Universal state orthogonalizer and qubit generator

The superposition principle is at the heart of quantum mechanics and at the root of many paradoxes arising when trying to extend its predictions to our everyday world. Schroedinger's cat is the prototype of such paradoxes and here, in contrast to many others, we choose to investigate it from the operational point of view. We experimentally demonstrate a universal strategy for producing an unambiguously distinguishable type of superposition, that of an arbitrary pure state and its orthogonal. It relies on only a limited amount of information about the input state to first generate its orthogonal one. Then, a simple change in the experimental parameters is used to produce arbitrary superpositions of the mutually orthogonal states. Constituting a sort of Schroedinger's black box, able to turn a whole zoo of input states into coherent superpositions, our scheme can produce arbitrary continuous-variable optical qubits, which may prove practical for implementing quantum technologies and measurement tasks.

In the quantum mechanical context, states jψi and jψ ⊥ i are said to be orthogonal when the overlap between the two state vectors is zero, i.e., hψjψ ⊥ i ¼ 0. Two states, which are orthogonal to each other, are then maximally discernible [1,2]. In a classical binary system, states denoted by 0 and 1 are orthogonal to and perfectly distinct from each other, and the NOT operation will switch between them. In quantum mechanics, a simple swap of 0 and 1 does not bring one state to its orthogonal state, due to the superposition principle. As an extreme case, a superposition state of 1= ffiffi ffi 2 p ðj0i þ j1iÞ is unchanged by such a swap operation. The universal quantum NOT operation, which is defined as an operation to bring a state to its orthogonal one, is indeed not possible without some prior knowledge of the state [3], just like it is impossible to perfectly and deterministically clone or amplify a quantum state [4,5] without prior information. An orthogonalizer with minimal information required was demonstrated for states living in a limited two-dimensional Hilbert space [6,7].
As far as a general quantum state in an infinitedimensional Hilbert space is concerned, the situation is even more complicated because a given state will then have infinite orthogonal states. Despite this fundamental limitation, it has been recently proposed for an optomechanical system by Vanner et al. [8] that a perfect orthogonalizer can be in principle realized even if only some very limited preliminary information about the input state is available. One can generalize the proposal of Ref. [8] as follows: given an arbitrary operatorĈ, as we know its mean value hĈi for the input state jψi, the operationÔ converts the original state to an orthogonal state, where1 is the identity operator. It is straightforward to see that when O C is applied to the input state the resulting state jψ ⊥ i ¼ NÔ C ðjψiÞ, where N is the normalization factor, is orthogonal to jψi. This is equally applicable to pure states existing within finite or infinite Hilbert spaces, but it does not generally work for mixed states. Although the operatorĈ can be in principle arbitrary, the above procedure cannot be applied if the input states are among its eigenstates because the success probability of the operation drops to zero [9]. Here, we propose and demonstrate the generalized orthogonalization procedure of Eq. (1) to infinitedimensional, continuous-variable (CV) states of light. In addition to this, we show that this approach also naturally leads to a method for producing arbitrary coherent superpositions of orthogonal quantum states out of any input pure one. For any complex number c, the (unnormalized) superposition is realized by applying the same operation of Eq. (1) to the input state, but with an appropriate change in the weight of the identity operator. Therefore, once the orthogonalizer is in operation, any quantum superposition of jψi and jψ ⊥ i, which constitutes a general arbitrary CV qubit, can also be straightforwardly realized.
In principle, a full tomographic reconstruction of a given input state would allow one to design a specific setup to generate the orthogonal one (and, possibly, a superposition of the two). However, such a strategy would be very inefficient and far from "universal." State tomography involves the Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. measurement of observable probabilities for a large number of experimental settings, thereby requiring many identical copies of the input state for an accurate estimation after a numerical processing of the measured data sets. Furthermore, once each state is reconstructed, one should design an "ad hoc" experimental scheme to generate the orthogonal one, and this might in general be far from trivial.
