Optimal Measurements for Quantum Fidelity between Gaussian States
Abstract
Quantum fidelity is a measure to quantify the closeness of two quantum states. In an operational sense, it is defined as the minimal overlap between the probability distributions of measurement outcomes and the minimum is taken over all possible positiveoperator valued measures (POVMs). Quantum fidelity has been investigated in various scientific fields, but the identification of associated optimal measurements has often been overlooked despite its great importance for practical purposes. We find here the optimal POVMs for quantum fidelity between multimode Gaussian states in a closed analytical form. Our general finding is specified for selected singlemode Gaussian states of particular interest and we identify three types of optimal measurements: a numberresolving detection, a projection on the eigenbasis of operator , and a quadrature detection, each of which is applied to distinct types of singlemode Gaussian states. We also show the equivalence between optimal measurements for quantum fidelity and those for quantum parameter estimation, enabling one to easily find the optimal measurements for displacement, phase, squeezing, and loss parameter estimations using Gaussian states.
I introduction
Quantification of the similarity of quantum states is of utmost importance in quantum information processing such as quantum error correction and quantum communication nielsen2000 (); wilde2017 (); weedbrook2012 (); braunstein2005 (). There are various measures of the closeness of two quantum states such as trace distance helstrom1976 (), quantum Chernoff bound audenaert2007 (); audenaert2008 (), and quantum relative entropy vedral2002 (). Among the diverse measures, one of the most common measures is quantum fidelity uhlmann1976 (). Theoretically, it is defined as the minimal overlap of the probability distributions obtained by an optimal positiveoperator valued measure (POVM) performed on two states. It has also been widely employed to verify how close actual states are to target states in experiments leibfried2004 (); lu2007 (); ourjoumtsev2007 (), practically assessing quantum information processing protocols such as quantum teleportation bennett1993 (); bouwmeester1997 (); braunstein1998 (); furusawa1998 () and quantum cloning buzek1996 (); lindblad2000 (); cerf2000 (); braunstein2001 (); fiurasek2001 (). It has been known that the quantum fidelity not only plays a crucial role in quantum parameter estimation helstrom1976 (); braunstein1994 (), but also sets a bound for quantum hypothesis testing helstrom1976 (); fuchs1999 () and quantum Chernoff bound audenaert2007 (); audenaert2008 ().
In general, quantum fidelity can be measured in two different, but equivalent ways in an experiment: One from the full knowledge of two quantum states, and the other from the probability distributions obtained by an optimal POVM. The first approach is experimentally very demanding due to the requirement of full state tomography, which necessitates a number of measurement settings and computationally laborious postprocessing for highdimensional states. The second approach, on the other hand, requires one to measure just the probability distributions with an optimally chosen POVM. The latter is thus more preferred and illustrated in Fig. 1. The experimental evaluation of quantum fidelity is straightforwardly attainable as long as the optimal measurement is known and experimentally implementable. One could employ alternative approaches that have been proposed to directly measure quantum fidelity between two quantum states filip2002 (); ekert2002 (); hendrych2003 (); bartkiewicz2013 (), but they are not universal to systems and even require an interaction between the states to be involved. Therefore, finding optimal measurements for quantum fidelity offers the simplest way to efficiently measure quantum fidelity.
One useful platform for quantum information processing is continuous variable systems, such as optical fields with indefinite photon numbers weedbrook2012 (). Especially, bosonic Gaussian states are practical resources because they are relatively less demanding to generate and manipulate in experiments ferraro2005 (); wang2007 (); weedbrook2012 (); adesso2014 (); serafini2017 (). Due to the importance of quantum fidelity between Gaussian states, there have been numerous attempts to find the analytical formula between constrained Gaussian states twamley1996 (); nha2005 (); olivares2006 (); scutaru1998 (); marian2012 (); marian2003 (); marian2008 (); spedalieri2013 (); paraoanu2000 (); banchi2015 (), but only recently two arbitrary Gaussian states have been implemented in a computable analytical formula of quantum fidelity banchi2015 (). One may experimentally measure the quantum fidelity between Gaussian states with the optimal POVM, but the optimal measurement setting has not yet been found although a general method of finding the optimal measurement for two given quantum states is known fuchs1995 ().
