Bell violation for unknown continuous-variable states

We describe a new Bell test for two-particle entangled systems that engages an unbounded continuous variable. The continuous variable state is allowed to be arbitrary and inaccessible to direct measurements. A systematic method is introduced to perform the required measurements indirectly. Our results provide new perspectives on both the study of local realistic theory for continuous-variable systems and on the nonlocal control theory of quantum information.

The issue of incompatibility between local realism and the completeness of quantum mechanics was originally raised for unbounded continuous variables in two-party systems by Einstein, Podolsky, and Rosen (EPR) [1]. Experiments to test local realism based on inequalities proposed by Bell [2] and his followers [3] imply, as is well known, that classical realism must be discarded as the basis for a universal theory. This has been repeatedly demonstrated in experiments with discrete variable systems [4][5][6][7][8].
Methods for testing local realism in continuous-variable systems have been proposed in order to advance the goal of reaching a completely loophole-free conclusion, and experimental tests on continuous-variable systems have been carried out [9][10][11][12][13]. However, these tests and all continuous-variable proposals to date [9][10][11][12][13][14][15][16][17][18][19][20] fall short because they rely on advance knowledge of the state under test. These methods fail whenever the state under test is unknown because then there is no basis by which measurement strategies can be guaranteed effective. One reason is that non-local correlations present in the original state can evade detection under dimensional reduction [21], as may happen, for example, in pursuing pseudo-spin [13,18,19] or binning [20] methods. An exceptiona l approach by E.G. Cavalcanti et al [22] leads to a continuous multipartite inequality that doesn't rely on advance knowledge of the state under test. However, to construct their inequality, operator commutation relations must be ignored, which also eliminates a large category of local realistic theories from test -see Q. Sun et al [22]. Additionally, violation of these inequalities may not be possible with only two parties -see A. Salles et al [22].
Thus two obstacles that have not yet been overcome are these: to derive a standard Bell-CHSH inequality [3] for an arbitrary and unknown bipartite input state in an unbounded continuous-variable state space, and to describe a currently feasible experimental method for its test. There are significant fundamental and practical reasons for solving this problem. On the fundamental side, a clear understanding of the domains of continuous-variable space which are incompatible with local realism remains to be achieved. More practically, in recent years paradigm-shifting quantum technologies have been developed which depend upon Bell non-locality in theory, and in some cases require the experimental violation of a Bell inequality of an unknown state [23]. Methods which permit Bell-CHSH inequalities to be formed and then tested on unknown states in continuous-variable systems may aid in the development and implementation of these technologies.
In this Letter we take a significant step in the way to overcoming both obstacles. To provide easy visualization, we address both issues in a specific scenario using the following two-photon down-conversion state: where | q B is one of a continuum of delta-normalized one-photon transverse momentum states of photon B, and |H A and |V A denote horizontally and vertically polarized quantum states of photon A. We assume that the transverse momentum state of photon A and the polarization of photon B factor out of the quantum state, and therefore need not be indicated. The sin θ and cos θ factors are included in writing |ψ AB to preserve its unit normalization, as the complex continuum amplitudes h( q) and v( q) are assumed to be unit-normalized, i.e., d q |h( q)| 2 = d q |v( q)| 2 = 1. Beyond normalization, nothing else is assumed about h( q) and v( q), including the value of their generally non-zero scalar product, The two-photon state in (1) has an important freedom in the amplitude functions h( q) and v( q), which are arbitrary superpositions of the modes in continuous q space. In the following we will use the term bundle to refer to an arbitrary superposition of | q states. Note that this means that it is impossible to fully determine the state (infinitely many measurements would be required). This point is crucial because it is the stopping point for attempts up to the present time to fully engage a continuous degree of freedom in Bell inequality analysis. We have overcome this roadblock, as we describe below.
It is natural to use the Schmidt analysis in considering two-party pure state entanglement, whether discrete or continuous. The Schmidt decomposition [24] reformulates the state (1) as where the sets {|u n A } and {|f n B } are superpositions of A's polarization states and B's momentum states respectively, and are derivable as the eigenvectors of A's discrete and B's continuous reduced density matrices. The κ 2 n are the associated eigenvalues, which are always the same for the two reduced density matrices.
