Introduction

Quantum optics is, at its core, the study of the distributions of photons in modes of the electromagnetic field. These distributions can exhibit fundamental physical features such as photon antibunching1,2,3 and photon number squeezing4,5, which cannot be explained using classical assumptions about the photon distribution. When we consider the joint distribution of photons across multiple modes, further nonclassical phenomena emerge, such as the presence of nonclassical correlations in the number of photons measured in each mode6,7,8,9,10,11,12,13,14,15. For an ideal two-mode squeezed vacuum state, the number of photons in each mode is always equal—the outcome of photon-number measurements are completely correlated16. By contrast, when two indistinguishable photons are incident on different ports of a balanced beam splitter, bosonic bunching dictates that both photons will exit the same port17, such that photon number measurements are anti-correlated. Here we introduce discorrelation, in which the joint probability Pn,n of measuring n photons in each mode is precisely zero for all n, but the marginal distributions Pn are nonzero for all n. Discorrelation is distinct from correlation, anti-correlation, and decorrelation, extending our understanding of quantum correlations. It is an infinite-dimensional version of “exclusive correlations”, here analysed as an effect of photon statistics rather than in the context of generalised Bell states18. In this work we show ways to generate discorrelated states using commonly available input states and standard quantum optical techniques, and analyse the entanglement properties and loss behaviour of these states.

Correlation has been studied extensively in quantum optics in the context of communication, namely as a means to share common randomness between two parties19,20,21,22,23,24,25. In this context, discorrelation can be used to share unique randomness between parties, complementary to conventional quantum communication protocols. Unique randomness, where each party has a random number that is distinct from the other parties’, could be useful in distributed voting schemes26 or for fairly dealing cards27 in the area of cryptographic study known as “mental poker”28. Mental poker and distributed voting involve the allotment of cards, voter identifiers, or other pieces of information fairly and secretly. By ensuring each individual receives a unique random number without knowledge of any others, discorrelation could remove the need for a trusted third party, replacing it with the fundamental randomness of quantum superpositions.

We propose two methods for generating discorrelation. The first is the displacement of a single photon by a coherent state on a beam splitter, producing an entangled state29,30,31,32. In this case, the discorrelated state is generated locally and then shared between parties. Although various aspects of these and related states have been analysed33,34,35, for example in the context of micro-macro entanglement36 and N00N-state generation37,38, the joint photon number distribution and the discorrelation therein, has not.

In the second case, we show how discorrelated multidimensional photon statistics can be generated nonlocally using a single shared two-dimensional state. This method is based on the coherent superposition of photon addition and subtraction, which has been proposed for generating nonclassical states39,40,41,42,43,44 and distilling entanglement in continuous-variable quantum states45,46. As with related photon subtraction and addition experiments47,48,49,50,51,52,53,54, and in photon-number-difference filtering55, photon detections are used to perform nonlinear operations, but in our case the input states can begin separable and become entangled by the discorrelation operation. The two methods for creating discorrelation are closely related as they both rely on the modification of photon number distributions due to the interference of Fock states with other continuous-variable states.

Results

The simplest way to create a discorrelated state (Fig. 1) is to impinge a single photon |1〉 and a coherent state |α〉 on two ports of a 50:50 beam splitter. The coherent state displaces the single photon34, resulting in the entangled two-mode state

Figure 1
figure 1

Interfering a single photon with a coherent state () on a 50:50 beam splitter (inset) produces discorrelation: the photon number at the two output ports m and n (shown as a heatmap) can take any value individually, but together the two ports can never produce m = n.

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where the coefficients are given by

and the normalization factor is . Thus the two output modes will have the same marginal photon number distribution, but since photon number measurements on the two modes will never give the result n = m, the output state is discorrelated. We show this discorrelated photon number distribution in Fig. 1 as a heatmap of the joint photon number detection probabilities.

Nonlocal, adaptable discorrelation

One can also implement the discorrelation operation in a nonlocal manner, whereby particular projective measurements can be used to herald states whose photon distributions depend on the parameters of the interaction. We employ an entangled HOM state17, created by the interference of two single photons at a 50:50 beam splitter, to distribute entanglement between two parties. As seen in Fig. 2, the HOM state interferes with two other (separable) multiphoton states, expressed in the photon number basis as and , where the coefficients , are normalized and the states share a phase reference. Following projective measurements of single photons at one output of each beam splitter (of transmissivity t, reflectivity r), the entanglement in the HOM state is mapped to the (pure) output state . This allows the two parties to share a state whose discorrelation depends on the input states and beam splitter parameters, without interacting directly. The coefficients of the output state equation (1) are now given by

Figure 2: Schematic for generating discorrelated states nonlocally.
figure 2

Two independent multiphoton states and each interfere with one mode of the entangled state , generated by HOM interference of single photons. The resulting two-mode state may become discorrelated dependent on the phase relationship between the two input stats.

