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Two-photon imaging and quantum holography

It has been claimed that ``the use of entangled photons in an imaging system can exhibit effects that cannot be mimicked by any other two-photon source, whatever strength of the correlations between the two photons'' [A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 87, 123602 (2001)]. While we believe that the cited statement is true, we show that the method proposed in that paper, with ``bucket detection'' of one of the photons, will give identical results for entangled states as for appropriately prepared classically correlated states.

It has been claimed that "the use of entangled photons in an imaging system can exhibit effects that cannot be mimicked by any other two-photon source, whatever strength of the correlations between the two photons" [A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 87, 123602 (2001)]. While we believe that the cited statement is true, we show that the method proposed in that paper, with "bucket detection" of one of the photons, will give identical results for entangled states as for appropriately prepared classically correlated states. Recently, there has been a constant focus of interest in quantum limited imaging [1,2,3] and lithography [4,5,6,7]. In the former of the applications, the quantum correlation between the amplitude fluctuations of an appropriately entangled multimode state of light is used to beat the standard quantum limit of spatial resolution. The latter application capitalizes on the short de Broglie wavelength of entangled multiphoton states and offers, in principle, a spatial resolution that is independent of the classical wavelength of the electromagnetic field [8].
In a recent paper by Abouraddy et al. [9], a method has been proposed where an entangled photon pair is used to image an object. The setup is depicted in Fig. 1. Each of the photons in a photon pair is sent through an object. The objects labeled 1 and 2 are described by the impulse response operatorsĥ 1 andĥ 2 , respectively. The claim by Abouraddy et al. is that, if the source emits an entangled two-photon multimode state, and if one of the photons emitted is detected by a "bucket detector" (to be defined below), information about a test object is obtained that cannot be obtained with any classical source of light. Abouraddy et al. have proposed two schemes, in the first, called distributed quantum imaging, the test object, that we want to characterize, is described byĥ 1 , whileĥ 2 is a (known) reference object [9]. The authors have also suggested a scheme where the roles of the objects 1 and 2 are reversed [10], so that the test object is described byĥ 2 and the photon that goes through it is detected by a bucket detector, while the photon going through the reference object, now described byĥ 1 , is detected by a detector array. This imaging method was named quantum holography.
The claim by Abouraddy et al. has been put in doubt by a recent analysis, and an experiment, by Bennink et al. [11]. Their analysis showed that if the test object in a quantum holographic setup is lossless, no information about the test object is obtained. In their experiment they used a lossy test object and a correlated classical source of light. Yet, using what Bennink et al. called "'two-photon' coincidence imaging" (where the authors' quotation marks indicate that the detection method mimics what Abouraddy et al. call quantum holography, but that classically correlated states were used instead of two-photon states) the information stored in the test object could be read out. In this paper, we extend the results of Bennink et al. by showing that distributed quantum imaging with a lossless reference object gives identically the same information about the test object as if an appropriate classical source were used. We also show that quantum holography with a lossless reference object (as in the experiment by Bennink et al.) can always be mimicked by a classical source.
We shall assume that the considered modes, both on the source side and on the detection side, are discrete. Apart from this difference, we shall try to keep the notation as similar as possible to the notation in Ref. [9]. Moreover, we will, just as Abouraddy et al. [9], assume that the two photons are spatially filtered in such a way that the photon emitted into unprimed modes does not interact with object 2, and the photon emitted into primed modes does not interact with object 1.
Consider a source that emits light in the pure twophoton state where we have used the shorthand notation |1 i ⊗ |1 j ′ ≡ |1 i , 1 j ′ . Let us first see what happens if we completely ignore the photons passing through object 2. Writingρ = |Ψ Ψ|, the state of the unprimed photon then becomes In general, this is a mixed state of a single photon. After The two-photon source emits one photon into the modes denoted i, which propagate through object 1 described by the transformationĥ1. The linear impulse response oper-atorĥ1 contains the information about object 1. Another photon is emitted into modes j ′ and is subjected to the oper-atorĥ2 describing object 2. On the output side, there is an array of detectors, each measuring the photon number of a specific output mode.
