matrices in the computational basis . Parameters satisfy the normalization condition . The composite -qubit map is given by the single-qubit map composition , where , with , corresponds to map (1) acting on the -th qubit. The focus of our attention throughout is in the situations where , , so that is close to local phase-damping map , which corresponds to . Probability measures the noise strength and gives also a convenient parametrization of time: refers to the initial time and refers to the asymptotic limit. Note that the factors in (1) are such that an exact fully-dephasing channel appears at and .

Let us begin by the phase-damping channel . We focus first on GHZ states

 |Φ+N⟩≐1√2(|0⟩⊗N+|1⟩⊗N). (2)

Under , all the entanglement in (2) decays (at slowest) exponentially with , as  aolita (). We show next that, for a fixed , the entanglement of

 |Φ+NT⟩≐H⊗N|Φ+N⟩=1√2(|+⟩⊗N+|−⟩⊗N), (3)

under is independent on . Operator stands for the Hadamard-gate rotation, defined by and , with . Transversal states (3) are thus local-unitarily equivalent to (2), possessing therefore the same amount and type of entanglement.

We consider here an arbitrary entanglement monotone , i.e. any function of which is non-increasing under local operations and classical communication. In Appendix A however, we extend the treatment to the relative entropy of quantumness Kavan (), which is not an entanglement monotone. Notice first that a single-qubit measurement on leaves the system in state or , with . Similarly, since it commutes with , a measurement on leaves the system in , or , with . Furthermore, it is immediate to see that and are local-unitarily equivalent. From this, and the monotonicity of under local measurements, it follows then that . Iterating this reasoning times and, for ease of notation, omitting the tensor-product factors, one obtains that

 E(ΛPD(|Φ+NT⟩)) ≥ E(ΛPD(|Φ+N−1T⟩))≥ … ≥ E(ΛPD(|Φ+2T⟩)). (4)

That is, transversal states (3) possess at least as much resistance as the two-qubit state , for all . As an example, consider the robustness of -party distillable GHZ-entanglement. The distillation of maximally entangled pairs between any pair of particles is sufficient to distill an -qubit GHZ state expdecayrobustdecay (). Bound (Robust multipartite quantum correlations without complex encodings), for the -party distillable two-qubit entanglement between any pair, implies that the distillation of entangled pairs (and consequently also of GHZ states) from the transversal states is at least as robust for qubits as for , in contrast to bare states (2). In Appendix B, we show that (Robust multipartite quantum correlations without complex encodings) holds not only for GHZ states but actually for arbitrary graph states, and even initially in the presence of global white noise. That is, the entanglement decay of generic graph states, encoded in appropriate transversal local bases, is bounded from below by that of a two-qubit graph state. The bound follows again from the fact that any connected -qubit graph state can be mapped into one of qubits by single-qubit or measurements. The encoding is again given by single-qubit Hadamard rotations, but applied only to the qubits measured in in the mapping (see Appendix for details).

Next, we probe bound (Robust multipartite quantum correlations without complex encodings) with a simple-to-calculate entanglement monotone: the negativity negativity (). In addition, to test their robustness against deviations from exact dephasing, we allow for maps with arbitrary , and . Specifically, we calculate analytically the negativity of any bipartition “one qubit versus the rest” of . The smaller is, the closer is the state to featuring no genuine -qubit entanglement expdecay (); expdecayrobustdecay (); aolita (). Due to the GHZ-diagonal structure (see, for instance, Ref. aolita ()) of , the calculation is enormously simplified, and reduces essentially to diagonalizing matrices of dimensions . One obtains

 N = ⌊N−12⌋∑μ=0(N−1μ)(max[0,f−μ+1−f+μ] (5) + max[0,f−μ−f+μ+1]),

where and , for even, or , for odd.

For the particular case addressed by (Robust multipartite quantum correlations without complex encodings), the negativity (5) simplifies (see Appendix C) to

 N(αZ=1) = (1−p)⌊N′2⌋∑μ=0(N′μ)[p2μ(1−p2)N′−μ (6) − p2N′−μ(1−p2)μ],

with . In Appendix C we show that as , for all . In addition, for all , we observe that the bound on given by (Robust multipartite quantum correlations without complex encodings) is too conservative, as it is not tight. Furthermore, the limit is approached the faster the smaller . This can be appreciated in Fig. 1 a), where for states (2) and (3) under the local-dephasing map is plotted as a function of and for different .

