PMCCPMCCPMCC

Search tips
Search criteria 

Advanced

 
Logo of scirepAboutEditorial BoardFor AuthorsScientific Reports
 
Sci Rep. 2016; 6: 27637.
Published online 2016 June 17. doi:  10.1038/srep27637
PMCID: PMC4911610

High-dimensional entanglement certification

Abstract

Quantum entanglement is the ability of joint quantum systems to possess global properties (correlation among systems) even when subsystems have no definite individual property. Whilst the 2-dimensional (qubit) case is well-understood, currently, tools to characterise entanglement in high dimensions are limited. We experimentally demonstrate a new procedure for entanglement certification that is suitable for large systems, based entirely on information-theoretics. It scales more efficiently than Bell’s inequality and entanglement witness. The method we developed works for arbitrarily large system dimension d and employs only two local measurements of complementary properties. This procedure can also certify whether the system is maximally entangled. We illustrate the protocol for families of bipartite states of qudits with dimension up to 32 composed of polarisation-entangled photon pairs.

As the dimension of investigated systems increases, it becomes more complicated to demonstrate their quantum effects1,2,3. Indeed, a full characterization (quantum tomography) becomes practically impossible already for systems of rather small dimensionality4,5. It is therefore important to explore new avenues to prove the presence of such effects, e.g. entanglement, for arbitrary dimensions. High-dimensional entangled states offer a larger code space, attracting interests for quantum key distribution6, teleportation7 and security-enhanced quantum cryptography8,9.

In this context, proving that one has achieved entanglement (entanglement certification) and detecting entanglement are different primitives. Indeed, entanglement detection methods10,11,12,13,14 must be as sensitive as possible and must be able to detect the largest possible class of entangled states. Often such methods are inapplicable to large system dimensions or scale poorly with increasing dimension as they entail increasingly complicated measurements and data analysis. Entanglement detection such as witness operators, for instance, typically requires a number of local measurement settings that scales linearly in d15. In contrast, entanglement certification refers to the fact that one has simply to prove that the system is entangled. To do entanglement certification, one can optimize the method to the specific entangled state that one is producing. It must fulfill different requirements: it must be as robust and simple as possible. In other words a good method for entanglement detection can also work for entanglement certification, but not vice versa.

Here, we present an entanglement certification procedure that is extremely simple to implement (the measurement of two local observables16,17 suffices), is compatible with current state-of-the-art experimental techniques, and can be easily scaled up to arbitrary dimension. In addition to certifying the production of entangled states, our procedure can also certify the production of maximal entanglement. To prove its simplicity, we present an experimental test that uses entangled systems with dimension up to d = 32, constructed by suitably grouping couples of entangled photon pairs. In the general case, for a two-qudit experiment with arbitrary d, two measures would still be sufficient to implement our method, but not sufficient for tomography or other entanglement detection methods. To certify the presence of entanglement, one has only to calculate the (classical) correlations among the measurement outcomes of the two observables, for example, through their mutual information. If such correlations are larger than some threshold, the state is guaranteed to be entangled (Fig. 1). If they attain their maximum value, the state must be maximally entangled.

Figure 1
Illustration of the procedure.

If A and C are complementary properties, the knowledge of A gives no information on C and vice versa. This happens whenever |left angle bracketa|cright angle bracket|2 = 1/d, for all |aright angle bracket and |cright angle bracket eigenstates of the observables A and C, where d is the Hilbert space dimension. This is equivalent to two sets of mutually unbiased bases (MUB’s). Consider the two-qubit maximally entangled state |Φ+right angle bracket [proportional, variant] |00right angle bracket + |11right angle bracket; it clearly has maximal correlation among results for the observables with eigenstates |0right angle bracket, |1right angle bracket: both qubits have the same value. This state has also maximal correlation among results for a complementary observable with eigenvalues An external file that holds a picture, illustration, etc.
Object name is srep27637-m1.jpg, since it can be written as |Φ+right angle bracket [proportional, variant] |+ +right angle bracket + |− right angle bracket. Thus, the mutual information between measurement outcomes on the two qubits is one bit per observable, summing to two bits. If one has only separable states, the sum cannot be larger than one: for example, the classically correlated state ρ [proportional, variant] |00right angle bracket left angle bracket00| + |11right angle bracket left angle bracket11| has one bit of mutual information for the outcomes of the first observable, but zero bits for the outcomes of the second.