interaction with object 1, the output state becomeŝ where the equality follows from the fact that we have assumed thatĥ 1 does not operate on the primed vectors. It may be prudent, at this point, to establish that the mode sets on the input and output side of object 1 or 2 do not have to be of the same type, as long as both are enumerable and orthonormal. For any given object, every specific pair of mode sets will lead to a unique transformationĥ 1 . On the detector side it is convenient to work in the eigenmodes of the detectors. Then, the probability p(q) of a photon detection in mode (detector) q is , the coefficients γ 1 (i, j) turn out to be the density operator coefficients 1 i |Tr 2 (ρ)|1 j , so that By inserting the resolution of the identity twice into Eq. (4), one arrives at where The expression above is the direct discrete-mode equivalent to Eq. (8) in Ref. [9]. If one permutes the indices q and q ′ , and lets the index 1 → 2 in Eqs. (4) and (6), one obtains the expressions for p 2 (q ′ ). Now, assume that both photons in the photon pair are considered. The output state after the interaction with the two objects becomesĥ 1ĥ2ρĥ † 2ĥ † 1 . Note that, sinceĥ 1 andĥ 2 operate in different vector spaces, they commute. The probability of registering a correlated detection event between detector q and detector q ′ can be expressed as A "bucket detector" is a multimode detector where all the modes propagating through an object are measured jointly. Hence, the information about the location of the detected photon (or equivalently, in what mode the photon was detected) is "erased." A schematic of such a detector is depicted in Fig. 2. In this case, the probability of photon detection in mode q (the marginal probability distribution of the photons interacting with object 1) becomesp If we assume that the state from the source is entangled such that ϕ(i, j) = ϕ(i)δ ij ′ , Eq. (8) can be rewritten as Following Ref. [9], we also introduce the parameter Replacing the sum over q ′ in Eq. (9) with this expression, we can writē This is the discrete counterpart of Eq. (12c) in Ref. [9]. Comparing expressions (6) and (8), Abouraddy et al. claim that "based on classical probability analysis one would intuitively expect that p 1 (q) would be equal tō p 1 (q). This is not always the case, however." From a strictly mathematical point of view it is evident that p 1 (q) andp 1 (q) are not equal for general functions h 1 (q, i) and h 2 (q ′ , i ′ ). However, we shall now prove that if the reference object is lossless, the two expressions are equal. Assume, therefore, that object 2, described byĥ 2 is lossless. This immediately implies that the operatorĥ 2 is unitary. Using this fact, we can write the expression for p 1 (q) as Furthermore, sinceĥ 1 andĥ 2 commute, we can recast this equation as Since the trace operation is basis invariant, the trace Tr 2 of any operatorÔ can be expressed either q ′ ∞ n=0 n q ′ |Ô|n q ′ or q ′ ∞ n=0 n q ′ |ĥ † 2Ôĥ 2 |n q ′ , whereĥ 2 is an arbitrary, unitary operator. Hence, from Eq. (12), we get This relation implies that Eqs. (4) and (8) are identical if the object 2 (the reference object) is lossless, irrespective ofρ andĥ 1 . Hence, in this case, nothing is gained by using distributed quantum imaging over single photon (per necessity, uncorrelated) imaging. The single-photon state Tr 2 (ρ) will give the same detection statistics asρ.