In turn, for the generic case , with the deviation from exact dephasing, we focus on the weak-noise regime. For sufficiently small , one can ignore all terms not linear in and approximate negativity (5) as . While this is no longer an -independent behavior, it clearly constitutes a remarkable robustness enhancement in comparison to , specially in the limit [see Fig. 1 b)]. In addition, numerical tests show that the approximation actually holds up to relatively large . For example, with and , the approximation by the exponential above is excellent up to . Furthermore, this in turn tends to as increases, an exponential decay as that of (2) but with the exponent damped by the factor . In all cases, the less the noise deviates from exact dephasing (), the slower is the decay of entanglement.

We emphasize that the enhancement is achieved without any overhead in complex logical-qubit encodings, just through local Hadamard rotations. Interestingly, these rotations correspond to a qubit-basis transversal to the one privileged by to the noise. In particular, for GHZ entanglement, the robust qubit-basis is exactly the one orthogonal to that defined by the only pure states immune to the noise. Numerical optimizations up to show that no other single-qubit basis yields slower decay of negativity than that of (3), although other single-qubit bases attain the same decay (up to numerical precision).

Quantum metrology with dephased resources.— For mixed states, higher entanglement does not necessarily imply better performance at fulfilling some physical task. In what follows, we show that our local-unitary protection enhances also the robustness against local dephasing of GHZ states as resources for phase estimation metrology () (For a frequency estimation model see for instance Chaves2013 ()).

We consider a Hamiltonian , with unknown parameter . The associated unitary evolution operator over a time is , with the phase to be estimated. A generic -qubit probe state accordingly transforms as . The statistical deviation in the estimation can be bounded as  CramerRao (). Here, represents the number of runs of the estimation, and is the quantum Fisher Information fisher () of , which measures how much information about can be extracted from . can be compute as

 (7)

and being respectively the eigenvalues and eigenvectors of the density matrix and the derivative in relation to the parameter . If is separable, is always limited as . However, entangled states can reach the maximal value , which yields a quadratic gain in precision. This is the maximal gain compatible with the uncertainty principle and is therefore known as the Heisenberg limit.

The Fisher information of locally dephased GHZ state is . Clearly, the quantum gain is obliterated by decoherence exponentially fast. However, if before dephasing takes place one locally rotates the resource GHZ state to (3), and then, before the estimation, undoes the rotation, the resulting Fisher information is

 FPDT=4N2(1−p)2+16N(1−p2)p2. (8)

The robustness-enhancement is thus exponential also for the accuracy in phase estimations (see Fig. 2).

Nonlocal computations with dephased resources.— Finally, we show that our local-unitary protection scheme enhances the robustness against local dephasing of the nonlocality of quantum correlations. Specifically, we focus on the performance of dephased GHZ states to assist in solving distributed-computing tasks, also known as communication-complexity problems (CCPs) CCP (); CCPBrukner ().

In the considered CCPs, distant users, assisted by some correlations and a restricted amount of communication, must locally calculate the value of a given function . For every Bell inequality there exists a CCP that can be solved with a higher probability of success with nonlocal correlations than with any classical resource if, and only if, the correlations violate the inequality CCPBrukner (). The probability of success in the CCP is , where is the Bell violation by the nonlocal correlations in the resource quantum state and the maximum violation over arbitrary nonlocal correlations.

As an example, we consider the CCP associated with the Mermin-Klyshko (MK) inequality for -bit correlations MK (). We obtain that violates the inequality for all , and its violation is bigger than that by . This leads to the enhancements of as the one plotted in Fig. 3. Also, in the inset of Fig. 3, we have plotted the dependence with of simple lower and upper bounds of the local fraction of the correlations, which quantifies the fraction of events describable by a local model epr2 (). The bounds were calculated as explained in Ref. chaves (). From these, one can see that while, for large , the local fraction of dephased bare GHZ states tends to unit exponentially fast, that of transversal GHZ states stays always below a constant value ().

