Fidelity threshold for long-range entanglement in quantum networks
A strategy to generate long-range entanglement in noisy quantum networks is presented. We consider a cubic lattice whose bonds are partially entangled mixed states of two qubits, and where quantum operations can be applied perfectly at the nodes. In contrast to protocols designed for one- or two-dimensional regular lattices, we find that entanglement can be created between arbitrarily distant qubits if the fidelity of the bonds is higher than a critical value, independent of the system size. Therefore, we show that a constant overhead of local resources, together with connections of finite fidelity, is sufficient to achieve long-distance quantum communication in noisy networks.
Quantum networks play a major role in quantum information processing Kimble (2008), as in distributed quantum computation or in quantum communication Cirac et al. (1999); Bennett et al. (1993). In fact they naturally describe the situation where neighboring stations (nodes) share partially entangled states of qubits (noisy links). One of the main tasks of quantum information processing is then to design protocols that establish entanglement between any pair of nodes, regardless of their distance in the network.
Quantum repeaters offer a first solution to this question: one can efficiently entangle the two extremities of a one-dimensional lattice of size by iterating purification steps and entanglement swappings Dür et al. (1999); Duan et al. (2001). This strategy needs qubits at each node and runs in a time that scales as . Though being very promising, their realization raises some technical problems, such as the need for reliable quantum memories Hartmann et al. (2007), or the difficulty in manipulating many qubits per station. The latter difficulty is surmounted in Childress et al. (2005), where only a constant number of qubits is required at each station. Various protocols improving the rate of long-distance quantum communication have been proposed over the past few years (see Sangouard et al. () and references therein), but either their time scaling remains polynomial in or they are based on rather complicated quantum error correcting codes Jiang et al. (2009).
Motivated by the discovery of powerful protocols in the case of two-dimensional pure-state networks Acín et al. (2007), another scheme for entanglement generation over long distance in noisy networks was presented in Perseguers et al. (2008). It exploits the higher connectivity of the nodes to gain information on the errors introduced by the noisy teleportations. This leads to a “one-shot” protocol where elementary entangled pairs are used only once, which thus relaxes the requirement of efficient quantum memories; see Fowler et al. () for the latest quantum communication protocol in square lattices. However, the overhead of local resources in these two-dimensional systems still slightly increases with (logarithmic dependence).
In this work, we show that entanglement generation over arbitrarily long distance and using the minimum amount of resources (constant number of qubits per node and quantum operations executed in a constant time) can be achieved in three-dimensional lattices. For this result to hold, the fidelity of the elementary links has to be larger than a threshold . We first provide an analytical upper bound on this value and then present a numerical estimate based on Monte Carlo simulations.
Ii Description of the model
We consider a cubic network that consists of vertices, each of them possessing six qubits (except the ones lying on the sides of the cube), on which arbitrary quantum operations can be applied perfectly. Nearest neighbors share one partially entangled state of the form
Such a state can be realized as follows: a station prepares locally
a maximally entangled pair of qubits, and sends one of them to a neighbor.
In the quantum channel, the traveling qubit undergoes random and independent
bit-flip and phase errors with probability . This channel describes a specific
physical process, but the generality of is in reality complete.
In fact, we show in App. A that any entangled state of two qubits
can be brought to this form by local quantum operations and classical communication.
Finally, all classical processes (communication and computation) are assumed to
take much less time than any quantum operation.
Physical implementations of three-dimensional lattices have been proposed in the context of quantum information processing and distributed quantum computation Brennen et al. (1999); Ionicioiu and Munro (). For practical reasons, however, it may be advantageous to realize the proposed construction in two dimensions, using a “slice-by-slice” generation similar to the techniques developed in Raussendorf and Harrington (2007). In that case note that the time required to run the protocol scales linearly with .
Recently, ideas of percolation theory have been applied successfully to the case of mixed states of rank two Broadfoot et al. (2009). In addition to the fact that the techniques are very different, our study is not restricted to amplitude damping channels, but considers full-rank mixed states which are robust against any small perturbations. In fact our protocol still works if dependent bit-flip and phase errors are present in the connections.