Starting from the theoretical suggestions of18, we extend the method from the qubit (d = 2) case to arbitrary high dimensions. The proposed mechanism to certify entanglement thus uses the following procedure:

  1. Identify two bipartite complementary observables AB and CD, where A, C are system 1 observables and B, D are system 2 observables (Fig. 1a).
  2. Measure the statistics of the outcomes of the two observables: local measurements on the two systems suffice.
  3. These measurements return the joint probabilities p(ao, bo) of obtaining outcome ao on system 1 for A and bo on system 2 for B, and p(co, do) of obtaining outcome co on system 1 and do on system 2. Use these results to calculate the mutual information IAB among measurement results for AB and ICD among results for CD, with
    An external file that holds a picture, illustration, etc.
Object name is srep27637-m29.jpg
  4. Certification: if IAB + ICD > log2 d, the two systems are entangled; if IAB + ICD = 2 log2 d, the two systems are maximally entangled.

The proof of these statements, based on Maassen and Uffink’s entropic uncertainty relation19 is given in18 (see also20,21,22).

In regards to the choice of observables, for pure states, the obvious choice would be the Schmidt bases and their respective Fourier bases. Whilst for mixed states, one can diagonalise the density matrix, identify the eigenvector with the largest weight and use the Schmidt basis (and its respective Fourier bases). Alternatively, one can choose the bases that diagonalises the reduced density matrices. While there is no guarantee that these choices will allow one to implement the procedure, they are the ones that may uncover the most correlations.

This method is simple to implement for systems of any dimensionality: it only entails independent measurements of two local observables on the two systems. Moreover, it is robust since, although one can optimize the choice of observables to maximize the sum of mutual information, the systems are guaranteed to be entangled if the above conditions are satisfied for any couple of complementary observables. It is interesting to note that, coherently performing sequential complementary measurements on the same system may generate the entanglement itself23.

To date, the most prominent way of producing higher dimensional entangled systems is via the orbital angular momentum degree of freedom of a photon6,24,25; schemes to produce three-level entangled states in trapped ions have also been proposed26,27. Our method, however, works for any d-dimensional system as long as the appropriate measurements can be performed.

Based on the theoretical work by Collins et al., other recent experiments25,28 have studied higher dimensional entangled system via generalised Bell’s inequalities, where the correlations between the two measurement settings on each qudit system are studied. The violation of Bell’s inequality is usually discussed in terms of quantum non-locality. In this context, the framework of hidden variable theories is in general specified by the 4 measurement settings, each with d outcomes, therefore needing d4 joint probabilites29 to describe the system globally. In this case25,28,29, examine two settings on each system, and how the observables on one system are correlated with the two observables on the second system, requiring a total of 2d × 2d joint outcomes. On the other hand, our method is based on entropic relations, requiring only two measurement settings, and is completely specified by 2d2 probabilities. Similarly to28, we used ensembles of individual entangled photon pairs to construct a higher dimensional state. All the subsystems that compose this state do not exist at the same time, but that is irrelevant to our current aims.

To demonstrate the method in practice, we performed an experiment using high-dimensional entangled systems. We certify the presence of entanglement for various families of states, by measuring pairs of complementary observables. These families are obtained by appropriately grouping couples of polarisation entangled photonic qubits, mapping qudits onto qubits:

An external file that holds a picture, illustration, etc.
Object name is srep27637-m3.jpg

where this mapping is obtained by expressing j in binary notation and rearranging the qubits in such a way that the ones placed in odd positions in the tensor products on the right hand side are assigned to system 1 while the ones in even positions to system 2. For example, a d = 4 maximally entangled state can be expressed as (|00right angle bracket + |11right angle bracket + |22right angle bracket + |33right angle bracket)/2 [equivalent] (|00 00right angle bracket + |01 01right angle bracket + |10 10right angle bracket + |11 11right angle bracket)/2 [equivalent] (|0right angle bracketα|0right angle bracketβ + |1right angle bracketα|1right angle bracketβ) [multiply sign in circle] (|0right angle bracketγ|0right angle bracketδ + |1right angle bracketγ|1right angle bracketδ)/2, where qubits labeled α and γ belong to system 1, while β and δ to system 2. It would also be extremely interesting to show that one could coherently manipulate the n entangled pairs locally, though that is not necessary to implement our method (which is also one of the main strengths of the method). An analogous mapping applies to the mixed states:

An external file that holds a picture, illustration, etc.
Object name is srep27637-m4.jpg

Regarding the necessary measurements, the observable AB corresponds to σz, namely the computational basis {|0right angle bracket, |1right angle bracket} for each qubit. As to the observable CD, we will consider two possibilities: the Fourier basis and the σx basis. The Fourier basis is defined as

An external file that holds a picture, illustration, etc.
Object name is srep27637-m5.jpg

with ω [equivalent] exp(2πi/d). It can be expressed as tensor products of single-qubit states (see Supplementary Material).

The Fourier basis in arbitrary dimension identifies an observable complementary to the computational basis. However, in the case considered here, there are complementary bases that are simpler to access experimentally, namely the bases where one measures σx or σy on each qubit. We consider σx here. The σx basis is given by |ckright angle bracket An external file that holds a picture, illustration, etc.
Object name is srep27637-m6.jpg obtained by expressing the binary digits of k in the {+, −} basis, i.e.

An external file that holds a picture, illustration, etc.
Object name is srep27637-m7.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep27637-m8.jpg are the bits of the number k. The σx basis is complementary to the computational basis, since |left angle bracketj|ckright angle bracket|2 = 1/2n = 1/d for all j, k.

We tested our entanglement certification procedure on several families of states. These particular states that we consider are arbitrarily chosen for their simplicity. For the specific examples, the entanglement of the d-dimensional entangled pair is exactly the same as the one obtained from consecutive entangled two-photon states. This happens only when d is a power of two, though the method works for any d. The first is

An external file that holds a picture, illustration, etc.
Object name is srep27637-m9.jpg

which mixes the maximally entangled state of equation (2) with the state in equation (3) with classical correlation only on the computational basis, the above state is always entangled for p  0.

The experimental test of this prediction is presented in Fig. 2, red circles (the markers are the data points and the connecting lines are the expected curves fitted to density matrices from quantum state tomography).

Figure 2
Experimental test of the method for d-dimensional systems.

The second family of states we consider are the Werner states: a mixture of a maximally entangled and a maximally mixed state,

An external file that holds a picture, illustration, etc.
Object name is srep27637-m10.jpg

where I is the identity d2 × d2 matrix. These states are entangled for p > 1/(d + 1) (e.g.30). Their mutual information is experimentally determined as the green triangles in Fig. 2.

The third family of states is

An external file that holds a picture, illustration, etc.
Object name is srep27637-m11.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep27637-m12.jpg. The experimental results are plotted in Fig. 2 as blue stars (when the CD observable is the σx basis |ckright angle bracket) and as purple diamonds (when CD is the Fourier basis |fjright angle bracket). Measurements in the Fourier basis and in the σx basis are performed for d = 4 and 8 only, due to the large number of projections required for higher dimensions where temporal phase instability would significantly affect the measurements. This state shows the difference between the complementary bases: its form implies that correlations are greater in the σx than the Fourier basis.

