Operational resource theory of imaginarity
Abstract
Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems, their dynamics, and interaction. Since the inception of quantum theory, it has been debated whether complex numbers are actually essential, or whether an alternative consistent formulation is possible using real numbers only. Here, we attack this long-standing problem both theoretically and experimentally, using the powerful tools of quantum resource theories. We show that – under reasonable assumptions – quantum states are easier to create and manipulate if they only have real elements. This gives an operational meaning to the resource theory of imaginarity, for which we identify and answer several important questions. This includes the state-conversion problem for all qubit states and all pure states of any dimension, and the approximate imaginarity distillation for all quantum states. As an application, we show that imaginarity plays a crucial role for state discrimination: there exist quantum states which can be perfectly distinguished via local operations and classical communication, but which cannot be distinguished with any nonzero probability if one of the parties has no access to imaginarity. This phenomenon proves that complex numbers are an indispensable part of quantum mechanics, and we also demonstrate it experimentally with linear optics.
Complex numbers, originated in mathematics, are widely used in mechanics, electrodynamics, and optics, allowing for an elegant formulation of the corresponding theory. The rise of quantum mechanics as a unified picture of waves and particles further strengthened the prominent role of complex number in physics. Indeed, the imaginary unit appears in many formulas in quantum theory, notably in Schrödinger equation describing the time evolution of a quantum system, and is of fundamental importance in quantum physics and quantum information science.
According to the postulates of quantum mechanics, a state of a quantum system is described by a wave function and phase . The wave-based point of view provides an important set of tools for the formulation and construction of quantum physics. Therefore it is natural to ask whether the complex arithmetic in quantum mechanics, arising from the imaginary part of is necessary to describe the fundamental properties and dynamics of a quantum system. In other words, can quantum physics be restated in a formalism using real numbers only? One approach to address this question is to use the standard rules of quantum mechanics, but to enforce all states and measurement operators to have real elements only Stückelberg (1960); Araki (1980); Hardy and Wootters (2012); Wootters (2012); Baez (2012); Aleksandrova et al. (2013); Wootters (2014, 2016); Barnum et al. (2016). The aim of this approach is then to find physical effects and applications, which are possible in standard quantum mechanics, but impossible in its version restricted to real numbers (Wootters, 1990; Liu et al., 2019). It has been noted that this real-vector-space quantum theory is fundamentally different from the standard one from various aspects, e.g., it is bilocally tomographic Hardy and Wootters (2012), a rebit (real qubit) can be maximally entangled with many rebits Wootters (2012); Aleksandrova et al. (2013); Wootters (2014), and it allows optimal transport of information from preparation to measurement Wootters (2016). with probability amplitude
Another reason to distinguish between complex and real quantum states is the effort to establish them in experimental setups. An important example is polarization-encoded photonic system, where we can realize an arbitrary rotation around the -axis by a single half-wave plate, while for a rotation around the -axis two additional quarter-wave plates are needed. The fact that a certain type of transformations is easy to perform is the basic feature of any quantum resource theory Horodecki and Oppenheim (2013); Brandão and Gour (2015); Chitambar and Gour (2019). This justifies the study of the resource theory of imaginarity (Hickey and Gour, 2018), using the framework of general quantum resource theories, which has been successfully applied to investigate basic properties and applications of quantum entanglement (Horodecki et al., 2009), quantum coherence (Streltsov et al., 2017a), and quantum thermodynamics (Goold et al., 2016; Lostaglio, 2019). This framework is based on the idea that there is a restriction on the operations on a quantum system, dictated by the physical setting. One then studies which conversions between states are possible under this restriction.
The aim of this work is twofold. Firstly, we provide the resource theory of imaginarity with an operational meaning, discussing the experimental role of complex and real operations, i.e., quantum operations which do not create imaginarity. Secondly, we identify and answer several important questions within this theory, and provide a concrete experimental application with linear optics. This includes the state-conversion problem, namely characterizing the interconversion between quantum states via real operations. We provide a complete solution to the single-shot state conversion problem between single-qubit mixed states. For arbitrary pure states, we evaluate the optimal probability of their interconversion. We also introduce the task of approximate imaginarity distillation, and present an optimal distillation procedure for all mixed states.