Conclusions.— There are many relevant situations in which multi-particle entanglement is subject to local noise, e.g. the distribution of entangled particles to many distant parties or the storage of these particles into different quantum memories. Here we have focused on the physically relevant case in which the noise has a privileged direction and have provided a simple and experimentally friendly recipe to enhance the robustness of quantum correlations. We have shown that a simple local change of bases, while preserving the correlation properties of the state, significantly improves its robustness. For general graph states, we have derived bounds on the decay entanglement and relative entropy of quantumness that are independent of . In the case of GHZ states, we have shown not only that the local-unitary encoding neutralizes their exponential decay with the system size, but also that an exponential improvement is still observed when there are deviations from the ideal case. In addition, the robustness of the usefulness of GHZ states as resources for parameter estimation and nonlocal computations is equivalently enhanced.

The enhancement introduces no cost at all in extra particles. We believe that the fact that an exponential enhancement is achieved through such an extremely simple scheme, makes the present passive-protection approach highly relevant to many current experimental platforms.

Acknowledgements.— This work was supported by the European ERC Starting grant PERCENT and the Q-Essence project, the Spanish FIS2010-14830 project and a Juan de la Cierva grant, and Caixa Catalunya.

## Appendix A Robustness law for the relative entropy of quantumness

The relative entropy of quantumness is a discord-like measure of quantum correlations (see Kavan () and references therein). That is, it encapsulates entanglement but does not restrict to it. Like all variants of discord, it is not non-increasing under local operations assisted by classical communication (LOCC). Indeed, it is not even non-increasing under general local operations Streltsov (). Still, in this appendix we show that a robustness law equivalent to (4) applies to it too.

The relative entropy of quantumness of an -qubit state is defined Kavan () as . The von Neumann relative entropy between states and measures how distinguishable they are. State is the closest classical state to , in the sense of minimizing the relative entropy with over all exclusively-classically correlated -qubit states , with any probability distribution and any -qubit basis. We show in what follows that

 QS(ΛPD(|Φ+NT⟩))≥…≥QS(ΛPD(|Φ+2T⟩)), (9)

for all .

As in the derivation of (Robust multipartite quantum correlations without complex encodings), we consider a single-qubit measurement acting on , which leaves the system in state . As said, cannot be guaranteed not to increase under generic local maps. However, for the particular case when the local maps are unital (those mapping the identity operator into itself), it was shown in Ref. Streltsov () that is non-increasing. Thus, since a single-qubit measurement is a local unital map, we have that

 QS(ΛPD(|Φ+NT⟩))≥ QS(12(|0⟩⟨0|⊗ϱ+N−1T+|1⟩⟨1|⊗ϱ−N−1T)). (10)

Besides, by definition it is , with the closest classical state to . Next, using the definition of the relative entropy in terms of traces, taking the partial trace over the measured qubit, and after a straightforward calculation, we obtain

 S(12(|0⟩⟨0|⊗ϱ+N−1T+|1⟩⟨1|⊗ϱ−N−1T)∣∣∣∣~ξNmin)= 12[S(ϱ+N−1T∣∣∣∣⟨0|~ξNmin|0⟩Tr[⟨0|~ξNmin|0⟩]) +S(ϱ−N−1T∣∣∣∣⟨1|~ξNmin|1⟩Tr[⟨1|~ξN% min|1⟩]) −log(2x)−log(2(1−x))], (11)

with and .

Now, from (A) we immediately obtain an explicit form for : It must be

 ~ξNmin=12(|0⟩⟨0|⊗ξminN−1++|1⟩⟨1|⊗ξmin% N−1−), (12)

with and the closest classical -qubit states to and , respectively. This is due to the following observations: (i) Clearly, with this form, both the first and second lines after the equality in (A) are minimized. (ii) The minimum of over is zero, which is precisely the value the form of above yields. This leads us to

 QS(12(|0⟩⟨0|⊗ϱ+N−1T+|1⟩⟨1|⊗ϱ−N−1T))= (13)

but, since states and are local-unitarily equivalent, they posses exactly the same amount and type of quantum correlations. Therefore, the last line of (A) equals and, together with (A), renders . Again as in the derivation of (4), iterating this reasoning times one arrives at (9).