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Now the normalization factor is related to the probability of the measurement of the two ancilla photons PH, calculated by

Given the projective measurement of single photons by the two ancilla detectors, the probability Pn,m of a particular joint measurement of photon number n, m on the remaining modes is given by . The condition for discorrelation between the two parties is that Pn,n = 0 for all n, therefore we seek solutions to which are independent of n. By setting m = n in equation (3) we obtain

This can be set to zero by setting either term in parentheses to zero. The term can be set to zero if . However, this does not satisfy our condition for discorrelation due to the dependence on n. Instead, it is the two-mode analogue of filtering out photonic Fock states for entanglement generation, which has been demonstrated for a single mode41.

The second term in parentheses in equation (5) is zero if

or alternatively

If this condition is met, there is precisely zero probability that the same photon number is measured on each of the two modes, corresponding to a completely discorrelated state. This condition is independent of the beam splitter transmissivity t, although we assume it is the same in both modes. Whether the output state is discorrelated therefore depends on the two initial states and . In fact, changing only one of the input states allows to turn on or off the discorrelation, as seen in the examples below.

Discorrelation with coherent states

The input coefficients for coherent states and are

where and are the complex amplitudes of each coherent state. To fulfill the condition in equation (7), we find that α2 = β2, i.e. that the two coherent states must have the same amplitude with integer multiples of π phase between them. The coefficient in equation (3) thus reduces to

from which it is clear from the last term in parentheses that when n = m, independent of all other parameters including the magnitude of the coherent state |α|.

We show in Fig. 3 the joint photon number probabilities corresponding to this interaction, which we calculated using the Quantum Optics Toolbox56. Depending on the relationship between the phases of the coherent states and the transmissivity of the beam splitter, a wide variety of exotic number distributions can be generated, and in particular discorrelation can be turned on or off by varying the relative phase between α and β. Tuning the beam splitter transmission also allows eliminating terms with due to the term in square brackets in Eq. (9). Even with no discorrelation as in Fig. 3(b), the state is still slightly entangled by the operation, despite the number distribution being extremely similar to the unentangled input states of Fig. 3(a). Unlike the discorrelated states in (c) and (d), however, the amount of entanglement as quantified by the logarithmic negativity57 depends on the coherent state strength α and beam splitter transmissivity t.

Figure 3
figure 3

Probability distributions of photon numbers (n, m) (a) of the input modes before the discorrelation procedure and (b–d) of the output modes ( in Fig. 2). Each case corresponds to different relationship between the phases of the coherent states and the beam splitter transmissivity: (b) , , (c) α = β, , (d) α = β, . In the latter two cases the terms with n = m are eliminated, and in the third case, terms with n + m = 14 are also removed. The logarithmic negativities of the four states (a measure of entanglement) are 0, 0.04, 1, and 1 respectively.

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Discorrelation with squeezed vacuum

We next consider two single-mode squeezed vacuum states |λA〉 and |λB〉, written in the number basis as

Here the are unnormalized, with the squeezing parameters 0 ≤ |λA,B| < 1. Substituting these in the discorrelation criterion equation (7) we find λA = λB; therefore if the two single-mode squeezed states have squeezing parameters with the same magnitude and phase, the resulting state will be discorrelated. In this case, the photon number distributions are shown in Fig. 4, and just as with the coherent states, terms with m = n are eliminated.

Figure 4
figure 4

Probability distribution of different combinations of photon numbers (n, m) (a) of the input modes before the discorrelation procedure and (b–d) of the output modes with single-mode squeezed vacuum inputs. Each case corresponds to different relationship between the phases of the SMSV and the beam splitter transmissivity: (b) λ1 = 1 = −λ2, , (c) λ1 = λ2, , (d) λ1 = λ2, . In the latter two cases the terms with n = m are eliminated, but here the components near tend to be amplified, rather than suppressed as for the coherent state input.

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We can also apply the discorrelation operation to an entangled state. Now the input state is no longer separable, and can be written as

With the same discorrelation procedure as before, we generate the state

Again, this state is discorrelated if there exisits zero probability of measuring |n, n〉 〈n, n|, which is the case when

We illustrate this with the case of a two-mode squeezed vacuum (TMSV) state. TMSV states exhibit perfect photon number correlations, which we modify with the discorrelation operation. Writing the TMSV state in the number basis as gives the coefficients cn,m = λnδn,m. The criterion equation (12) is fulfilled, since cn−1,n+1 = cn+1,n−1 = 0, independent of the squeezing parameter λ and any phases involved. The joint photon number distribution for this case is shown in Fig. 5. Now the outputs are still tightly correlated, but with photon numbers offset by 2. For example, if n = 2, m = 0 or m = 4, as opposed to the input TMSV state where n = m.