In this work, we find the optimal POVMs, in a closed analytical form, enabling to measure quantum fidelity between two multimode Gaussian states. Such general form of optimal POVMs allows us to classify optimal measurements for quantum fidelity between two singlemode Gaussian states of particular interest. In addition, we demonstrate the equivalence between optimal measurements for quantum fidelity and those for quantum Fisher information, upon which we discuss quantum parameter estimation in the context of singlemode Gaussian metrology, such as displacement estimation pinel2013 (); safranek2016 (), phase estimation pinel2013 (); safranek2016 (); oh2018 (), squeezing parameter estimation pinel2013 (); safranek2016 (), and loss parameter estimation monras2007 (); pinel2013 ().
Ii Optimal POVM for quantum fidelity
Let us consider two distinct probability distributions and with possible outcomes . One notable measure of statistical distinguishability of these distributions is the Bhattacharyya coefficient fuchs1995 (); bhattacharyya1943 (); wilde2017 (),
This quantity takes the maximum value of 1 if and only if two given probability distributions are equivalent, i.e., for all possible outcomes . This notion of distinguishability has been extended to quantum regime by minimizing over all possible POVMs {} performed to two given states and such that
Here the probability distributions are obtained by performing a given POVM , satisfying , on two states. The quantum fidelity reduces to a simpler form as uhlmann1976 ()
From the definition of quantum fidelity, it is obvious that finding the optimal POVM is crucial to maximally distinguish two given quantum states. It has been found that the optimal measurements have to satisfy
(1)  
(2) 
where is a unitary operator satisfying and is a constant fuchs1995 (). In the case of fullrank states and , the optimal measurement is unique and consists of projections onto the eigenbasis of a Hermitian operator, written by
(3) 
Thus, simplifying the operator to find its eigenbasis is the central task to determine the optimal measurement. We note a simple property of the operator ,
(4) 
where is a unitary operator.
Iii Optimal measurements for multimode Gaussian states
Let us consider bosonic modes described by quadrature operators which satisfy the canonical commutation relations arvind1995 ()
where is the identity matrix. Transformations of coordinates that preserve the canonical commutation relation can be represented by symplectic transformation matrices such that .
Gaussian states are a special class of continuous variables states. They are defined as the states whose Wigner function is a Gaussian distribution ferraro2005 (); wang2007 (); weedbrook2012 (); adesso2014 (); serafini2017 (). It is known that an arbitrary Gaussian state can be written in the Gibbsexponential form as banchi2015 (),
(5) 
where is the first moment vector, is the Gibbs matrix defined as with the covariance matrix , and is a normalization factor which we omit throughout this work for convenience. The Gibbsexponential form of Eq. (5) makes it easy to deal with the square root of density matrices.
After some algebra (see Appendix A for the detail), we find that the operator takes the exponential form, written up to an unimportant normalization factor as
(6) 
where the matrix is the solution of the equation
(7) 
and is the displacement operator, is a real vector, which can be explicitly expressed for particular cases as below. For equal covariance matrices , Eq. (7) has a trivial solution , allowing Eq. (6) to take a simpler form where . The eigenbasis of the operator is thus that of a quadrature operator followed by a unitary operator , which is also that of a quadrature operator. For , on the other hand, one can write
(8) 
where . The expression of is provided in Appendix A. Note that for equal displacements (). When and are diagonalized by the same symplectic matrix , all modes of the states can be completely decoupled to be a product of singlemode states by applying a unitary operation corresponding to . We thus investigate singlemode cases more intensively in the following section.
It is known that the Gibbs matrices are singular when symplectic eigenvalues of the covariance matrix are equal to banchi2015 (). The continuity of the above expression enables the singular case to be treated as a limiting case. To this end, we replace the singular symplectic eigenvalues by with a small positive , by which Eq. (7) is welldefined as
(9) 
In the limit , the unique solution of the above expression gives rise to the optimal measurements. It is worth emphasizing that when rankdeficient states are involved, optimal measurements may not be unique.