The entanglement certification method can be modified by using a different measure of correlations. Indeed, instead of the mutual information, one can also measure correlations with the Pearson coefficient

An external file that holds a picture, illustration, etc.
Object name is srep27637-m13.jpg

where left angle bracketXright angle bracket is the expectation value of X and An external file that holds a picture, illustration, etc.
Object name is srep27637-m14.jpg its variance. It measures the linear correlation between the outcomes of two observables A and B, and takes values An external file that holds a picture, illustration, etc.
Object name is srep27637-m15.jpg. It is conjectured18 that if CAB + CCD > 1, then the state is entangled. Moreover, it is known that if CAB + CCD = 2, the state is maximally entangled. This may be helpful in high-noise scenarios, since it allows for the certification of a larger class of entangled states. The Pearson correlation coefficient seems to be more robust against imbalanced probabilities or decoherence in the experiment (see Fig. 3), and numerical simulations suggest that it is more effective in detecting entanglement18. So, in addition to the sum of mutual information IAB + ICD, we can use CAB + CCD as an alternative way of certifying entanglement (modulo a conjecture).

Figure 3
Experimental comparison between mutual information (above) and Pearson coefficient (below).

We consider the two-qubit states

An external file that holds a picture, illustration, etc.
Object name is srep27637-m16.jpg
An external file that holds a picture, illustration, etc.
Object name is srep27637-m17.jpg

and the Werner state ρw of Eq. (7) with d = 2. In (10), An external file that holds a picture, illustration, etc.
Object name is srep27637-m18.jpg. The state ρb is entangled for p  1/2, whereas ρs is always separable and has zero discord only for p = 0, 1. For two-qubit states, the σx basis |ckright angle bracket coincides with the Fourier basis |fjright angle bracket, so the two possible observables CD we used above coincide here.

However, we preferred working with mutual information since the fact that the Pearson coefficient can be used to certify entanglement is still a conjecture, whereas it has been rigorously proved for the sum of mutual information.

Whilst it is true that the standard entanglement witness

An external file that holds a picture, illustration, etc.
Object name is srep27637-m19.jpg

out-performs our method for entanglement dection for the (d = 2 case) Werner state (detects entanglement at p  0.5), this is not true for arbitrary dimensions in general.

Finally, also comparing this against Bell’s inequalities using the Werner state, our method using mutual information with two complementary observables (computational and σx basis), for d = 2, the threshold of log2(d) is surpassed at p  0.78. However, if we increase this to three observables (indeed one can, and in this case we consider the σy basis), the entanglement is detected at p  0.65 (see supplementary material of18), which outperforms Bell’s inequalities at An external file that holds a picture, illustration, etc.
Object name is srep27637-m20.jpg11. Also, in the limit of large d, our method (see Eq. (S7) in the Supplementary Material) also out-performs the threshold in Bell’s inequality violation, given in29.

In summary, we have presented an entanglement certification method that is suitable for large dimensional systems. To our knowledge, there are no such methods that are as efficient to implement on a 32-dimensional system such as the one we experimentally study here. We have shown that entangled states are more correlated on the outcomes of complementary observables. One may think that this property is shared by other types of quantum correlations such as quantum discord31 but, surprisingly, this is not the case, at least when the correlations are gauged through mutual information. Indeed, the separable state ρs has highest IAB + ICD for p = 0, 1 (Fig. 3), the values for which its discord is null. In contrast, IAB + ICD is lowest for p = 1/2 which is where such state has highest discord. This unexpected property is lost when the correlation is gauged with the Pearson coefficient, as CAB + CCD = 1 for all values of p.

Methods

In order to generate polarisation-entangled photon pairs, a set up similar to32 and identical to33 is used. We use the polarisation encoding where |Hright angle bracket [equivalent] |0right angle bracket and |Vright angle bracket [equivalent] |1right angle bracket. The set up is shown in Fig. 4. A type-I nonlinear crystal (BiBO) is pumped with a vertically polarised cw laser (404 nm) at 80 mW, generating pairs of horizontally polarised single photons (808 nm). The photons are spectrally filtered using 3 nm top-hat filteres centred on 808 nm. The first half way plate HWP1 (optics axis set to π/8 radians) rotates the output photon pair from |HHright angle bracket to |+ +right angle bracket, where An external file that holds a picture, illustration, etc.
Object name is srep27637-m21.jpg. The polarisation in the top arm is matched and purified to that of the bottom arm using a polarisation beam splitter (PBS) rotated by π/4 radians, which transmits |+right angle bracket.