As an application, we show that imaginarity plays a crucial role for local quantum state discrimination, when complex numbers are allowed in the measurement. We show that there exist real bipartite states which can be perfectly distinguished via local operations and classical communication (LOCC), but which cannot be distinguished with any non-zero probability via LOCC restricted to real local measurements. In the context of quantum tomography, a similar effect has been observed previously in Wootters (1990). We provide a theoretical framework for studying local state discrimination, and find the advantage of local complex measurements which outperform real local measurements. By experimentally measuring the optimal distinguishing probability for different families of mixed states, our results clearly demonstrate that complex numbers play a distinguished role in quantum theory, allowing for phenomena which would not be possible with real quantum mechanics alone.
Results
Resource theory of imaginarity—The first step to formulating any resource theory is to identify the free states of the theory, i.e. quantum states which, within the theory under study, can be created at no cost. Similar to the resource theory of coherence Streltsov et al. (2017a); Baumgratz et al. (2014), we specify a particular basis , and a pure quantum state can be written as
(1) |
with complex coefficients which satisfy . The natural choice for free states in the theory of imaginarity are real states, i.e., quantum states with all coefficients being real (up to a non-observable overall phase) Hickey and Gour (2018). Mixed real states can be identified as convex combinations of real pure states ,
(2) |
The set of all real states will be denoted by . It can also be characterized as the set of states with a real density matrix Hickey and Gour (2018).
The formulation of a resource theory is completed by defining an appropriate set of free operations, corresponding to physical transformations of the quantum systems which are easy to implement. In general, quantum operations can be specified by a set of Kraus operators satisfying the completeness relation . In this way, it is guaranteed that describes a physical transformation, which can in principle be realized in nature. In the case of probabilistic transformations, the Kraus operators satisfy the more general condition .
As the free operations of imaginarity theory we identify quantum operations which admit a Kraus decomposition having only real elements in the free basis Hickey and Gour (2018):
(3) |
Such transformations are called real operations Hickey and Gour (2018). This definition guarantees that real operations cannot create imaginarity, even if interpreted as a general quantum measurement. In this case, the post-measurement state will be real for any real initial state, regardless of the measurement outcome.
A desirable feature of a quantum resource theory is the existence of a golden unit: a quantum state which can be converted into any other state via free operations. In the resource theory of imaginarity the golden unit is the maximally imaginary state . Interestingly, via real operations it is possible to convert into any state of arbitrary dimension Hickey and Gour (2018). Another maximally imaginary state is given by . In the Methods Section we discuss the main features of quantum resource theories, including resource quantifiers and state conversion properties under free operations.
Quantum state conversion—We will now present a complete solution for the conversion problem via real operations for all qubit states, characterizing when a qubit state can be converted into another qubit state via real operations. To this end, recall that any single-qubit state can be represented by a real 3-dimensional Bloch vector. Now, the transition is possible via real operations if and only if
(4a) | ||||
(4b) |
where and are the Bloch vectors of the initial and the target state, respectively. A figure illustrating the set of accessible states for different initial states is shown in the Methods Section.
Notably, there exist states which cannot be obtained from a given state via real operations. In this case, it might still be possible to achieve the conversion probabilistically. In the following, we present the optimal conversion probability via real operations for any two pure states.
Theorem 1.
The maximum probability for a pure state transformation via real operations is given by
(5) |
The proof of the theorem makes use of properties of general resource quantifiers, we refer to the Methods Section and Supplemental Material for more details.
Approximate imaginarity distillation—So far we have discussed exact transformations between quantum states via real operations, both deterministically and stochastically. We will now go one step further, and consider approximate transformations, in the cases when an exact transformation is impossible. The figure of merit in this case is the transformation fidelity ; we refer to the Methods Section for more details.
Typically, one aims to convert into the most valuable quantum state, which in the resource theory of imaginarity is the maximally imaginary state . This leads us to the fidelity of imaginarity, quantifying the maximal fidelity between a state and the maximally imaginary state, achievable via real operations: .
As we will see below, the fidelity of imaginarity is closely related to the robustness of imaginarity, defined as Hickey and Gour (2018)
(6) |
where is a (possibly non-real) quantum state. As we show in the Supplemental Material, it admits the following closed formula:
Theorem 2.
For any quantum state the fidelity of imaginarity is given as
(7) |
This result provides a closed formula for the fidelity of imaginarity, quantifying how well a quantum state can be converted into the maximally imaginary state via real operations.