## Appendix B Robustness law for generic (possibly mixed) graph states

Here, we extend bounds (4) and (9) first to arbitrary pure graph states, and then to globally-depolarized arbitrary graph states. To encompass both bounds with the same notation, we use in what follows to denote a generic measure of quantum correlations, wich can either be an entanglement monotone, , or the relative entropy of quantumness, , defined in App. A.

For every qubit of any connected -qubit graph state , a measurement in either the or the bases leaves the remaining qubits in a connected -qubit graph state (or in a state local-unitarily equivalent to it, depending on the measurrment outcome) graph_review (). One can thus apply the the same machinery used in the derivations of (4) and (9) and arrive at the size-independent bound

 C(ΛPD(|GNT⟩))≥…≥C(ΛPD(|G2T⟩)), (14)

for all . Here, is a two-qubit graph state (local-unitarily equivalent to ), and is obtained by applying single-qubit Hadamard rotations to some of the qubits in (those corresponding to the above-mentioned measurements). Finally, the same arguments hold even for imperfect initial states of the form , where is some visibility. The resulting robustness law is then

 C(ΛPD(v|GNT⟩⟨GNT|+(1−v)\openone2N))≥… ≥C(ΛPD(v|G2T⟩⟨G2T|+(1−v)\openone4)). (15)

## Appendix C Asymptotic value of negativity under exact dephasing

Here, we first show that negativity (5) reduces to (6) when , and then that the latter tends to the -independent value in the limit , for all .

First, taking and in (5) leads, through a simple and straightforward calculation, to

 N(αZ=1) = (1−p)⌊N′2⌋∑μ=0(N′μ)∣∣∣p2μ(1−p2)N′−μ (16) − p2N′−μ(1−p2)μ∣∣∣,

with . Let us see that the absolute value inside this summation can be removed. Notice that, for all , and for any , it is

 1−p2 ≥ p2⇒ (1−p2)N′−2μ ≥ p2N′−2μ⇒ p2μ(1−p2)N′−μ ≥ p2N′−μ(1−p2)μ.

Therefore , what implies

 N(αZ=1) = (1−p)⌊N′2⌋∑μ=0(N′μ)[p2μ(1−p2)N′−μ (17) − p2N′−μ(1−p2)μ].

Next we show that , for all . To this end, see first that

 ⌊N′2⌋∑μ=0(N′μ)p2N′−μ(1−p2)μ≡ N′∑μ=~N(N′μ)p2μ(1−p2)N′−μ, (18)

with

 ~N≐⎧⎨⎩⌊N′2⌋+1if N is even, ⌊N′2⌋if N is odd.

Using (C), one rewrites negativity (6) as , where and are the following sums:

 S1≐⌊N′2⌋∑μ=0(N′μ)p2μ(1−p2)N′−μ, (19a) S2≐N′∑μ=~N(N′μ)p2μ(1−p2)N′−μ. (19b)

Now, invoking the binomial theorem, we notice that

 S1+S2≡⎧⎨⎩1if N is even, 1+(N′N′/2)p2N′2(1−p2)N′2if N is odd.

Therefore, negativity ((6)) can be expressed as , when is even, and as , when is odd.

Finally, we show first that as , and then that also as , which finishes the proof. For sufficiently large , we can apply Stirling’s approximation for the factorial: . So, we obtain that, for ,

 (N′N′/2)p2N′2(1−p2)N′2 → √2πN′2N′p2N′2(1−p2)N′2 ≐ √2πγN′N′ → 0,

where the convergence of the last limit is guaranteed by the fact that , for all .

Now, from definition (19b) and the facts that and , for all , we see that sum is bounded from above as

 S2≤⎧⎪ ⎪⎨⎪ ⎪⎩N2(N′N′/2)p2N′2(1−p2)N′2if N is even, (N2+1)(N′N′/2)p2N′2(1−p2)N′2if N is odd.

Invoking once more Stirling’s approximation, using and the definition of above, we have that for large these bounds are approximately given by

 S2≤⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩√12πN2N−1γN−1if N is even, √12π(N+2)2N−1γN−1if N is odd.

As , both bounds tend to the quantity , whose limiting value is 0 again because .

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