Iii A mapping to noisy cluster states
It was shown in Perseguers et al. (2008) how to create and propagate a large Greenberger-Horne-Zeilinger (GHZ) state in a noisy square lattice. This state is robust against bit flips if their rate is not too high but is very fragile against phase errors. Any of them indeed destroys the coherence of the GHZ state. Therefore, an encoding of the qubits is required, which leads to a logarithmic scaling of the physical resources per node. Since we are looking for a fidelity threshold, we want to create a large state that has the ability to correct both bit-flip and phase errors. Cluster states thus arise as a natural choice. In fact they have been shown to possess an intrinsic capability of error correction, so that long-range entanglement between two faces of an infinite noisy cubic cluster state is indeed possible Raussendorf et al. (2005). Our protocol is based on this construction, with two radical differences, however: first, the settings are distinct, and second, we allow only local quantum operations on all the nodes.
A cluster state, which is an instance of graph states, can be constructed by inserting a qubit in the state at each vertex of the graph and by applying a control phase between all neighboring pairs Hein et al. (2006). In our setting we cannot perform these control phases since they are non-local quantum operations, but we can add an ancillary qubit and perform joint measurements at each node such that the resulting state is a cluster state. This method has been described in Verstraete and Cirac (2004) in the case of perfect links, which can be interpreted as the virtual components of a large valence-bond state, and has been generalized to imperfect connections in Raussendorf et al. (2005). Nevertheless, let us describe here an explicit (and slightly different) construction, mainly for completeness sake but also for relating precisely the error rate in the quantum networks with the one in the noisy cluster state.
At each node, we add a qubit and use the noisy links to indirectly perform the control phases. Let us first describe how this is achieved if all connections are perfect, i.e. their qubits are in the state . We consider two nodes of the lattice, with two qubits and in the states and , and a connection between two qubits and , see Fig. 1. We start by applying, on the qubits of the first node, the measurement operators
with , which are followed by a bit flip on if the outcome is . The resulting state on and reads . We then apply the second measurement
followed by the matrix on if we get as outcome. Finally, and are left in the (entangled) state
which is the result of a control phase between and . Clearly, if , the state is the cluster state on two qubits. Now, let us determine which errors occur if we blindly perform the very same operations but using another Bell state. It is straightforward to compute the results of these operations if one uses , , or for the connections: one gets , , or , respectively. Since the matrices commute with the control phases, it follows that errors do not propagate while constructing a (noisy) cluster state from the cubic quantum network. Moreover, because of the specific choice of coefficients in Eq. (1), errors appear independently at the nodes. Since a node of the lattice has degree six at most, and two errors cancel each other, the vertices of the resulting cluster state suffer an error with a probability at most equal to
This expression reduces to in the regime of small error rates.
Therefore, we are exactly in the setting of Raussendorf et al. (2005), where thermal
fluctuations in the cluster state induce independent local errors with rate .
Iv Long-range entanglement in noisy cluster states
In this section, we mainly follow the construction and the notation proposed in Raussendorf et al. (2005), namely, the measurement of the qubits of according to a specific pattern of local bases. The outcomes of the measurements are random, but the choice of the bases establishes some parity constraints on them. Any violation of these constraints indicates an error, and a classical processing of all collected “syndromes” allows one to reliably identify the typical errors. This correction works perfectly for small error rates, but it breaks down at Ohno et al. (2004). The difference between the present method and that given in Raussendorf et al. (2005) is that no non-local quantum operation is allowed. This obliges us to design a more elaborated error correction, leading to a different type of long-distance entanglement. In fact we are not going to create a pure and perfect Bell pair of logical qubits, but rather a mixture of two entangled physical qubits.