Iv Optimal measurements for singlemode Gaussian states
Any singlemode Gaussian state can be written as
where is a thermal state with the average number of thermal quanta , and is a squeezing operator with a squeezing parameter . Note that when , the Gibbs matrix is written as
Let us consider two singlemode Gaussian states characterized by and . With introducing a symplectic matrix that diagonalizes such that with banchi2015 (), the twoGaussian states can be written in a more compact way as
where is a unitary operator satisfying . Thus, without loss of generality, the above symplectic transformation simplifies the initial problem of distinguishing between two arbitrary Gaussian states, so that we focus on distinguishing between one squeezed state and a thermal state, up to Gaussian unitary operations and . Furthermore, since thermal states are invariant under rotation, we can further simplify the problem to one between a thermal state and a squeezed state along or direction. In order to do that we introduce a rotation operator and a corresponding orthogonal matrix defined such that and , so that the two singlemode Gaussian states can be decomposed as
where . Since the quantum fidelity is invariant under unitary operations by definition, without loss of generality, the matrix for arbitrary two singlemode states can be expressed by for a general Gaussian state that is squeezed by a squeezing parameter along or axis, , and a thermal state, , under a transformation by . Such simplification enables the matrix to take the form of
(10) 
Consider the case that and are fullrank states, i.e., . For the states with , one can easily show that where and its eigenbasis is that of a quadrature operator, as in the multimode case. When , on the other hand, the operator of Eq. (10) can be expressed as
(11) 
for which Eqs. (6) and (8) are taken into account. Here, is obtained by solving Eq. (7) with replaced by . Let us now simplify the matrix of Eq. (11) and find its eigenbasis.
The effect of the first moments and is contained in the displacement vector whose full expression is shown in Appendix A. The crucial step to obtain the optimal measurements is thus the diagonalization of the operator . From the form of , one can see that the eigenbasis of is classified by the signs of the eigenvalues, and , of .

If the signs of eigenvalues of are the same (), i.e., is positivedefinite or negativedefinite, the eigenbasis of is that of the number operator followed by Gaussian unitary operations including and a squeezing operation that makes the magnitude of eigenvalues same.

If the signs of eigenvalues are different (), the eigenbasis of is that of followed by Gaussian unitary operations.

If only one of the eigenvalues is zero (, but ), the eigenbasis of is that of a quadrature operator along a certain direction.
In summary, once the signs of the eigenvalues of are known, the optimal measurement can be determined by the above classification. It can also be represented as a function of and for a given , as shown in Fig. 2, where the regions are distinguished by the spectrum of the matrix (see Appendix B to get the spectrum).
It is worth discussing special cases, when each type is optimal. First, consider the case that when is also a thermal state, so that , and and are diagonalized by the same symplectic transformation. In this case, Eq. (7) leads to , and the eigenbasis of is the number basis followed by and . Hence, type(i) is optimal. This result can also be inferred by the fact that the same unitary operation diagonalizes the both states into thermal states, and their eigenbasis is the number state. Second, consider the case when and has distinct eigenvalues, i.e., is a squeezed state. It renders the signs of and being different regardless of and , i.e., type(ii) is optimal. Third, consider the case that either of or is zero. When , Eq. (7) has a solution only when
(12) 
and the operator is simply written as . Thus, type(iii) with the quadrature measurement of is optimal, reproducing the same results in Ref. nha2005, . Similarly, when , Eq. (7) has a solution only when
(13) 
and type(iii) with the quadrature measurement of is optimal.
Now consider the case of rankdeficient Gaussian states. Since all rankdeficient Gaussian states are a pure state and the inverse of a pure state does not exist, of Eq. (3) needs to be treated with care. Assuming is a pure state without loss of generality, one can write the operator of Eq. (3) with projecting and into the support of , where the inverse can be defined, as wilde2017 ()
where is the projector onto the support of . For and consequently , it is therefore clear that . The same result can also be derived by considering pure states as a limiting case of zerotemperature (see Appendix C for the detail). Thus, an optimal POVM set is , and can be implemented by applying the Gaussian unitary transformation that transforms to a vacuum state followed by performing on/off detection. It is worth emphasizing again that the optimal measurement offered by the operator when pure states are involved is not unique, so that the suggested setup is merely one of the optimal measurements, all satisfying the conditions of Eqs. (1) and (2).