Applications—We will now discuss applications of imaginarity as a resource for discrimination of quantum states and quantum channels. Channel discrimination can be seen as a game, where one has access to a “black box” with the promise that it implements a quantum channel with probability . The goal of the game is to guess by applying the black box to a quantum state , followed up by a suitably chosen positive operator valued measure (POVM) which satisfies . The measurement outcome then serves as a basis for guessing that the black box has implemented the channel in the corresponding realization of the experiment. The probability of correctly guessing an ensemble of channels in this procedure is given by
Recently, it has been shown that any quantum resource provides an operational advantage in some channel discrimination task Takagi et al. (2019); Takagi and Regula (2019). Specifically, for the resource theory of imaginarity, it holds that
(8) |
Eq. (8) implies that for any quantum state which has non-real elements there exists a set of channels such that the optimal guessing probability is strictly larger than for any .
Another, closely related task is quantum state discrimination Chefles (2000), where one aims to distinguish between quantum states , each given with probability . To this end, one performs quantum measurements, described by a POVM . The average probability for correctly guessing the state is
To make the role of imaginarity in this task explicit, we extend the robustness of imaginarity from states to measurements. To this end, let be the set of all real POVMs , i.e., all have only real elements in the fixed basis. We define the robustness of imaginarity of a measurement as
(9) |
where the minimum is taken over all POVMs . Following Takagi and Regula (2019), we find a result analogous to Eq. (8):
(10) |
For any POVM which is outside of , there exists an ensemble of states and probabilities leading to a better performance of , when compared to any measurement with real POVM elements.
Going one step further, we will now show that complex numbers play an indispensable role in local state discrimination Bergia et al. (1980); Wootters (1990). Assume that the states to be discriminated are shared by two distant parties, Alice and Bob. It was shown in Walgate et al. (2000) that any pair of pure orthogonal states can be perfectly distinguished via local operations and classical communication (LOCC). To perfectly distinguish the states via LOCC, there must exist a POVM with elements of the form and the property for all and . If the states are real, we are particularly interested in perfect discrimination with local real operations and classical communication (LRCC), where all must be real and symmetric. Indeed, if two states are pure, orthogonal, and real, such perfect LRCC discrimination is possible; see Supplemental Materials for more details.
For some real mixed states, instead, the situation is radically different. Consider the states
(11) | ||||
with the Bell states , and . These states can be perfectly distinguished via LOCC if Alice and Bob perform local measurements in the basis and share their measurement outcomes via a classical channel, see Methods Section for more details. On the other hand, the states (11) cannot be distinguished via LRCC with any nonzero probability. Too see this, note that the two states can be written as , and for any real symmetric matrix . It follows that for any POVM element with real symmetric matrices it holds .
The states (11) show the role of imaginarity for quantum state discrimination in an extreme way. The two states are completely indistinguishable via LRCC, even if we consider imperfect state discrimination with finite error. It is clear from the above discussion that this effect is also observed if only one of the parties is limited to real operations, and the other party has access to all quantum operations locally. Nevertheless, the states can be perfectly distinguished by LOCC, if both Alice and Bob can perform general quantum measurements locally.
These results further highlight the relevance of complex numbers in quantum mechanics. Note that the states (11) have real elements in the computational basis. This means that they are also valid states in “real quantum theory” Stückelberg (1960); Araki (1980); Hardy and Wootters (2012); Wootters (2012); Baez (2012); Aleksandrova et al. (2013); Wootters (2014, 2016); Barnum et al. (2016), which is the restriction of quantum theory to real states and real measurements. In such a theory, two remote parties would not be able to distinguish these states with any non-zero probability, whereas they are actually perfectly distinguishable in reality.
Experimental relevance of imaginarity—Here, we perform a comparison of real operations and general quantum operations in optical experiments, focusing on the single-photon interferometer set-up with half-(quarter-) wave plates and polarizing beam splitters as the building blocks. Moreover, we refer to a wave plate with unset optical axis as unset wave plate. Note that a combination of polarizing beam splitter and half-wave plate plays a similar role as a variable beamsplitter Reck et al. (1994); Kok et al. (2007); Carolan et al. (2015). The details of the analysis are given in the Methods Section.
Under above assumptions, a real quantum operation acting on path degree of a dimensional system can be implemented with unset wave plates, whereas a general quantum operation requires at least unset wave plates for the implementation. We assume that both operations are implemented via a unitary dilation, see Methods Section for more details. For large , this implementation allows one to reduce the number of unset wave plates by a factor of , if real operations are used instead of general quantum operations.