iv.1 Measurement pattern and long-distance quantum correlations
Let us define a finite three-dimensional cluster state on the cube
and select two qubits and centered in two opposite faces and . The coordinates of these qubits are and , with . For a reason that will soon become clear, we consider lattices of size (mod 4), so that is odd and even. Let us also introduce two disjoint sublattices and with double spacing, where and stand for odd and even. Their vertices are
and their edges are given by the sets
We also define the planes
and denote by and the planes that contain and . These planes will be used to derive the Bell correlations of the future long-distance entangled state (we first consider that no error occurs, and then extend the results to noisy cluster states). Qubits that belong to the vertices of and are measured in the basis, while all other qubits are measured in the basis. There are, however, some exceptions in and (see Fig. 2): First, the central qubit is not measured, since it will be part of the long-distance entangled state. Second, qubits with coordinates are measured in the basis in order to create the right quantum correlations, as explained in the following paragraph. Finally, we measure in the basis all qubits whose first two coordinates are or and which lie in the shaded areas; these outcomes will be important for the error correction.
To compute the effect of the measurements on the quantum correlations between and , we use the fact that a perfect cluster state obeys the eigenvalue equation for all , where is the stabilizer
with the neighborhood of . If we let the products of stabilizers and act on the cluster state, we find that and are indeed maximally entangled:
with . The eigenvalues are calculated from the measurement outcomes and :
where , , , and .
iv.2 Error correction
As already mentioned, measurement outcomes are random but not independent. It is thus possible to assign to most vertices , with or , the parity syndrome
where designates the neighborhood of in .
Since this equation arises from a product of stabilizers, ,
we have that if no error occurs on the qubits of . The key point
of the construction is that a error on any edge of changes the sign
of the two syndromes at its extremities. This is due to the fact that errors do not
commute with measurements, while outcomes are not affected by
them. The sublattices are treated separately, but in a similar way. We
refer the reader to Raussendorf et al. (2005); Dennis et al. (2002) for a detailed discussion of the error
recovery or to App. B for the basics to understand our protocol.
In contrast with Raussendorf et al. (2005), and apart from the rough faces present in any surface code,
we also suffer a lack of syndrome information in and .
We cannot have a perfect and complete syndrome pattern for both and
in these faces; for this to happen one should be able to measure both and
eigenvalues of the concerned qubits, which is impossible, or apply non-local quantum
operations, which we do not allow. Actually, useful long-distance quantum correlations can
still be created if one performs the measurements depicted in Fig. 2b:
half outcomes are used to gain information on , and symmetrically for ,
see Figs. 3 and 4.
As an example of the effect of the unknown syndromes in , let us consider that an error occurred on the center qubit , and that all other qubits did not suffer any error. Since we do not know the syndromes of that lie directly below and above , we are not able to restore the correlation. This occurs with probability . From this fact, one finds that the final state on and is a mixed state of the form
with and . This state is known to be distillable, and thus useful from a quantum information perspective, whenever its fidelity is larger that one-half Deutsch et al. (1996). This can be achieved when the error rate is smaller than a threshold . In the next paragraphs, we first prove a lower bound on this value, , and then present numerical results, showing that the real threshold is indeed much larger: .
iv.2.1 Correlation loss due to the missing syndromes in
Paths of errors, which we generically denote by , have a non-trivial effect on the correlation if they cross the plane an odd number of times, as depicted in Fig. 4. Moreover, the number of errors which actually occur on a path is at least , where denotes its length. This is the case because our error correction always leads to a minimum pairing of the syndromes . We now follow Chap. V in Dennis et al. (2002) to find an upper bound on the probability of inferring the wrong quantum correlation:
where and stand for the bottom and top faces. Note that we already took into account the symmetries of the problem in this expression. For convenience, let us now introduce a new coordinate system for the vertices of , such that , , and , with , see also Fig. 3. In this coordinate system, paths of errors travel a distance and can start from missing syndromes in (lower triangle in ). Because for each vertex there are, in a cubic lattice, at most self-avoiding walks pointing upward, we find that the last term of Eq. (10) is upper bounded by
where denotes the smallest integer not less than . The sum over , together with the binomial coefficients, counts all possible paths of errors that appear in a given walk. One can check that the bound tends to 0 in the limit if , i.e. if . The same result holds for the paths and ; note that this value is about three times smaller than the real critical point . Similar considerations for the paths finally yield, for ,
This bound never tends to zero, but still converges if is small enough. Before computing a threshold for , however, we first have to consider the errors made in the other sublattice.