Note that since party A has only two dimensions it has only two eigenvalues, and this forces all but two of B's infinitely many Schmidt eigenvalues to vanish. Thus the infinite n sum in (3) has only two non-zero terms, which we write: where we have dropped the A and B labels because it will be easy to remember that the discrete states belong to photon A and the continuous states to photon B. Here |u 1 and |u 2 are merely rotations of the original polarization states |H and |V , and |f 1 and |f 2 are unknown bundles of B's momentum states | q , and we write them as |f 1 = d q ϕ 1 ( q)| q and |f 2 = d q ϕ 2 ( q)| q , with a key orthogonality property: guaranteed by the Schmidt rearrangement [25]. κ 1 and κ 2 are real positive coefficients analogous to the sin θ and cos θ in (1), with κ 2 1 + κ 2 2 = 1. We note that because h( q) and v( q) are unknown, then {κ 1 , κ 2 } are also unknown. Lastly, for simplicity in the following derivation, we assume that z is real-valued which ensures that |u i is linearly polarized.
The Schmidt theorem provides an optimum result in three ways. First, as partners for the rotated polarization states it makes two bundles of momentum states from the (presumed unknown) amplitudes h( q) and v( q). Second, it guarantees that those state bundles are orthogonal, and so we have a pair of orthonormality relations u i |u j = f i |f j = δ ij . Third, independent of the makeup of the two bundles, the Schmidt states |f 1 and |f 2 define a plane in the infinite dimensional | q space.
We are now much closer to Bell Inequality territory because rotations in planes in A and B spaces are what the CHSH inequality demands. But the bundles of continuum states making up the two Schmidt states |f 1 and |f 2 are mysterious because the original functions h( q) and v( q) were unknown. There are no operators available in continuum B space to make the rotations required by the Bell-CHSH analysis. We will describe below how to make measurements in a rotated basis in the continuum space without rotation operators for the space, but first let us reproduce the Bell-CHSH Inequality analysis, under the assumption that rotations in the |f 1 -|f 2 plane can be controlled.
With ordinary optical components one can always undertake a rotation of the Schmidt basis in photon A's polarization space, i.e., where α defines the arbitrary rotation angle. A rotated basis |f β 1 , |f β 2 of momentum space bundles for photon B can be defined similarly with β as the rotation angle in q space, while the practical matter of accomplishing such a rotation remains temporarily an open question. However, given these rotations, the conventional CHSH analysis of local hidden variable theory [3] can be employed. One considers the Bell operator B and finds B ≤ 2, where B is defined as Here C(α, β) is the CHSH correlation between photons A and B when the measurements are set for the angles α and β, and P ij (α, β) are the joint probabilities of finding photon A in state |u α i and photon B in state |f β j , with i, j = 1, 2. That is, According to quantum mechanics, the joint probability is given as which is a joint projection in the state spaces of both photons and has the potential to violate the CHSH inequality. Then the Bell operator B can be calculated to be B = 2κ 1 κ 2 sin 2α(sin 2β − sin 2β ′ ) + sin 2α ′ (sin 2β + sin 2β ′ ) + cos 2α(cos 2β − cos 2β ′ ) + cos 2α ′ (cos 2β + cos 2β ′ ).
As described above, the central hurdle to be overcome is the lack of a method to measure the Schmidt bundles in the continuous | q space of photon B. As we now demonstrate, a specially engineered auxiliary photon is sufficient to accomplish this. The requisite auxiliary photon can be easily created using an auxiliary entangled state which is identical to the original state. Practical techniques for generating pairs of identical entangled biphotons are available, as discussed in the Supplemental Information, so we proceed with the setup sketched in Fig. 1.