Similar results can be found in implementing a real -outcome generalized measurement on a single polarization-encoded qubit Zhao et al. (2015), when compared to the corresponding general qubit measurement. Even in this case, using real measurements instead of general measurements reduces the number of unset wave plates by in the limit of large .
These results show that real states are easier to create and real operations are easier to perform when compared to general states and operations in a single-photon interferometer set-up. This justifies the choice of real states (operations) as the free states (operations) of the resource theory of imaginarity.
Experimental local state discrimination—As discussed earlier, imaginarity plays an important role for quantum state discrimination. We devised an experimental setup for local state discrimination using two entangled photons. The experimental protocol and setup are shown in Fig. 1. Our setup allows us to prepare a class of two-qubit states, and determine the optimal guessing probability under LOCC.
Our experiment consists of two parts. In the first part we consider the discrimination of the two states in Eq. (11). We recall that these states can be perfectly distinguished if Alice and Bob perform local projective measurements in the maximally imaginary basis and share the outcomes of their measurement via a classical channel. The experimental results are shown in Fig. 2 (a). From the experimental data we obtain for the success probability Clarke et al. (2001); Mosley et al. (2006); Solís-Prosser et al. (2017). To demonstrate that the states cannot be distinguished if the local measurements are real, we also show the experimentally measured probabilities of the and measurements. In this case the output is nearly a uniform distribution. From the right side of Fig. 2 (a), we extract experimentally determined guessing probabilities under different local Pauli measurements and classical communications. We can see that imaginarity is necessary in measurement of both subsystems for improving the guessing probability, since any real projective measurements can be written as a combination of , , and .
In Fig. 2 (b-e), we show the experimental results for distinguishing different families of mixed states. The exact expression for the states is given in the caption of Fig. 2. All dashed lines represent theoretically derived maximum probability for distinguishing the states via local projective measurement and classical communication; dotted lines represent the aforementioned probabilities under local real measurements, we refer to the caption of Fig. 2 and the Supplemental Material for more details. Note that all states considered here have only real elements in the computational basis.
These results clearly demonstrate the relevance of imaginarity to local state discrimination. The ability to perform measurements with non-real POVM elements can significantly improve the success probability for state discrimination, even if the states to be distinguished have real density matrices.
Discussion
In this work, we investigate the resource theory of imaginarity, studying the role of complex numbers in quantum mechanics in an operational way. We formulate and completely solve several questions within this theory, including the deterministic conversion for all qubit states, the probabilistic conversion for all pure states, and the single-shot imaginarity distillation for all states of arbitrary dimension.
Our methods can be readily applied to study the role of complex number in quantum information processing tasks. We demonstrate this for local state discrimination, where two remote parties aim to distinguish states by applying local operations and classical communication. We show – both theoretically and experimentally – that there exist real quantum states which can be perfectly distinguished in this setup if imaginarity is used in the local measurements. However, when restricting to only real measurements, the states cannot be distinguished with any nonzero probability. This demonstrates that complex numbers are an essential ingredient of quantum mechanics.
The usefulness of complex number in quantum mechanics is worth an in-depth study also in the light of the recent advances in quantum technologies. An important example is quantum computers, which can solve certain problems of interest significantly faster than any classical computer Bravyi et al. (2018); Arute et al. (2019). As of today, the reason for this quantum advantage is not completely understood, especially when it comes to quantum computers operating on noisy states Knill and Laflamme (1998); Datta et al. (2005, 2008); Dakić et al. (2010); Matera et al. (2016). A quantitative analysis of imaginarity in quantum computers can shed new light on the quantum features required for the quantum speedup. Another promising line of research is the relation between different quantum resources Streltsov et al. (2015); Ma et al. (2016); Wu et al. (2018); Wang et al. (2019); Sparaciari et al. (2020), which will give us new insights into quantum resource consumption in quantum technological tasks, such as quantum metrology and quantum communication.
Methods
Main features of quantum resource theories
One of the main questions in any quantum resource theory is whether for two given quantum states and there exists a free operation transforming into :
(12) |
The existence of such a transformation immediately implies that is more resourceful than , and in particular
(13) |
for any resource measure .