iv.2.2 Loss of correlation in and fidelity of the final state
The situation for the correlation is very similar to the previous case, since the measurement pattern is symmetric. Nonetheless there is a small difference: the missing syndromes do not lie on the vertices of the sublattice, but rather on its outer edges and . This creates additional parts of rough faces in . It follows that the corresponding paths of errors have their first and last edges pointing in the direction, so that a slightly better bound for the error can be derived:
Combining the two bounds on and we find
which corresponds, via Eq. (4), to an error rate in the initial connections. This value is quite small, mainly because our counting of paths of errors is very crude. Note that there are only few such paths of small length, and therefore this analytical bound could be increased by carefully computing its smallest orders in . At this point, however, we prefer to turn to Monte Carlo simulations to find a much better estimate of the error threshold. We refer the reader to App. C for a description of the algorithm; in particular we propose an intuitive and efficient method, even if not optimal, to infer the value of the missing syndromes. The result of these simulations is plotted in Fig. 5: long-distance entanglement is achieved for error rates smaller than
i.e. for the original lattice. Before concluding, let us comment on these thresholds:
The values of the unknown syndromes are not optimally inferred in our algorithm, and therefore a higher value of may be found. However, it is clear that it cannot exceed the critical error rate .
One could get a higher threshold by directly computing as a function of . In fact, errors in the faces and do not appear with probability , but only with probability .
Finally, as suggested in (Raussendorf et al., 2005, Rem. 2), lattices of size may also be appropriate for generating long-distance entanglement. This result also holds in our setting, because additional errors only appear in the faces and and not in the bulk of the lattice.
We have investigated the problem of generating long-distance entanglement in
noisy quantum networks. We have focused on three-dimensional regular lattices,
whose edges are full-rank mixed states of two qubits. We have proven that
entanglement can be established between two infinitely distant qubits
if the fidelity of the connections is large enough. Our protocol starts by
transforming the quantum network into a thermal cluster state. Then, all but two distant qubits
are measured according to a specific pattern of local bases, and a syndrome-based
error correction is performed. The error recovery is very similar to the one
used for planar codes, with the difference being that our setting does not allow one to
get complete information on the syndromes. Nevertheless, useful quantum correlations can
be created between the two unmeasured qubits if the error rate is
smaller than a critical value. We have given both an analytical lower bound on this value
and a numerical estimation (about 2%) based on Monte Carlo simulations.
In conclusion, we have shown that a constant overhead of local resources is sufficient to achieve long-distance communication in quantum networks. This contrasts with previous one- or two-dimensional strategies, in which the physical resources per station increase with the distance. Our protocol requires perfect local quantum operations, which is somehow justified by the fact that most errors occur while sending quantum information between stations. It would nevertheless be of fundamental interest to design fault-tolerant protocols, in two or three dimensions, for which the overhead of local resources is as small as possible.
Acknowledgements.The author thanks Ignacio Cirac and Antonio Acín for initiating the project and for useful discussions. This work has been supported by the QCCC program of the Elite Network of Bavaria.
Appendix A Elementary entangled pairs of qubits
We show in this appendix that there is no loss of generality in choosing the elementary links to be described by Eq. (1). First, it is well known that any two-qubit entangled state can be brought to the rotationally symmetric mixture
In fact, this is achieved by applying random bilateral rotations, locally, to each qubit. This state is called a Werner state Werner (1989) and has the same fidelity as the initial state from which it derives. Let us denote by its components in the Bell basis, , and define the two unitaries
One can check that the separable operation applied on a Bell-diagonal state
switches its coefficients and , while the coefficients
and are unaltered. A similar result holds for
, which only switches the components and . Suppose now
that an entangled pair , with , has been created
between two neighboring nodes. This already sets the coefficient to
the desired value . Then, apply with probability
, and with probability on . This
leads to the state . Finally,
repeat the operation by applying with probability one-half.
Both coefficients and are set to , which
proves that the state given in Eq. (1) is indeed general.