On the contrary, even if the measurement of the mean value of operatorĈ in our scheme ideally requires a large number of identical copies of the input state, a measure of accuracy may be found. Specifically, when there are only N copies, the estimation of the mean value is accurate by where ΔC is the standard deviation of the measurement outcomes for the expectation value of the operatorĈ. Thus, even though we do not claim that this approach requires the least amount of resources, it is certainly much more resource efficient than those based on a full reconstruction (see a discussion on the level of observation versus the amount of information about a state in Ref. [10]). More importantly, a single universal experimental apparatus, only depending on a single parameter (the mean value hĈi), is necessary for our scheme to orthogonalize arbitrary states and produce arbitrary quantum superpositions. In the particular case of a set of input states for which hĈi is the same, no change at all is needed in the apparatus to process all the elements of the set.
Both the orthogonalizer and the procedure for producing coherent superpositions are general enough to work based on any operatorĈ and with any pure input state jψi. Choosing a particular operator implies designing the experimental setup accordingly, but a specific apparatus would then work for arbitrary input states, with the proper adjustment of its parameters. To concisely demonstrate the functioning of the above methods and their generality, in the following we present the experimental analysis of specific examples based on two differentĈ operators (therefore implying two completely different experimental setups). We experimentally test both systems with one kind of input state, but we also show that they are able to deal equally well with arbitrary ones.
A particularly simple and interesting case is obtained whenĈ ¼â † , the bosonic creation operator, which has no eigenstates and can thus be safely applied independently of the arbitrary state at the input. Here, one just needs to know the mean value ofâ † on the particular input state to construct theÔ a † ≡â † − hâ † i1 orthogonalizer or a general superposition as prescribed by Eq. (3). Both of these operations can be experimentally implemented by extending some of the tools recently developed in our group. A simplified scheme of the experimental setup is shown in Fig. 1(a) (also see Ref. [9]). In particular, the photon creation operator can be conditionally realized by means of stimulated parametric down-conversion (PDC) in a nonlinear crystal seeded by the optical input state in the signal mode [12][13][14]. The coherent superposition of this operation and the identity can be realized by mixing the (herald) idler PDC mode with a coherent light field on an unbalanced beam splitter that erases the information about the origin of a click in the heralding single-photon detector at one of its outputs. By simply controlling both the relative phase between the input and the coherent state impinging on the beam splitter, and the reflectivity of the latter, different superpositions ofâ † and1 can be obtained, in particular those corresponding to the orthogonalizer and to arbitrary CV superposition states. Similar techniques, involving phase-space displacement on the herald mode of conditional state generation, have been recently used for quantum state engineering up to two photons [15], and for the FIG . 1. (a) Conceptual experimental scheme of the orthogonalizer and CV qubit generator based on photon addition by heralded stimulated PDC. A click in the single-photon-counting module (SPCM) normally heralds a single photon addition to the generic input jψi state. However, if the PDC idler mode is mixed with a coherent state jβi on a beam splitter (BS) prior to detection, a superposition of the photon creation operator and the identity with adjustable weights and phases can be obtained. In the actual experiments we used coherent states jαi as the input states and the operator superposition was implemented by using polarization modes (see the Supplemental Material [9] for details). HD is a time-domain homodyne detector triggered by SPCM clicks [11]. (b) Raw measured x quadrature distributions (marginals of the Wigner function) for the input coherent states (ribbon-style curves) and for the corresponding results of the orthogonalization procedure (shaded areas) with jαj ¼ 0.5, 1.0, 2.0. generation of optical CV qubits made of superpositions of squeezed vacuum and squeezed single-photon states [16].
We tested the concept presented above by using coherent states jαi ¼DðαÞj0i as the input [whereDðαÞ is the displacement operator [17]]. In this special case, the orthogonalizer operator based on photon creation becomeŝ and it is easy to see that, when applied to jαi, this results in the displaced Fock stateDðαÞj1i, which is clearly orthogonal to jαi [9]. Figure 1(b) illustrates the result of the application of the orthogonalizer to coherent states of different initial amplitudes. The x quadrature distributions show that the Wigner functions of the orthogonal states are differently displaced versions of a single-photon Wigner function [18]. A full tomography of the input and output states for the case with jαj ¼ 1.0 results in a mutual fidelity of 0.4. The discrepancy from an ideal value of zero comes from different sources of experimental imperfections that limit the purity of the prepared and measured states and, as a consequence, their orthogonality (a detailed discussion is presented in the Supplemental Material [9]). A simple adjustment of the parameters in the beam splitter placed in the idler mode allows one to produce various CV qubit states. In Fig. 2 we show the measured Wigner functions for different equal-weight superposition states of an input coherent state with jαj ¼ 1.0 and its orthogonal state. In the different plots, the phase of the resulting CV qubit is simply varied by properly controlling the relative phase between the input and the displacement coherent states and the reflectivity of the idler beam splitter. The fidelities of the reconstructed states to the ideal superpositions of jαi and jα ⊥ i are all quite large, and of the order of 90%.