Let us now consider the case where the test object is described byĥ 2 , that is, it is the modes through the test object that are detected with the "bucket detector." This imaging principle is called "quantum holography" [10] or "two-photon coincidence imaging" [11]. Abouraddy et al. [10] claimed that it is possible to use entanglement between the states in the primed and unprimed modes to read out holographic information about the test objectĥ 2 even if the photons traversing it are detected with a bucket detector, such as a sphere coated on the inside with a photosensitive film that surrounds it. However, Bennink et al. showed that if the test object is lossless, then the bucket detector always clicks, effectively tracing out all information about the test object. Hence, quantum holography of such test objects does not work. However, assume that the test object (object 2) is lossy. It is clear that in this case the reference object (object 1) should be lossless, as losses will only introduce additional randomness in the measurement. Sinceĥ 1 andĥ 2 commute, this randomness corresponds to loss of information aboutĥ 2 . If the reference object is lossless, it is rather obvious that if a two-photon (possibly) entangled state |Ψ is used, its quantum holography detection statistics will be exactly mimicked by the classically correlated statê Hence, it seems to us that entangled-state holography with bucket detection of any object is of limited use, since identical results can be obtained using classical states. This was what Bennink et al. [11] demonstrated experimentally. The statistical distributionP 1 (q) in their experiment very clearly brought out the information encoded in the mask although all the photons interacting with the (lossy) mask were detected by a bucket detector. Above, we have shown that distributed quantum imaging and quantum holography with a lossless reference object (but, in general, a lossy test object) will not offer anything imaging with classical states cannot provide. Bennink et al. [11] proved that quantum holography of a lossless test object also can be mimicked by classical states. Now, we will treat the case when both object 1 and 2 are lossy. The standard way of including losses is to extend the two original sets of modes with auxiliary modes. We shall do so by assuming that the respective detector eigenmodes constitute only a subset of the primed and unprimed modes. To be realistic we can assume that only the photons in N (N ′ ) of the unprimed (primed) modes are detected, and that these modes are labeled 1, 2, . . . , N (1 ′ , 2 ′ , . . . , N ′ ). The probability of detecting a photon in mode q then becomes From Eq. (15), we see that photodetection probability P 1 (q) is simply the sum over all the detection events q ′ of the two-photon joint probability distributionP 1 (q) = N ′ q ′ =1 P (q, q ′ ) and the probability of detecting one photon in mode q and no photon in the modes q ′ = 1 ′ , 2 ′ , . . . , N ′ . This is just what one expects intuitively. For lossy reference objects, P 1 (q) differs fromP 1 (q). Now, denote the probability of not detecting the photon in the primed modes P 0 = q P 1,0 (q) and consider, e.g., the product state where M ′ > N ′ . It is quite obvious that this state, that lacks any correlation between the photon in the primed and unprimed modes, will yield an identical photodetection statistics distributionP 1 (q) to the corresponding distribution of the stateρ (that may be entangled). Note that this is true irrespective ofρ,ĥ 1 , andĥ 2 . The result depends critically on the fact thatĥ 2 is an operation local in the primed modes. However, this is, in general, not a physically accessible state since it may contain excitation in modes with M ′ > N ′ (the modes corresponding to the loss). If only the modes q ′ = 1 ′ , . . . , N ′ can be excited, then one cannot, in general, mimic the photodetection statistics of the two lossy objects illuminated by an two-photon entangled state, with a classically correlated state. However, even if this is not the case, we do not see what could be gained by using entanglement, because, in general, loss will destroy the entanglement of a quantum state more rapidly than it will reduce the correlations in a classical state. Since we have shown that in the case when the reference object is lossless, entanglement combined with bucket detection offers no improvement for neither two-photon imaging nor for quantum holography, we conjecture that if bucket detection of one of the photons is used, entangled two-photon imaging never offers any advantage over (classically) correlated-photon imaging.
An effect similar to "quantum holography," but not to be confused with it, has been demonstrated in Refs. [12,13], where a diffraction pattern could be read out trough the joint probability distribution of a two-photon state, although the marginal probability distributions did not show any diffraction pattern. This is an effect due to entanglement and the fact that bucket detection was not used. To model such an effect in the simplest possible fashion, consider the two-photon, four-mode, entangled state Assume thatĥ 1 =1 1 and h 2 (1, 1) = h 2 (1, 2) = h 2 (2, 1) = −h 2 (2, 2) = 1/ √ 2. One can then readily show that p(1, 1 ′ ) = p(2, 2 ′ ) = 1/2 and p(1, 2 ′ ) = p(2, 1 ′ ) = 0. That is, which one of the (two) detectors behind each object that will "click" is totally unpredictable. The statistics of either detector pair alone do not contain any information about object 2. This is so regardless if one simply ignores the outcome of the other detector pair, or if one uses a bucket detector (that, in this case, always will "click"). Due to the input-state entanglement, there is, however, a perfect correlation between the joint detection events. The correlation statistics hence reveal (some) information about the object.
In summary, we have shown that although entangled two-photon imaging may bring out effects that cannot be mimicked by any classical source, the advantage with this imaging method is lost if "bucket detection" of one of the photons is employed, at least if the reference object is lossless (and this ideal case is what one should strive for, since it gives the optimal information about the test object). To capitalize on the unique properties of two-photon imaging, the detection joint probability distribution needs to be retained.