If cannot be converted into via a free operation, e.g. if , it might still be possible to achieve the conversion probabilistically, if the corresponding resource theory allows for stochastic free operations, with free Kraus operators such that . The maximal probability for converting into is then defined as
(14) |
with probabilities , and the maximum is taken over all (possibly incomplete) sets of free Kraus operators, i.e., . The existence of a deterministic free operation between and as in Eq. (12) is then equivalent to .
If two states and do not allow for deterministic neither stochastic transformations [i.e., ], there remains the possibility to perform the transformation approximately. The figure of merit in this case is the maximal transformation fidelity
(15) |
with fidelity , and the maximum is taken over all free operations .
Any resource measure is monotonic under free operations, see Eq. (13). For resource theories which allow for stochastic conversion, one typically requires a stronger constraint on the resource measure, to be monotonic on average under free operations:
(16) |
Here, the states arise from by applying a free operation: with free Kraus operators , and is the corresponding probability: . Quantifiers satisfying Eq. (16) are also called strong resource monotones. If is additionally convex, i.e., , then strong monotonicity (16) implies the weaker condition (13).
A powerful upper bound on the conversion probability (14) can be obtained from any resource quantifier which is convex and strongly monotonic under free operations. For any such resource quantifier , it holds Wu et al. (2020):
(17) |
An important resource quantifier is the robustness with respect to set of free states :
(18) |
where the minimum is taken over all quantum states . For any quantum resource theory, the robustness is closely related to the success probability in channel discrimination tasks Takagi et al. (2019); Takagi and Regula (2019):
(19) |
Eq. (19) implies that for any resource state (i.e. a quantum state which is not element of ) there exist a set of channels and a probability distribution such that the optimal guessing probability is strictly larger than for any .
State transformations via real operations
For single-qubit states, possible transformations via real operations are fully characterized by Eqs. (4). The proof of this result makes use of methods developed earlier within the resource theory of quantum coherence Streltsov et al. (2017b); Chitambar and Gour (2016a, b), and more details can be found in the Supplemental Material. In Fig. 3 we show the --projection of the accessible region for three different initial states. The complete region can be obtained by rotation around the -axis.
Role of real operations in optical experiments
In standard linear optics using the polarization and path degrees of freedom of photons, real operations can be implemented more economically, compared to general quantum operations. We begin with a simple observation, that implementing a general unitary on photon polarization requires to control at least wave plates (this is due to a qubit unitary being specified by parameters), whereas only one half-wave plate is needed if the unitary has only real components, e.g., rotation about the -axis. As pointed out before, when restricting the optical elements to half (quarter)-wave plates, a rotation about the -axis needs two additional quarter-wave plates compared to a rotation about the -axis. This observation is the first evidence that the set of real operations is potentially easier to implement in terms of the number of optical elements, compared to the set of complex quantum operations.
We then consider single-qubit measurement with outcomes. As illustrated in Fig. 4, any such measurement can be implemented with unset wave plates. If , we have two Kraus operators and with
(20) |
By singular value decomposition, there are unitaries and such that , and are diagonal matrices with nonnegative entries. By Eq. (20) we obtain
(21) |
which implies that and . In summary, a general two-outcome measurement can be performed by applying a unitary , followed by a two-outcome measurement with diagonal Kraus operators and , and — depending on the measurement outcome — completed by a conditional unitary or . A setup realizing this procedure on photon polarization is shown in Fig. 4.
The unitaries , , and on the polarization-encoded qubit can be realized by wave plates per unitary, while the measurement with diagonal Kraus operators can be realized with beam displacers and wave plates, of which 2 are unset. This amounts to unset wave plates in total. By using the same procedure repeatedly, this setup can be extended to Kraus operators, see also Ahnert and Payne (2005). For each additional Kraus operator we need unset wave plates, giving unset wave plates in total, as claimed.
If all Kraus operators are real, fewer wave plates are needed. This can be seen from the fact that the singular value decomposition of each can be done with real and . Thus, a real measurement with two outcomes can be implemented with unset wave plates, and each additional real Kraus operator requires additional wave plates, see also Fig. 4. The number is optimal, since it corresponds to the number of independent real parameters for real Kraus operators. Compared to unset wave plates for a general -outcome measurement via the method presented above, in the limit we can save approximately half of the optical elements if we restrict ourselves to real measurements.