Source S t emits a pair of photons in the desired discrete-continuum entangled state, of which the Schmidt form is The discretely (polarization) entangled photon in mode t is heading northwest (NW) and the continuously (momentum) entangled photon in modet is heading southeast (SE), illustrated by the red paths in Fig. 1. The goal of our following analysis is to propose a Bell test, namely, measuring various correlations in terms of joint probabilities, for such a discrete-continuum entangled state regardless of what is known or not known about the continuous-space photon in modet and whether it is accessible or not to direct measurement. A polarization projection on basis |u α 1 for the photon in mode t can be realized with a polarizerû α 1 that passes the |u α 1 component into mode T , i.e., where c α and s α stand for cos α and sin α. The probability of this measurement outcome being realized is given by P 1 (α) = tt ψ|u α 1 t t u α 1 |ψ tt = κ 2 1 c 2 α + κ 2 2 s 2 α , and can be determined experimentally by recording the number of coincidences detected during a fixed time window in modes (T,t) and (t,t) for polarizer angle α, where N α (T,t) and N (t,t) are the number of coincidences in their corresponding modes. This also gives the value of κ 1 and κ 2 since κ 2 1 + κ 2 2 = 1 as stated after (5). To determine joint probabilities, one needs to measure the continuum space in a basis rotated by the angle β as well, so we now express the state in the rotated basis, which we rewrite again as Here c ij with i, j = 1, 2, are normalized amplitude coefficients, and they relate to joint probabilities in an obvious way: P ij (α, β) = |c ij | 2 P i (α). Now that the probability P i (α) can be measured easily, as is shown above, the value of joint probability P ij (α, β) can be determined by measuring only the coefficients |c ij | 2 . This can be realized with the help of the auxiliary photon pair |ψ aā , which is generated by source S a to have exactly the same form as the state under test, i.e., |ψ aā = κ 1 |u 1 a |f 1 ā + κ 2 |u 2 a |f 2 ā , with the discretely entangled photon in mode a heading SW and the continuously entangled photon in modeā heading NE, illustrated by the blue paths in Fig. 1. The auxiliary photon pair allows us to perform an indirect measurement in the continuous-variable space of the photon in modet. First, the mode a photon of the auxiliary pair is projected (by a polarizerû s 1 ) onto the polarization basis |u s 1 , where angle s is chosen to strip off the |f β 2 component from the photon in modeā. A glance at (15) shows how a stripping in continuum space by action in polarization space works. In (15), by choosing α such that κ 1 tan β = κ 2 tan α, the |f β 2 component would be eliminated. In the case of auxiliary photon a, we choose s such that κ 1 tan β = κ 2 tan s and obtain |u s 1 a a u s with P 1 (s) = aā ψ|u s 1 a a u s 1 |ψ aā . P i (s) is determined experimentally in exactly the same way as P i (α). The photon enters mode A from mode a after passing the stripping polarizerû s 1 , as shown in Fig. 1. Then the four-photon state after the two polarization projections in modes t and a is given by Next, as shown in Fig. 1, the modet photon is combined with the modeā photon (which is in the continuous variable state |f β 1 ) by a 50:50 beam splitter (BS). The outcome modes are denoted asT (NE) andĀ (SE). The effect of the BS can be expressed as As a result of Hong-Ou-Mandel (HOM) interference [4], the coincidence of the outcome photons in modesT andĀ determines the degree of distinguishability between the photons in modest andā. To be more specific, the contributing component of the modet photon in Eq. (19) to the coincidences after the BS is c 12 |f β 2 t, which is the distinguishable component of the photon in modeā. This amounts to a filtering or projecting operation of the photon in modet onto the continuous variable basis |f β 2 . With the above operations, a joint projection is realized for testing the entangled photon pair |ψ tt . It is then straightforward to achieve the joint probability P 12 (α, β). The four-photon coincidence probability in modes T,T , A,Ā is given as where N αβ (T, A,T ,Ā) and N (t, a,t,ā) are four-photon coincidence counts of the corresponding modes for polarization angles α and β. The individual probabilities can be determined using (14). Consequently, the joint probability can be written in terms of measurable quantities, Measurement of the other joint probabilities P 11 (α, β) and P 2j (α, β) are accomplished by appropriately rotating the angles α and β by π/2. In this way the correlation function C(α, β) can be achieved straightforwardly. Other correlations can be obtained similarly with other choices of angles α and β. To achieve the Bell violation given in (11) the orientation of the stripping polarizer is determined as tan s = (κ 1 /κ 2 )( √ 2 − 1) and tan s ′ = (κ 1 /κ 2 )( √ 2 + 1). Beyond the Bell violation issue, it is important to note that our method of measuring the continuous-variable space is an example of non-local quantum control [28]. It provides a new perspective on indirect measurement of a system state which is not directly accessible experimentally. We have shown explicitly how, by manipulating a discrete and controllable entangled partner, measurements of a continuum system may be made. Apart from increased measurement capabilities, this type of indirect measurement may be useful for transferring or encoding information into continuous-variable spaces which are difficult to detect or probe directly. Therefore, with proper design, it may be possible to construct communication protocols which impede potential eavesdroppers from obtaining the encoded information.