Figure 4
Scheme for polarisation entangled source.

The unitary operation realised by the sequence of wave plates QWP1 + HWP2(θ) + QWP2, as a function θ of the HWP2 angle, applies a phase to photon in V state, effectively that of a phase gate: An external file that holds a picture, illustration, etc.
Object name is srep27637-m22.jpg. Denoting the input photon in the top (bottom) arm with subscript 1 (2), the two-photon state collected into the polarisation mainting fibre (PMF) is therefore An external file that holds a picture, illustration, etc.
Object name is srep27637-m23.jpg.

We use a silicon avalanche single photon counter and measure in coincidence with a time window of 2.5 ns. At the fibre coupled PBS, H is transmitted and V is reflected. After the fibre PBS crystal, the top (bottom) arm fibre contains the components (V1 and  H2) (H1 and  e V2). However, the design of the fibre PBS is such that the output coupler flips the polarisation state in the bottom arm. Without this flip, the registered coincidence will be either e |Vright angle bracket1 |Vright angle bracket2 or |Hright angle bracket1 |Hright angle bracket2. After the flip the detector will register either e |Vright angle bracket1 |Hright angle bracket2 or |Hright angle bracket2 |Vright angle bracket1, and these two components are made spatially indistinguishable by adjusting the position of the fibre, giving the post-selected state An external file that holds a picture, illustration, etc.
Object name is srep27637-m24.jpg. HWP3 and HWP4 apply the appropriate bit flip operations on each single qubit. QWP3 + HWP5 and QWP4 + HWP6 along with the final two PBSs project each photon onto any polarisation state.

The polarisation maintaining fibres have a beat length (H and V delayed with respect to each other by 2π) of 24 mm; the coherence length of the photon pairs is 250 μm (extracted from a Hong-Ou-Mandel dip of 99% visibility), which means that if uncompensated, 1.0 m of fibre will spatially separate the H and V components of a photon out of coherence, turning a Bell state An external file that holds a picture, illustration, etc.
Object name is srep27637-m25.jpg into the classically correlated mixture An external file that holds a picture, illustration, etc.
Object name is srep27637-m26.jpg. When producing a Bell state, this spatial decoherence was compensated by crossing the slow axis of the PMF and tuning the stage position of the fibre (and vice versa when the mixed state is required).

Additional Information

How to cite this article: Huang, Z. et al. High-dimensional entanglement certification. Sci. Rep. 6, 27637; doi: 10.1038/srep27637 (2016).

Supplementary Material

Supplementary Information:

Acknowledgments

A.P. acknowledges an Australian Research Council Discovery Early Career Researcher Award under project number DE140101700 and an RMIT University Vice-Chancellor’s Senior Research Fellowship.

Footnotes

Author Contributions Z.H., L.M., C.M. and A.P. designed the experiment. Z.H. performed the experiment and analysed the data. All authors contributed to writing the manuscript.