V Optimal POVM for quantum Fisher information
Quantum parameter estimation is an informational task to estimate an unknown parameter of interest by using quantum systems helstrom1976 (). In a standard scenario of quantum parameter estimation, independent copies of quantum states that contain information about the unknown parameter are measured by a POVM, and the estimation is performed by manipulating the measurement data. The ultimate precision bound of the estimation is governed by quantum CramérRao inequality, stating that the mean square error of any unbiased estimator is lowerbounded by the inverse of quantum Fisher information multiplied by the number of copies helstrom1976 (). Thus, quantum Fisher information is the most crucial quantity which determines the ultimate precision of estimation braunstein1994 (), which is written as
where is the symmetric logarithmic derivative (SLD) operator satisfying .
The quantum Fisher information can be written in terms of quantum fidelity as banchi2015 ()
It implies that quantum parameter estimation is related to distinguishing two infinitesimally close states and . Indeed, similar to the quantum fidelity, quantum Fisher information is defined as the maximal classical Fisher information over all possible POVMs, and the optimal POVM has to satisfy braunstein1994 ()
(14)  
(15) 
It is known that the projection onto the eigenbasis of is the optimal measurement for quantum Fisher information braunstein1998 (). This means that the SLD operator plays the same role as the operator does for quantum fidelity. We prove that the above conditions are indeed equivalent to the conditions of Eqs. (1) and (2), resulting in the relation /2 for infinitesimal (see Appendix D for the proof). This indicates that the optimal POVM for quantum fidelity between and offers the optimal measurement for quantum parameter estimation that reaches the quantum Fisher information.
Especially for Gaussian states, since the matrix and the vector are infinitesimal for and and thus
the SLD operator is simply written as
(16) 
where can be determined from . Taking an infinitesimal limit in Eq. (7), one can show that for an infinitesimal is the solution of
(17) 
and is formally written in a basisindependent form as
(18) 
and . Here and are the first moment vector and the covariance matrix of , respectively, and . The derivation of and is provided in Appendix E. The relation of and the SLD operator and the expression of and enable to find SLD operators directly from the operator . Finally, from the SLD operator one can easily derive the expression of the quantum Fisher information:
(19) 
The derivation is provided in Appendix D. As a remark, note that the expressions of , and quantum Fisher information are equivalent to those found in Refs. serafini2017, ; jiang2014, , but our derivation based on quantum fidelity is significantly simpler and straightforward. Furthermore, by replacing a singleparameter by a multiparameter and defining the SLD operators by , the expression of quantum Fisher information matrix can be easily derived by using a similar method nichols2018 (); safranek2019 ().
In the following subsections, we find optimal measurements for displacement, phase, squeezing, and loss parameter estimation in relation to our results for quantum fidelity.
v.1 Displacement parameter estimation
For a singlemode Gaussian probe state , the displacement operation only changes the first moment while keeping the second moments fixed:
where is assumed without loss of generality. Therefore, the first moment vectors and the covariance matrices of and are related as
respectively. Since the covariance matrix is invariant, one can immediately see that the optimal measurement for quantum fidelity between and is type(iii), so that the optimal measurement for estimation of the displacement parameter is also type(iii). Explicitly, using the expression of , one can easily obtain the SLD operator and quantum Fisher information:
v.2 Phase parameter estimation
Let us consider a singlemode Gaussian probe state that undergoes a phase shifter with a phase parameter to be estimated. Since the displacement operation performed to the probe state does not change the type of optimal measurement, we focus on only the state with zeromean for simplicity, i.e.,
where is a rotation operator. The relevant states under investigation are and , but the full expressions with an arbitrary angle get involved without altering the type of optimal measurement. We thus consider the states and at , and assume to be the squeezed thermal state and is a rotated squeezed thermal state without loss of generality. Let us proceed with and first, and then take the limit at the end. The covariance matrices of and are respectively written as
where the proportionality becomes an equality when adding a prefactor of . Since the average numbers of thermal quanta are the same between the above two states, one may immediately infer that the optimal measurement is type(ii). Let us see if this is indeed the case. For the states and , it can be shown that
where a constant is given such that . The matrix satisfies Eq. (7), and indicates that the optimal measurement for quantum fidelity between and is type(iii). To apply this to quantum Fisher information, we take the limit , resulting in
Hence,
(20) 
where is the SLD operator in phase estimation oh2018 (). This reveals that the operators and have the common eigenbasis. It is now clear that the optimal measurement for phase parameter estimation is type(ii), as also recently found via the SLD operator in Ref. oh2018, . Also note that while the above result is derived by an explicit optimal measurement for quantum fidelity, the same result can be easily derived by using Eq. (18).