In summary, we have addressed the two obstacles mentioned in paragraph 3, obtaining a resolution with the aid of a new approach to continuous-variable measurement. Specifically, we have devised a Bell-CHSH inequality for the two-particle case in which one particle is defined by an unbounded continuous variable in a unknown state of arbitrary complexity, and we have sketched a currently feasible measurement approach for its implementation. This technique may expand further the systems in which Bell non-locality may be used for practical applications [23].

Supplemental Material
At least two approaches are open for generating a discrete-continuum (e.g., polarization-spectrum) entangled state to perform the Bell test proposed in the text. One setup is illustrated in Fig. 2, where the entangled photon pairs are produced in a pair of spontaneous parametric down-conversion (SPDC) crystals, a combination of type I and type II, pumped with an ultra-short UV laser pulse in a double pass configuration [1]. The photon pair produced by the first passage can be written in general as where Φ(ω 1 , ω 2 ), Ψ(ω 1 , ω 2 ) are two different amplitude functions relating to the field-crystal interaction parameters, and ω 1 , ω 2 represent the frequency of the signal (mode t) and idler (modet) photons respectively. Here |ω 1 , H represents a single photon state with frequency ω 1 and polarization H. The first and second terms in Eq. (24) are generated by the type I and type II crystals, respectively. The propagation directions of the two down-converted photons are determined by the phase-matching conditions of the SPDC crystals. Schematic illustration of producing two identical discrete-continuum entangled photon pairs with two-passage spontaneous parametric down conversion. The ultra-short UV laser pulse passes through the combination of type I and type II crystals and creates the first photon pair |ψ ′ tt (in red paths) with the signal and idler photons propagating in modes t andt respectively. The UV pulse pulse is then reflected back by the mirror M and passes through the two-crystal structure again to create the second entangled photon pair |ψ ′ aā (in blue paths) with the signal and idler photons propagating in modes a andā respectively. The spectrum of the signal photons in modes t and a are filtered by the interference filters (IF) so that the desired discrete-continuum (polarization-spectrum) entangled states |ψ tt and |ψ aā are achieved respectively.
Then one can insert an interference filter (IF) centered at ω 0 in front of the signal photon (as shown in Fig. 2). After the filter the two-photon state is left in a desired discrete-continuum (polarization-spectrum) entangled state, i.e., ω 2 ), and f (ω 1 − ω 0 ) is the spectral response function of the filter. Here we have omitted the factorable components, i.e., the spectral state of the photon in mode t and the polarization state of photon in modet. While this scheme is capable in principle of generating the required discrete-continuous entanglement, it is likely that the spontaneous parametric down-conversion sources will have to be specially engineered to achieve a large degree of entanglement. This is because the degree of entanglement of the state in (2) is directly related to the degree of orthogonality of the conditional wave functions Φ ω0 (ω 2 ), and Ψ ω0 (ω 2 ). When dω 2 Φ * ω0 (ω 2 )Ψ ω0 (ω 2 ) ≃ 0, the degree of entanglement will approach the maximal value possible. In practice, engineering the sources to achieve this may be difficult since the output of both crystals will have very similar biphoton wave functions, differing only in the crystal phase-matching functions. Regardless, the extreme control over the biphoton wave function in spontaneous parametric down-conversion which has been demonstrated in previous studies gives some optimism that this obstacle may be overcome [2,3].
After the first passage, the UV laser pulse is reflected back by a mirror (M) and then passes through the two-crystal structure again to create the second desired discrete-continuum entangled photon pair |ψ aā . As shown in Fig. 2, the two down-converted photons propagate in blue paths with the signal photon in mode a and the idler photon in modē a. Again the spectrum of the signal photon is filtered by an identical IF centered at ω 0 .
Then the two photons in modet andā can be combined by a 50:50 beam splitter as proposed in the text to perform the Bell test measurement. To ensure the temporal indistinguishability of the two photons arriving at the beam splitter, one needs to make sure that the laser pulse length is much shorter than the coherence time of the down-converted photon [1]. By adjusting the distance between the two-crystal structure and the mirror one can achieve the Hong-Ou-Mandel effect [4], and thus realize the necessary temporal indistinguishability.
Another approach for realizing the necessary temporal indistinguishability is to produce SPDC photon pairs with very long coherence times by using a very narrow-band filter as demonstrated in Ref. [5]. The temporal indistinguishability is then provided by appropriate post-selection of coincident detection events in fast single-photon detectors. In this case the second (auxiliary) discrete-continuum entangled pair can be generated from an identical yet independent two-crystal structure.