In order to demonstrate the effectiveness and generality of the proposed approach, we now consider another scheme to realize the orthogonalizer. Since the mean number of photons in a state is a parameter often easy to determine experimentally, one may insert such a mean photon number n and the number operatorn ≡â †â into Eq. (1), which thus becomesÔ We put this scheme to an experimental test with the setup of Fig. 3(a), which is similar to the one first developed for testing the bosonic commutation relation [19] (see the Supplemental Material [9] for a more detailed description). Such a setup conditionally produces the arbitrary superposition of operators which is seen to be proportional ton þ ½B=ðA þ BÞ1 using the bosonic commutation relation. Therefore, a state orthogonal to a generic one of mean photon numbern can be straightforwardly obtained by adjusting the setup of Fig. 3(a) so that B=ðA þ BÞ ¼ −n. It is seen that, when used in combination to input coherent states jαi, this scheme results in the same orthogonal state as for the previous example. Note that we do not assume any knowledge about the initial state except its mean photon number. Thus, simply choosing another coherent state jα 0 i as the orthogonal one is not a valid alternative. In any case, the overlap of two coherent states jhαjα 0 ij 2 ¼ e −jα−α 0 j 2 is never zero, especially in a regime of low intensities. Figures 3(b) and 3(c) show the measured quadrature distributions and the reconstructed Wigner functions for such input and orthogonal output states, whose measured mutual fidelity is F ¼ 0.34, while the fidelities F c to an ideal coherent state jα ¼ 0.78i are 0.96 and 0.18, respectively (see the Supplemental Material [9] for more details). To summarize, we have shown the first experimental application of a universal orthogonalization procedure to arbitrary CV optical states. Relying on a very limited amount of preliminary information about the input states, we verified the effectiveness and generality of this powerful technique through two illustrative examples based on the photon creation (â † ) and number (n) operators. Simple modifications in the experimental parameters also allowed us to produce CV qubits based on the superposition of an arbitrary input state jψi and its orthogonal jψ ⊥ i. Note that our goal is not that of producing a specific quantum state, which could be prepared with simpler methods anyhow, but rather to demonstrate a universal scheme for producing orthogonal and CV qubit states starting from arbitrary inputs. Coherent states were used here only for their ease of preparation. Adjusting a single parameter (for example, the reflectivity of a beam splitter) to accommodate for a different hĈi in the chosen experimental setup is sufficient for orthogonalizing and creating quantum superpositions out of any input pure state.
To better illustrate this, we give in the following (theoretical) examples of the generation of orthogonal states using the same orthogonalizing operatorsÔ a † and O n . We simulate the output orthogonal states using exactly the same schemes analyzed in our experiments for two other initial states that are representative cases of highly nonclassical states: an even cat state jαi þ j − αi and a squeezed vacuumŜðζÞj0i (normalization is omitted in the following for simplicity), where the single-mode squeezing operator is given byŜ with ζ ¼ re iϕ the squeezing parameter. Figure 4 shows the theoretical simulation of the Wigner functions for the orthogonal states generated by our experimental setups. Note that, while the application of a generic orthogonalizing operator will always take any input state to its orthogonal version, in general it is not the case that different choices of the operatorĈ used in theÔ C orthogonalizer map to the same orthogonal state. This is evident from the  This technique can become a useful tool for quantumstate engineering, to produce custom-made quantum states. Future goals should include testing the system with different input states and extending its experimental application to larger intensity input fields. Beyond pure photonics, our scheme can also be applied to various physical systems including phononic states of ions in a trap and nanomechanical oscillators [8]. Even though our demonstration here is for relatively small-amplitude coherent states, the method itself can be applied to bigger systems, and can be thus used for the study of foundational issues, like the test of quantum-to-classical transition models [20]. Additional data can be obtained from the corresponding authors.