v.3 Squeezing parameter estimation
We consider squeezing parameter estimation with an arbitrary Gaussian state as a probe state,
where we assume for simplicity. It corresponds to the case when we estimate the strength of squeezing parameter along the axis. Since that and have different squeezing parameters under the same average number of thermal quanta, just like the case of phase estimation, the optimal measurement is type(ii). Indeed, one can derive the SLD operator using Eq. (18),
which is clearly type(ii) because the signs of eigenvalues of are different. Quantum Fisher information can also be easily obtained
v.4 Loss parameter estimation
Consider a singlemode Gaussian probe state that undergoes a phaseinsensitive loss channel, and the dynamics of the state is described by the quantum master equation as
(21) 
where is the annihilation operator and is the loss rate to be estimated. The solution of the above differential equation for a singlemode Gaussian probe state can be given in terms of the first moment vector and the covariance matrix as ferraro2005 ()
Note that the dynamics of the covariance matrix does not change the symplectic transformation diagonalizing the covariance matrix. It is thus clear that the optimal parameter for quantum fidelity between and is type(i), so the optimal measurement for loss parameter estimation is also type(i). Specifically, one can easily obtain that
where we have defined and and zeromean input states are assumed for simplicity. The matrix is obviously negativedefinite; thus it corresponds to type(i). This reproduces the result in Ref. monras2007, ; pinel2013, . The optimality of type(i) holds also for other phaseinsensitive loss parameter estimations as long as the symplectic matrix that diagonalizes the covariance matrix does not change.
Vi Conclusions
We have found the optimal POVMs for quantum fidelity between two multimode Gaussian states in a closed analytical form. The full generality of our result has allowed us to further elaborate on the case of singlemode Gaussian states in depth. We have demonstrated that there exist only three different types of optimal measurements, along with Gaussian operations including a unitary operation that transforms and to squeezed states along either or and a thermal state, respectively, and that arises due to the difference in displacements. The number counting measurement is optimal when the covariance matrices of the states are diagonalized by the same symplectic matrix, while the projection onto the eigenbasis of is optimal when the average number of thermal quanta of two quantum states is the same. Optimality of quadrature measurement holds for two cases: when the covariance matrices are the same or when two Gaussian states satisfy the conditions of Eqs. (12) and (13). We have also applied our results to various parameter estimation scenarios in Gaussian metrology. We have proven the equivalence between the optimal measurement for quantum fidelity and that for quantum Fisher information, enabling to readily derive optimal measurements for quantum parameter estimation. We expect our approach to pave the way to further investigate the quantum parameter estimation.
While the number resolving detection and the quadrature measurement are experimentally feasible with current technology, the measurement setup projecting onto the eigenbasis of the POVM is not yet known. We hope that an appropriate measurement setup will be constructed in the near future in response to the significance arising from this work and the recent study for phase estimation oh2018 (). We also leave further classification of optimal measurements for multimode Gaussian states as future work, which can be made straightforwardly from our results at the expense of increased complexity.
Acknowledgements.
C.O. and H.J. are supported by a National Research Foundation of Korea grant funded by the Korea government (MSIP) (No. 20100018295) and by the KIST Institutional Program (No. 2E2780018P043). L.B. was supported by the UK EPSRC grant EP/K034480/1. S.Y.L. is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2018R1D1A1B07048633).Appendix
Appendix A Simplification of the operator
Here, we simplify the operator with and . Note that with , which is frequently used in this section. Simplifying in the following way,
with , one can have