Appendix B Basics of syndrome-based error correction
Let us consider the sublattices and described in the text, in which errors occur independently on each edge with probability , and assign to each vertex the syndrome if it is connected to an even number of erroneous edges, and otherwise. In the case of perfect and complete syndrome information, one knows exactly where all paths of errors start and end: this occurs at syndromes . In the regime of small error rate , it turns out that the best error recovery strategy is to pair these syndromes such that the total length of all pairings is minimized. Then, one connects any two paired syndromes by a path of minimum length, and artificially introduces “errors” along these paths. This creates loops of errors in the cluster state, which, however, do not cause any damage to the long-distance quantum correlations. In fact these loops either do not intersect the planes and or cross them twice, and consequently do not modify the eigenvalues in Eq. (6).
Problems arise because some syndromes are unknown. For instance, consider the edges that have
only one extremity in or : their coordinates are and
in , and and in , see Fig. 3.
These are the rough faces described in Raussendorf et al. (2005), and errors on these edges change the sign of
only one syndrome (and not two) in the corresponding sublattice. An equivalent viewpoint
is that both extremities of these edges indeed belong to or , but we
do not have access to their outer syndrome. The consequence of this lack of information
is that some paths of errors are not closed anymore, but rather originate from a
missing syndrome and terminate at another. Typically, these open paths enter only
superficially the lattice if the error rate is small, but they start stretching
from one side to another as soon as exceeds the value .
In the latter case, paths of errors can cross an odd number of times the planes
of correlations, which results in a complete loss of long-distance entanglement
in the limit .
Appendix C Monte Carlo simulations
We now describe the algorithm used for computing the data of Fig. 5. The correction procedure is very similar for the two sublattices and , and consists of two main parts. First, given a lattice with random errors, we infer the value of the syndromes for which we have no information. Second, we proceed with the usual error recovery. The program outputs 1 if the correction is successful, and 0 otherwise.
c.0.1 Inferring the missing syndromes
We propose a very simple way of assigning the value or to the missing syndromes, so that a good approximation of the optimal configuration is found. To that end, it is helpful to consider a typical realization of a noisy cluster state in the regime of small error rates, as depicted in Fig. 6. Our algorithm reads
Initialize all unknown syndromes to .
Using nearest-neighbor site percolation, find all clusters of syndromes . Keep only the clusters of odd size, and for each compute the minimum distance to a closest unknown syndrome . 111Several such syndromes may exist; choose the one that lie in the plane parallel to or that contains . This avoids unnecessary crossings of the plane of correlation. Let denote the number of such clusters. Note that we do not consider clusters of even size, since good pairings can be found for them, individually.
Calculate the distance between all pairs of clusters and . This distance could be the length of the shortest path from to , but in practice it is much easier to calculate the distance between their “centers of mass.”
Find and such that , and create a pair . Remove and from the list of clusters, and repeat the procedure until no cluster is left. In case of odd , add an extra “pair” for the remain cluster, with . This creates the list .
For each , check if . If this inequality holds, inverse the value of the corresponding missing syndromes: and .
The proposed algorithm is optimal in the regime of very dilute errors, but this
is not true for high error rates anymore (even if results are good for
all ). Note that there exist optimal algorithms which are based on minimal perfect
matchings in weighted graphs and are run in a polynomial time (see Wang et al. (), Chap. 4).
For three-dimensional lattices, however, the number of edges in these
graphs scales as [they are nearly complete graphs on vertices],
and therefore these algorithms are not so efficient in practice.
Nevertheless, it would be very interesting to implement an optimal algorithm
and decide whether the unknown syndromes are responsible for the threshold,
or whether the equality holds.
c.0.2 Error recovery
We use the well-known and efficient algorithm described by J. Edmonds in Edmonds (1965)
to find an optimal pairing of the syndromes . The error correction is
successful if the parity of paths of errors crossing the plane of correlation
is even. Simulations of the error corrections have been performed for various
lattice sizes (up to nodes), and for both and correlations.