References

  • Arndt M. & Hornberger K. Testing the limits of quantum mechanical superpositions. Nat. Phys. 10, 271–277 (2014).
  • Sciarrino F. Quantum optics: Micro meets macro. Nat. Phys. 9, 529–529 (2013).
  • Friedman J. R., Patel V., Chen W., Tolpygo S. & Lukens J. E. Quantum superposition of distinct macroscopic states. Nat. 406, 43–46 (2000). [PubMed]
  • Gühne O. et al. . Detection of entanglement with few local measurements. Phys. Rev. A 66, 062305 (2002).
  • Artiles L., Gill R. et al. . An invitation to quantum tomography. J. Roy. Statist. Soc. Ser. B 67, 109–134 (2005).
  • Mair A., Vaziri A., Weihs G. & Zeilinger A. Entanglement of the orbital angular momentum states of photons. Nat. 412, 313–316 (2001). [PubMed]
  • Bennett C. H. & Wiesner S. J. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881–2884 (1992). [PubMed]
  • Bruß D. & Macchiavello C. Optimal eavesdropping in cryptography with three-dimensional quantum states. Phys. Rev. Lett. 88, 127901 (2002). [PubMed]
  • Durt T., Kaszlikowski D., Chen J.-L. & Kwek L. C. Security of quantum key distributions with entangled qudits. Phys. Rev. A 69, 032313 (2004).
  • Bruß D. Characterizing entanglement. J. Math. Phys. 43, 4237–4251 (2002).
  • Horodecki R., Horodecki P., Horodecki M. & Horodecki K. Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009).
  • Gühne O. & Tóth G. Entanglement detection. Phys. Rep. 474, 1–75 (2009).
  • Rudnicki Ł., Horodecki P., and Życzkowski K. Collective Uncertainty Entanglement Test. Phys. Rev. Lett . 107, 150502 (2011). [PubMed]
  • Rudnicki Ł., Puchała Z., Horodecki P., and Życzkowski K. Collectibility for mixed quantum states. Phys. Rev. A 86, 062329 (2012).
  • Gühne O. et al. . Experimental detection of entanglement via witness operators and local measurements. J. Modern Opt. 50, 1079–1102 (2003).
  • Kothe C. & Björk G. Entanglement quantification through local observable correlations. Phys. Rev. A 75, 012336 (2007).
  • Abascal I. S. & Björk G. Bipartite entanglement measure based on covariance. Phys. Rev. A 75, 062317 (2007).
  • Maccone L., Bruß D. & Macchiavello C. Complementarity and correlations. Phys. Rev. Lett. 114, 130401 (2015). [PubMed]
  • Maassen H. & Uffink J. B. M. Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988). [PubMed]
  • Schneeloch J., Broadbent C. J. & Howell J. C. Uncertainty relation for mutual information. Phys. Rev. A 90, 062119 (2014).
  • Coles P. J. & Piani M. Improved entropic uncertainty relations and information exclusion relations. Phys. Rev. A 89, 022112 (2014)
  • Coles P. J., Colbeck R., Yu L. & Zwolak M. Uncertainty relations from simple entropic properties. Phys. Rev. Lett. 108, 210405 (2012). [PubMed]
  • Coles P. J. & Piani M. Complementary sequential measurements generate entanglement. Phys. Rev. A 89, 010302 (2014).
  • Fickler R. et al. . Quantum entanglement of high angular momenta. Sci . 338, 640–643 (2012). [PubMed]
  • Dada A. C., Leach J., Buller G. S., Padgett M. J. & Andersson E. Experimental high-dimensional two-photon entanglement and violations of generalized bell inequalities. Nat. Phys. 7, 677–680 (2011).
  • Zheng S.-B. Generation of entangled states for many multilevel atoms in a thermal cavity and ions in thermal motion. Phys. Rev. A 68, 035801 (2003).
  • Ye S.-Y., Zhong Z.-R. & Zheng S.-B. Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities. Phys. Rev. A 77, 014303 (2008).
  • Lo H.-P. et al. . Experimental violation of bell inequalities for multi-dimensional systems. arXiv preprint arXiv:1501.06429 (2015).
  • Collins D., Gisin N., Linden N., Massar S. & Popescu S. Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett. 88, 040404 (2002). [PubMed]
  • Kendon V. M., Życzkowski K. & Munro W. J. Bounds on entanglement in qudit subsystems. Phys. Rev. A 66, 062310 (2002).
  • Ollivier H. & Zurek W. H. Quantum discord: A measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001). [PubMed]
  • Matthews J. C. et al. . Observing fermionic statistics with photons in arbitrary processes. Sci. Rep. 3, 1539 (2013). [PMC free article] [PubMed]
  • Chapman R. J. et al. .Experimental perfect state transfer of an entangled photonic qubit. Nat. Commun . 7:11339 doi: 10.1038/ncomms11339 (2016). [PMC free article] [PubMed]

Articles from Scientific Reports are provided here courtesy of Nature Publishing Group