For implementing a real operation of arbitrary dimension, fewer optical elements are needed compared with the corresponding number for a general quantum operation, due to the upper bound of number of parameters to specify these operations. Note that every real operation acting on a system of dimension has a real dilation Hickey and Gour (2018):
(22) |
where is a real orthogonal matrix. Correspondingly, a general quantum operation admits a dilation with a general unitary matrix. Implementing an unitary in path degree requires at least unset wave plates, corresponding to the number of real parameters of the unitary. On the other hand, a real orthogonal matrix can be decomposed into real orthogonal matrices, each acting on two levels. Since a real orthogonal two-level matrix can be implemented with a single wave plate, any real orthogonal matrix can be implemented by using unset wave plates. Thus, implementing a real operation can be achieved with unset wave plates. Instead, implementing a general quantum operation in the same way requires at least unset wave plates. For large , restricting ourselves to real operations reduces the number of unset wave plates by , when compared to the number of wave plates for a general quantum operations implemented via a unitary dilation.
Imaginarity in local state discrimination
For two mixed states and to be perfectly distinguishable via LOCC, there must exist a POVM with elements of the form
(23) |
with Hermitian and , and moreover
(24a) | ||||
(24b) |
Correspondingly, if the states are distinguishable via LRCC, and must be real and symmetric.
The states in Eq. (11) are distinguishable via LOCC if Alice and Bob perform local measurements in the basis and share their measurement outcomes via a classical channel. The corresponding POVM elements are given as
(25a) | ||||
(25b) |
We verify that Eqs. (24) are satisfied, implying that this POVM perfectly disctiminates the states (11). As is explained in the main text, the states (11) cannot be distriminated via LRCC with any non-zero probability.
Acknowledgments
T.V.K., S.R., and A.S. acknowledge financial support by the “Quantum Optical Technologies” project, carried out within the International Research Agendas programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund. C.M.S. acknowledges the hospitality of the Centre for Quantum Optical Technologies at the University of Warsaw, and financial support by the Pacific Institute for the Mathematical Sciences (PIMS) and a Faculty of Science Grand Challenge award at the University of Calgary.
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Supplemental Material
.1 Proof of Eqs. (4)
Here we will study deterministic state conversion via real operations, resorting to the large amount of tools developed within the resource theory of coherence Streltsov et al. (2017a). To use this analogy in an optimal way, we introduce a new set of operations, which we term --preserving operations and denote by . They correspond to single-qubit quantum operations which map the - plane of the Bloch space onto itself, i.e., if a state has a Bloch vector in the - plane, then also has this property. In the same way, --preserving operations map the set of real states onto itself. Similarly, -preserving operations map diagonal states onto diagonal states, thus corresponding to maximally incoherent operations (MIO) Aberg (2006).
In the following we will prove two lemmas, thus demonstrating a close relation between the resource theories of coherence and imaginarity.
Lemma 1.
Let and be qubit states with Bloch vectors in the - plane. If there exists a --preserving operation such that , there also exists a -preserving operation such that .
Proof.
Since is - preserving, it converts both states and into states and with Bloch vectors in the - plane, i.e., and have purely imaginary off-diagonal elements. This implies that any convex combination of and is also converted into a state with purely imaginary off-diagonal elements.
Let now be the Kraus operators of . We introduce another transformation
(26) |
with Kraus operators . It is straightforward to verify that is indeed a valid set of Kraus operators:
(27) |
Moreover, when applied to any state in the - plane, we obtain
(28) |
where in the last step we used the fact that is Hermitian. It follows that
(29a) | ||||
(29b) | ||||
(29c) |
In the next step, we introduce the transformation
(30) |
Recalling that the states and have purely imaginary off-diagonal elements, we further obtain
(31a) | ||||
(31b) | ||||
(31c) |
This implies that is a -preserving operation transforming onto . ∎
In the next step, we will use Lemma 1 to characterize the set of real states achievable from a given real state via --preserving operations.
Lemma 2.
Let and be qubit states in the - plane of the Bloch sphere. Then, there exists a --preserving operation such that if and only if
(32) | ||||
(33) |
where and denote the Bloch vectors of and , respectively.
Proof.
We will first prove that a --preserving operation violating Eq. (32) and/or Eq. (33) does not exist. Assume – by contradiction – that there exists a --preserving operation violating Eq. (32) and/or Eq. (33). Then, by Lemma 1 there must also exist a -preserving (i.e. MIO) operation such that . Such a transformation does not exist due to results in Streltsov et al. (2017b); Chitambar and Gour (2016a, b).