The extrapolation to infinite lattices is done by fitting the data with an
exponential function, see Fig. 7. Results are
plotted in Fig. 5.
Finally, let us present evidence that our algorithm gives correct and optimal results in the regime of small error rates. Considering the series expansions of at first order in , one sees that only three edges of may degrade the correlation: these are the bonds in that cross and whose first coordinate belongs to , see Fig. 3a. At second order, one can check that the probability to infer the wrong correlation due to the missing syndromes in is . Therefore, by symmetry, the fidelity reads
It is easy to see that a single error in cannot damage the correlation, and a careful counting of configurations with two errors yields . Consequently we find
These two series expansions are plotted (dashed lines) in Fig. 5: they
agree perfectly with the results of the Monte Carlo simulations and thus
validate our algorithm.
- Kimble (2008) H. J. Kimble, Nature 453, 1023 (2008).
- Cirac et al. (1999) J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, Phys. Rev. A 59, 4249 (1999).
- Bennett et al. (1993) C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
- Dür et al. (1999) W. Dür, H.-J. Briegel, J. I. Cirac, and P. Zoller, Phys. Rev. A 59, 169 (1999).
- Duan et al. (2001) L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature 414, 413 (2001).
- Hartmann et al. (2007) L. Hartmann, B. Kraus, H.-J. Briegel, and W. Dür, Phys. Rev. A 75, 032310 (2007).
- Childress et al. (2005) L. I. Childress, J. M. Taylor, A. S. Sørensen, and M. D. Lukin, Phys. Rev. A 72, 052330 (2005).
- (8) N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, eprint arXiv:quant-ph/0906.2699.
- Jiang et al. (2009) L. Jiang, J. M. Taylor, K. Nemoto, W. J. Munro, R. V. Meter, and M. D. Lukin, Phys. Rev. A 79, 032325 (2009).
- Acín et al. (2007) A. Acín, J. I. Cirac, and M. Lewenstein, Nature Phys. 3, 256 (2007).
- Perseguers et al. (2008) S. Perseguers, L. Jiang, N. Schuch, F. Verstraete, M. Lukin, J. Cirac, and K. Vollbrecht, Phys. Rev. A 78, 062324 (2008).
- (12) A. G. Fowler, D. S. Wang, T. D. Ladd, R. V. Meter, and L. C. L. Hollenberg, eprint arXiv:quant-ph/0910.4074.
- Brennen et al. (1999) G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, Phys. Rev. Lett. 82, 1060 (1999).
- (14) R. Ionicioiu and W. J. Munro, eprint arXiv:quant-ph/0906.1727.
- Raussendorf and Harrington (2007) R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, 190504 (2007).
- Broadfoot et al. (2009) S. Broadfoot, U. Dorner, and D. Jaksch, EuroPhys. Lett. 88, 50002 (2009).
- Raussendorf et al. (2005) R. Raussendorf, S. Bravyi, and J. Harrington, Phys. Rev. A 71, 062313 (2005).
- Hein et al. (2006) M. Hein, W. Dür, R. Raussendorf, M. V. den Nest, and H.-J. Briegel, in Quantum Computer, Algorithms and Chaos (IOS, Amsterdam, 2006), vol. 162 of International School of Physics Enrico Fermi, edited by G. Casati, D. L. Shepelyansky, P. Zoller, and G. Benenti.
- Verstraete and Cirac (2004) F. Verstraete and J. I. Cirac, Phys. Rev. A 70, 060302(R) (2004).
- Ohno et al. (2004) T. Ohno, G. Arakawa, I. Ichinose, and T. Matsui, Nucl. Phys. B697, 462 (2004).
- Dennis et al. (2002) E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys. 43, 4452 (2002).
- Deutsch et al. (1996) D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818 (1996).
- Werner (1989) R. F. Werner, Phys. Rev. A 40, 4277 (1989).
- (24) D. S. Wang, A. G. Fowler, A. M. Stephens, and L. C. L. Hollenberg, eprint arXiv:quant-ph/0905.0531.
- Edmonds (1965) J. Edmonds, Canad. J. Math. 17, 449 (1965).