Figure 5
figure 5

Probability distribution of different combinations of photon numbers (n, m) (a) at the input modes and (b,c) output modes for a two-mode squeezed vacuum input with λ = 1. The latter two cases correspond to different beam splitter transmissivity: (b) , (c) . In both cases terms with m = n are shifted to n = m ± 2, and the latter also removes n + m = 4.

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Loss dependence

As in all protocols involving photon number statistics, it is important to study the effects of loss in realistic implementations58. We show in Fig. 6(a) the entanglement of three types of discorrelated states via the logarithmic negativity57 of the joint output state as a function of loss applied symmetrically to both modes of the state. All three states start with a logarithmic negativity of 1, and decay identically with loss. For comparison we show the two-mode squeezed vacuum with the squeezing parameter for a logarithmic negativity of 1 with no loss, and the “not-discorrelated” state from Fig. 3(b) (but with t = 0.5), whose logarithmic negativity is not always 1 in the lossless case, but instead depends on the strength of the coherent states and the beam splitter transmissivity. For example, this not-discorrelated state has logarithmic negativity ≈1 when t = 2/α2, and ≈0 when .

Figure 6
figure 6

(a) Logarithmic negativity versus loss for two-mode discorrelated states and other entangled states. The HOM state, a discorrelated state based on a displaced single photon (Fig. 1), and a discorrelated state based on displacing the HOM state (Fig. 3(c)) all show the same loss scaling, with the TMSV slightly better. The not-discorrelated state (Fig. 3(b), here with t = 0.5) has less entanglement, as it is closer to the unentangled input states. (b) Discorrelation versus loss for the three discorrelated states. A discorrelation D < 1 indicates a nonzero probability of measuring the same outcome, and D < 0 means the outcomes are more likely to be correlated than uncorrelated coherent states. The discorrelation for the TMSV is not plotted because it is always more correlated than the reference state, and thus has a large negative discorrelation (≈−3.5). (c) Discorrelation versus loss at three different points in the circuit. In each case the loss is applied symmetrically to both arms. Discorrelation is most sensitive to loss in the ancilla photons before interference, and least sensitive to loss in the heralding detectors.

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As another quantification of the effect of loss, we introduce the discorrelation

defined by the probability of observing the same photon number between the two parties compared to this probability for uncorrelated coherent states. Thus when Pm=n,discorr = 0, D = 1 and when Pm=n,discorr = Pm=n,uncorr, D = 0. Both probabilities are calculated from normalized states, with the uncorrelated state additionally lossless. A negative discorrelation means the output state is more likely than uncorrelated states to produce a correlated result. For high loss, the “correlated result” is just the vacuum. We show in Fig. 6(b) the loss behavior of discorrelation for three initally discorrelated states. We also analyse the effect of loss at various points in the discorrelation circuit: in the ancilla preparation, before the single-photon detectors, and after discorrelation. In the lossless case the state is perfectly discorrelated as seen in Fig. 6(c), which falls off as loss is added.

Discussion

We have presented a form of quantum correlation, discorrelation, with the property that joint measurements of the photon number by two parties never yield the same result. Discorrelation can be produced by interfering a single photon with a coherent state, or by interfering two coherent states with a Hong-Ou-Mandel entangled state plus photon detection, the latter of which allows tuning additional properties of the state.

Discorrelation is a multideminsional phenomenon that maps the entanglement of the two-photon HOM state to much larger states. In fact the statistics of discorrelation can be interpreted as a generalised HOM-type bosonic bunching effect for higher numbers of photons, or as a displacement of the HOM state , which retains a similar quantum signature in the photon-number basis. Our procedure is bears a practical resemblance to generating discrete-continuous hybrid entanglement59,60,61, although here we map, rather than swap, discrete-variable path entanglement to continuous-variable entanglement.

It may be possible to extend discorrelation to more than two modes, wherein each mode has a large distribution over photon number, but no two modes can have the same measurement result. This form of discorrelation could be used to share unique random numbers across many parties, which may be useful in an untrusted card dealer scenario. However, the security implications need further analysis, along with considerations on the scaling of such a protocol.

Our results indicate that discorrelated states can be generated from a variety of input states and with a variety of output statistics. However these discorrelated states, similar to HOM states, are not robust against photon loss. In addition, it will be important in experiments to consider the quality of the single photon ancillae and indistinguishability of the four modes.

Additional Information

How to cite this article: Meyer-Scott, E. et al. Discorrelated quantum states. Sci. Rep. 7, 41622; doi: 10.1038/srep41622 (2017).

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