We will now show that a --preserving operation exists if Eqs. (32) and (33) are fulfilled. Note that and any rotation around the -axis are --preserving operations. Thus, we can restrict ourselves to the positive part of the Bloch space, i.e., all Bloch coordinates considered in the following are non-negative. Moreover, we are interested in the boundary of the achievable region, characterized by the maximal for a given .
If , Eq. (33) guarantees that Eq. (32) is satisfied. A --preserving operation fulfilling Eq. (33) with equality is given by the Kraus operators
(34) |
where the parameters and are chosen as
(35a) | ||||
(35b) |
with and parameter is in the range . By varying it is possible to attain any value for in the range . This proves that for the boundary of the achievable region is characterzied by Eq. (33).
Equipped with these results, we are now ready to prove Eqs. (4) of the main text. Since rotations around the -axis correspond to real unitaries, we can without loss of generality assume that the initial and the final state have Bloch vectors in the - plane. It is thus enough to prove the statement for
(37a) | ||||
(37b) |
The proof of the theorem now directly follows from Lemma 2 by symmetry, exchanging the and directions.
.2 Proof of Theorem 1
The proof will use properties of pure states within imaginarity theory (see Section .4) and geometric imaginarity (see Section .5). Since is a strong imaginarity monotone, the transition probability is bounded as [see also Eq. (17) in the main text]:
(38) |
In the case of pure states we can use the results from Section .5 to obtain
(39) |
We will now consider the case
(40) |
and show that there exists a real operation saturating the bound (39). To see this, we first apply a real orthogonal transformation to the state , bringing it into the form
(41) |
see Section .4. Then, we apply a real operation with the Kraus operators
(42) |
where is defined as
(43) |
Note that by Eq. (40). As it can be verified by inspection, the Kraus operator transforms into the state
(44) |
with probability
(45) |
Note that is equivalent to the desired state up to a real orthogonal transformation, see Section .4.
For the remaining case , the transformation can be achieved with unit probability Hickey and Gour (2018).
.3 Proof of Theorem 2
We will now prove that for any quantum state the fidelity of imaginarity can be written as
(46) |
Here, is the robustness of imaginarity, defined as
(47) |
where the minimum is taken over all quantum states .
From the definition of , we can write as
(48) |
with some quantum state and a real state . By applying a real operation on both sides we obtain
(49) |
Since is a real operation, we have . Noting that for any real state it holds that , we have
(50) |
which proves the bound
(51) |
We will now show that this bound is achievable by a real operation . Given a general quantum state, it is always possible to decompose it as Hickey and Gour (2018)
(52) |
where is a real quantum state and is a real anti-symmetric matrix. By spectral theorem has an even rank and there is a real orthogonal matrix such that is block-diagonal (Horn and Johnson, 2012, p. 136):
(53) |
where .
If the dimension is even, we now define a real operation via the real Kraus operators
(54) |
For odd dimension, the Kraus operators are defined in the same way for , and we further define
(55) |
Let now be a real orthogonal matrix such that is block-diagonal as in Eq. (53). We see that is a real single-qubit state, which implies that
(56) |
Moreover, we have
(57) |
The fidelity of the final state with the maximally imaginary state can now be evaluated as follows:
(58) | ||||
To complete the proof, note that the robustness of imaginarity can be expressed as (see Section .6)
(59) |
Using this result in Eq. (58), we obtain
(60) |
In summary, we proved that the fidelity is upper-bounded by Eq. (51), and that this upper bound is achievable for any state with a suitably chosen real operation . This completes the proof of the theorem.
.4 Pure states in imaginarity theory
We will now show that in the resource theory of imaginarity any pure state can be expressed in a simple form. Recalling that the definition of imaginarity is basis dependent, we define complex conjugation of a state as follows:
(61) |
where is the reference basis, and . The states and can also be expressed as
(62a) | ||||
(62b) |
where and are real numbers with , and are real states. Equipped with these tools, we are now ready to prove the following proposition.
Proposition 1.
For any pure state there exists a real orthogonal matrix such that
(63) |
Proof.
In the first step, note that for any two real states and there exists a real orthogonal matrix such that
(64) | ||||
(65) |
where . Applying to the state gives us
(66) |
Since the state is effectively a single-qubit state, we can associate a Bloch vector with it, with coordinates
(67) | |||
Let now be a real orthogonal transformation, such that the Bloch vector of the state is in the positive - plane. Since , we can give the coordinates of as follows: