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the probabilistic nature of quantum mechanics imposes fundamental limits to hypothesis testing and parameter estimation @xcite . such limits are relevant to many metrological applications , such as optical interferometry , optomechanical sensing , gravitational - wave detection @xcite , optical imaging @xcite , magnetometry , gyroscopy , and atomic clocks @xcite . the ultimate quantum limits to parameter estimation have been studied extensively in recent years , as they imply that a minimum amount of resource , such as the average photon number for optical phase estimation , is needed to achieve a desired precision , regardless of the measurement method .    for the measurement of a single optical phase parameter , the ultimate quantum limit to the mean - square error scales as @xmath4 , where @xmath5 is the average photon number of the field which undergoes that phase shift . this scaling is often called the heisenberg limit . after years of speculation and debate @xcite , the heisenberg limit for single - parameter linear phase estimation has only recently been proven @xcite . although decoherence , such as optical loss and dephasing , can impose stricter limitations @xcite , the heisenberg limit is a more fundamental bound and will be increasingly relevant as quantum technologies continue to improve and decoherence effects are further reduced . many real - world tasks , such as optical imaging @xcite , quantum tomography and system identification @xcite , and waveform estimation ( e.g.  estimating a signal that varies continuously in time ) @xcite , require the estimation of multiple parameters . multiparameter quantum cramr - rao bounds have been known since the 1970s @xcite , but efforts to derive multiparameter heisenberg limits from these bounds have not been successful . this is not surprising , since even in the case of single - parameter phase estimation , it is not possible , without additional assumptions on the state , to derive the heisenberg limit from the quantum cramr - rao bound . ( the latter gives a lower bound on the mean - square error of @xmath6 , which does not imply the heisenberg limit of @xmath4 , as can be seen from the state @xmath7 which has @xmath8 but divergent @xmath9 . )    in ref . @xcite , some of us recently proposed a heisenberg - style limit for the estimation of an optical phase waveform with stationary gaussian prior statistics and a power - law spectrum . however , that limit , being derived from a quantum cramr - rao bound , requires additional assumptions : it applies only to the specific class of optical beams described by gaussian fields , with statistics that are both stationary and time - symmetric . a very different approach was that of ref . @xcite , which derives a multiparameter heisenberg limit for independent parameters by applying the single - parameter heisenberg limit to each parameter . in practice , multiple parameters often have nontrivial prior correlations , particularly in the case of continuous waveform estimation , where the correlations are crucial to pose the problem @xcite . thus the existence of general heisenberg limits for such cases has remained an open question .    in this paper , we derive new quantum bounds on multiparameter estimation by developing quantum versions of the bell - ziv - zakai bounds . the bell - ziv - zakai bounds were proposed in 1997 by bell _ et al . _ @xcite , building upon the ziv - zakai bound @xcite , and futher generalized by basu and bresler @xcite . we then apply our bounds to the notable task of quantum optical phase waveform estimation . here , the waveform to be estimated is a time - varying phase shift signal , @xmath10 , applied to an optical beam . for a waveform @xmath10 with stationary gaussian prior statistics and a power - law spectrum ( @xmath11 ) , we prove a lower bound on the mean - square error with a @xmath2 scaling , where @xmath3 is the mean photon flux . this proof confirms that the scaling previously proposed in ref . @xcite is valid for arbitrary quantum states . moreover , we show that this scaling is achievable for all @xmath1 . previously , achievability has been shown only numerically , and only for @xmath12  @xcite . by contrast the results in the current paper are completely rigorous heisenberg bounds , being both applicable to arbitrary field states , and achievable , for all @xmath1 . this paper is separated into two main parts . the first part , sec . [ qzzb ] to sec . [ sec : ach ] , assumes unbounded parameters and focuses on the mean - square error as the distortion measure ( i.e.  the figure of merit for the accuracy of the estimation ) . the second part , sec . [ sec : per ] and sec . [ sec : ach2 ] , focuses on periodic distortion functions , which are more appropriate for periodic parameters such as phase or orientation angles for gyroscopy . they are also insensitive to phase - wrap errors and enable us to rigorously prove that our bounds are achievable . first we summarize known results for the classical estimation problem , then present quantum versions of the bounds in sec . [ sec : qbzzb ] . let @xmath13 be a column vector of unknown real parameters , @xmath14 be the prior probability density , and @xmath15 be the likelihood function with observation @xmath16 . both @xmath13 and @xmath16 are random variables . note that @xmath16 need not be the same dimension as @xmath13 . further , let @xmath17 be the estimator of @xmath13 from @xmath16 . ( we use @xmath18 rather than @xmath19 , as is common in statistics , to avoid possible confusion with quantum operators . ) we also define the error vector as @xmath20 to characterise the performance of the estimate , we consider a distortion function of the form @xmath21 , where @xmath22 is a given but arbitrary real column vector that defines the error components of interest , and @xmath23 denotes the transpose . for example , the mean - square error for a particular component @xmath24 is the expected value of a distortion function @xmath25 with @xmath26 , so @xmath27}^2 . \label{point}\ ] ] suppose that the distortion function is symmetric [ that is , @xmath28 , nondecreasing on @xmath29 , differentiable , and has @xmath30 . then the expected distortion is , from eq . ( 44 ) of ref . @xcite , @xmath31 } & = \frac{1}{2 } \int_0^\infty d\tau \ , \dot d{\left(\frac{\tau}{2}\right ) } \textrm{pr}{\left(|{\boldsymbol{u}}^\top{\boldsymbol{\epsilon}}| \ge \frac{\tau}{2}\right ) } , \label{d_bound}\end{aligned}\ ] ] where @xmath32 is the derivative of @xmath33 , @xmath34 denotes expectation over @xmath13 and @xmath16 , and pr is the probability for the boolean function of these random variables to be true . for a general mean - square error criterion , the expected distortion can be expressed in terms of the error covariance matrix @xmath35 as @xmath31 } & =    { \mathbb e}{\left[({\boldsymbol{u}}^\top{\boldsymbol{\epsilon}})^2\right ] }    = { \boldsymbol{u}}^\top{\boldsymbol{\sigma}}{\boldsymbol{u } } , { \nonumber \\}{\boldsymbol{\sigma } } & : = { \mathbb e}{\left({\boldsymbol{\epsilon}}{\boldsymbol{\epsilon}}^\top\right)}. \label{quad_d}\end{aligned}\ ] ] since @xmath36 is assumed to be nonnegative , a lower bound on the expected distortion can be obtained by lower - bounding the probability @xmath37 . using eqs . ( 31 ) and ( 35 ) of @xcite to bound @xmath37 and noting property 1 of @xcite , yields the bell - ziv - zakai bounds @xcite : @xmath38 }   \ge   \frac{1}{2 } \int_0^\infty d\tau \ , \dot d{\left(\frac{\tau}{2}\right ) } \mathcal v\bigg\ { \max_{{\boldsymbol{v}}:{\boldsymbol{u}}^\top{\boldsymbol{v } } = 1 } \int d{\boldsymbol{x } } { \nonumber \\}&\quad\times { \left[p_{{\boldsymbol{x}}}({\boldsymbol{x}})+p_{{\boldsymbol{x}}}({\boldsymbol{x}}+{\boldsymbol{v}}\tau)\right ] } p_e({\boldsymbol{x}},{\boldsymbol{x}}+{\boldsymbol{v}}\tau)\bigg\ } , \label{bzzb1 } \\ & \ge \int_0^\infty d\tau \ , \dot d{\left(\frac{\tau}{2}\right ) } \mathcal v\bigg\{\max_{{\boldsymbol{v } } : { \boldsymbol{u}}^\top { \boldsymbol{v } } = 1 } \int d{\boldsymbol{x } } { \nonumber \\}&\quad\times \min{\left[p_{{\boldsymbol{x}}}({\boldsymbol{x}}),p_{{\boldsymbol{x}}}({\boldsymbol{x } } + { \boldsymbol{v}}\tau)\right ] } p_e^{\rm el}({\boldsymbol{x}},{\boldsymbol{x}}+{\boldsymbol{v}}\tau)\bigg\ } . \label{bzzb2}\end{aligned}\ ] ] here @xmath39 is the valley - filling function defined as @xmath40 and @xmath41 is the minimum error probability for the bayesian binary hypothesis testing problem with hypotheses defined as @xmath42 and @xmath43 , observation probability densities given by @xmath44 and @xmath45 , and prior probabilities given by @xmath46 to be explicit @xcite , @xmath47 @xmath48 is defined in the same way as @xmath49 except that the prior probabilities are equal ( @xmath50 ) .      for the quantum parameter estimation problem , let @xmath51 be the density operator that describes the state of a quantum probe as a function of the unknown parameter @xmath52 , and @xmath53 be the positive operator - valued measure ( povm ) that describes the measurement with outcome @xmath54 @xcite . the likelihood function becomes @xmath55,\end{aligned}\ ] ] with @xmath56 denoting the operator trace . it is known @xcite that , for any povm , @xmath57 } , \label{f_bound}\end{aligned}\ ] ] where @xmath58 is the uhlmann fidelity . equations   and , together with the first bell - ziv - zakai bound given by eq . ( [ bzzb1 ] ) , then give quantum lower bounds on the estimation error : @xmath38 }   \ge   \frac{1}{4 } \int_0^\infty d\tau \ , \dot d{\left(\frac{\tau}{2}\right ) } { \nonumber \\}&\quad\times \mathcal v\bigg\ { \max_{{\boldsymbol{v}}:{\boldsymbol{u}}^\top{\boldsymbol{v } } = 1 } \int d{\boldsymbol{x}}{\left[p_{{\boldsymbol{x}}}({\boldsymbol{x}})+p_{{\boldsymbol{x}}}({\boldsymbol{x}}+{\boldsymbol{v}}\tau)\right ] } { \nonumber \\}&\quad\times \left(1 - \big{|}\big{|}\pi_0\rho_{{\boldsymbol{x}}}- \pi_1\rho_{{\boldsymbol{x}}+{\boldsymbol{v}}\tau}\big{|}\big{|}_1\right ) \bigg\ } \label{qbzzb1 } \\ & \ge   \frac{1}{4 } \int_0^\infty d\tau \ , \dot d{\left(\frac{\tau}{2}\right ) } { \nonumber \\}&\quad\times \mathcal v\bigg\ { \max_{{\boldsymbol{v}}:{\boldsymbol{u}}^\top{\boldsymbol{v } } = 1 } \int d{\boldsymbol{x}}{\left[p_{{\boldsymbol{x}}}({\boldsymbol{x}})+p_{{\boldsymbol{x}}}({\boldsymbol{x}}+{\boldsymbol{v}}\tau)\right ] } { \nonumber \\}&\quad\times { \left[1 - \sqrt{1 - 4\pi_0\pi_1f({\boldsymbol{x}},{\boldsymbol{x}}+{\boldsymbol{v}}\tau)}\right ] } \bigg\}.\end{aligned}\ ] ] similarly , eq .  ( [ bzzb2 ] ) and quantum bounds on @xmath48 via eqs . ( [ trace_bound ] ) and ( [ f_bound ] ) lead to the lower bounds @xmath38 }   \ge \frac{1}{2 } \int_0^\infty d\tau \ , \dot d{\left(\frac{\tau}{2}\right ) } { \nonumber \\}&\quad\times \mathcal v\bigg\{\max_{{\boldsymbol{v } } : { \boldsymbol{u}}^\top { \boldsymbol{v } } = 1 } \int d{\boldsymbol{x } } \min{\left[p_{{\boldsymbol{x}}}({\boldsymbol{x}}),p_{{\boldsymbol{x}}}({\boldsymbol{x } } + { \boldsymbol{v}}\tau)\right ] } { \nonumber \\}&\quad\times \left(1 - \frac{1}{2}\big{|}\big{|}\rho_{{\boldsymbol{x}}}- \rho_{{\boldsymbol{x}}+{\boldsymbol{v}}\tau}\big{|}\big{|}_1\right ) \bigg\ } \\ & \ge \frac{1}{2 } \int_0^\infty d\tau \ , \dot d{\left(\frac{\tau}{2}\right ) } { \nonumber \\}&\quad\times \mathcal v\bigg\{\max_{{\boldsymbol{v } } : { \boldsymbol{u}}^\top { \boldsymbol{v } } = 1 } \int d{\boldsymbol{x } } \min{\left[p_{{\boldsymbol{x}}}({\boldsymbol{x}}),p_{{\boldsymbol{x}}}({\boldsymbol{x } } + { \boldsymbol{v}}\tau)\right ] } { \nonumber \\}&\quad\times { \left[1 - \sqrt{1-f({\boldsymbol{x}},{\boldsymbol{x}}+{\boldsymbol{v}}\tau)}\right ] } \bigg\}. \label{qbzzb4}\end{aligned}\ ] ] we call eqs . ( [ qbzzb1])([qbzzb4 ] ) the quantum bell - ziv - zakai bounds .    to derive further analytic results , we focus on the fidelity bound given by eq . ( [ qbzzb4 ] ) . it can be further simplified if @xmath59 does not depend on @xmath52 and the prior @xmath14 is a multivariate gaussian distribution . the integral with respect to @xmath52 then becomes , using eqs . ( a2 ) and ( a10 ) of ref . @xcite , @xmath60 } =   { \operatorname{erfc}}{\left(\frac{\tau}{\tau_0}\right ) } , \label{erfc}\end{aligned}\ ] ] where @xmath61 and @xmath62 is the prior covariance matrix . a convenient lower bound on the erfc function is @xmath63 where @xmath64 is the triangle function @xmath65 , as shown in fig . [ fig : erfc ] . may be entangled between the modes , and each mode passes through a phase shift @xmath66 . the output may be measured via some general joint measurement @xmath53.,scaledwidth=48.0% ]    we now consider the problem of phase estimation from the measurement of quantum optical modes , as illustrated in fig . [ phasemod ] . the output quantum state is @xmath67 where @xmath68 is the initial quantum state and @xmath69 is a column vector of photon number operators for the optical modes . we use a hat to distinguish the number operator from other uses of @xmath70 as an integer . we will not otherwise use a hat to indicate operators . purifying @xmath51 to @xmath71 , and taking the purification of @xmath72 to be @xmath73 , uhlmann s theorem @xcite yields a lower bound on @xmath74 given by @xcite @xmath75},\end{aligned}\ ] ] where we have defined @xmath76 , and @xmath77 is the photon - number distribution of the initial quantum state , with @xmath78 an eigenstate of @xmath69 .        to derive a bound on @xmath59 in terms of the average photon numbers , the following bound on cosine is useful : @xmath79 where @xmath80 is a solution of @xmath81 , as shown in fig . [ cosine_bound ] . this leads to @xmath82 } { \nonumber \\}&\ge   \sum_{{\boldsymbol{n}},{\boldsymbol{m}}}c({\boldsymbol{n}})c({\boldsymbol{m } } ) { \left[1-\lambda\tau{\left(|{\boldsymbol{v}}^\top{\boldsymbol{n}}|+|{\boldsymbol{v}}^\top{\boldsymbol{m}}|\right)}\right ] } { \nonumber \\}&\geq   \sum_{{\boldsymbol{n}},{\boldsymbol{m}}}c({\boldsymbol{n}})c({\boldsymbol{m } } ) { \left[1-\lambda\tau{\left(|{\boldsymbol{v}}|^\top{\boldsymbol{n}}+|{\boldsymbol{v}}|^\top{\boldsymbol{m}}\right)}\right ] } { \nonumber \\ } &   =    1 - 2\lambda\tau |{\boldsymbol{v}}|^\top{\left\langle\hat{{\boldsymbol{n}}}\right\rangle},\end{aligned}\ ] ] where @xmath83 means taking the absolute value of each element of @xmath84 . since @xmath85 , a tighter bound is @xmath86 a slightly tighter bound may be obtained using the method in refs . @xcite , but the scaling would remain the same .    focusing on the mean - square error , putting eqs . ( [ quad_d ] ) , ( [ qbzzb4 ] ) , ( [ erfc ] ) , ( [ erfc_bound ] ) , and ( [ speed_limit ] ) together , and using @xmath87 , @xmath88 the maximization of @xmath89 , subject to the constraint @xmath90 , gives the tightest bound , but it is difficult to perform analytically . in the next section , we shall focus on waveform estimation and discover that an appropriate choice of @xmath84 , though suboptimal , can still lead to a reasonably tight bound . we now consider phase modulation that varies in time , as illustrated in fig . [ waveform ] . define discrete time as @xmath91 @xmath92 being an integer . each parameter @xmath93 corresponds to a phase at time @xmath94 : @xmath95 and each photon - number operator @xmath96 is related to the photon - flux operator @xmath97 by @xmath98 other quantities are redefined as follows : @xmath99 in the continuous - time limit @xmath100 , the mean - square error becomes @xmath101 and the constraint in eq . ( [ qbzzb ] ) becomes @xmath102 to evaluate the bound @xmath89 given by eq . ( [ z ] ) in this limit , we need to compute @xmath103 given by eq . ( [ tau0 ] ) and @xmath104 given by eq . ( [ speed_limit ] ) . they depend on the following : @xmath105 is defined by @xmath106 assume now that the prior statistics of @xmath10 are stationary . this means that we can define a prior power spectral density @xmath107 such that @xmath108,\end{aligned}\ ] ] and the inverse of @xmath109 is given by @xmath110.\end{aligned}\ ] ] we then obtain @xmath111    we are particularly interested in the estimation error at a particular time @xmath112 , in which case @xmath113 and from eq . ( [ conuv ] ) , @xmath114 . we will see below that the choice @xmath115 , so @xmath116 where @xmath117 is a convenient one for deriving a lower bound , for a suitable choice of characteristic time @xmath118 . it gives latexmath:[\ ] ] where @xmath437 is the riemann zeta function . this means that the sum is exponentially small in @xmath3 , which is why it can be omitted in eq .  . 67ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12  & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop _ ( ,  ,  ) link:\doibase    10.1126/science.1104149 [ * * ,   ( ) ] http://dx.doi.org/10.1038/nphoton.2011.35 [ * * ,   ( ) ] http://www.degruyter.com/view/product/180320[__ ]  ( ,  ,  ) @noop _ _ ( ,  ,  ) link:\doibase 10.1103/physrevlett.106.090401 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.86.042115 [ * * ,   ( ) ] link:\doibase    10.1088/1367 - 2630/15/7/073005 [ * * ,   ( ) ] link:\doibase    10.1126/science.1086489 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.102.253601 [ * * ,   ( ) ] http://dx.doi.org/10.1038/nphoton.2012.346 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.54.r4649 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.33.4033 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.75.2944 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.77.2352 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.105.180402 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.107.059904 [ * * ,   ( ) ] http://stacks.iop.org/1367-2630/14/i=9/a=093052 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.87.034101 [ * * ,   ( ) ] link:\doibase    10.1016/j.aop.2012.12.004 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.104.103602 [ * * ,   ( ) ] http://stacks.iop.org/1751-8121/46/i=3/a=035302 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.108.230401 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.108.260405 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.86.053813 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.85.041802 [ * * ,   ( ) ] @noop ( ) ,   link:\doibase    10.1103/physrevlett.108.210404 [ * * ,   ( ) ] link:\doibase 10.1103/physrevx.2.041006 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.83.021804 [ * * ,   ( ) ] link:\doibase 10.1038/nphys1958 [ * * ,   ( ) ] link:\doibase 10.1007/s13538 - 011 - 0037-y [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.109.190404 [ * * ,   ( ) ] @noop ( ) ,   link:\doibase 10.1038/ncomms2067 [ * * ,   ( ) ] , @noop ( ) ,   link:\doibase    10.1103/revmodphys.71.1539 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.111.070403 [ * * ,   ( ) ]  and ,  eds . , @noop _ _  ( , ,  ) link:\doibase    10.1103/physreva.79.062301 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.111.113601 [ * * ,   ( ) ] link:\doibase    10.1103/physrevlett.104.093601 [ * * ,   ( ) ] link:\doibase 10.1126/science.1225258 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.111.163602 [ * * ,   ( ) ] link:\doibase    10.1109/tit.1973.1055103 [ * * , ( ) ] link:\doibase    10.1109/tit.1974.1055173 [ * * , ( ) ] link:\doibase    10.1142/s0219749909004839 [ * * ,   ( ) ] @noop ( ) ,   http://onlinelibrary.wiley.com/book/10.1002/0471221082[__ ]  ( ,  ,  ) link:\doibase 10.1109/18.556118 [ * * ,   ( ) ] link:\doibase 10.1109/tit.1969.1054301 [ * * ,   ( ) ] link:\doibase 10.1109/18.841198 [ * * , ( ) ] link:\doibase 10.1103/physreva.87.019901 [ * * ,   ( ) ]  and ,  eds . , @noop _ _  ( ,  ,  ) link:\doibase 10.1109/tcom.1967.1089532 [ * * ,   ( ) ] @noop * * ,   ( ) link:\doibase 10.1016/0034 - 4877(76)90060 - 4 [ * * ,   ( ) ] link:\doibase 10.1103/physreva.67.052109 [ * * ,   ( ) ] http://stacks.iop.org/0954-8998/3/i=5/a=003 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.75.4587 [ * * ,   ( ) ] link:\doibase 10.1103/physrevlett.89.133602 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.73.063824 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.73.063824 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.78.053820 [ * * ,   ( ) ] link:\doibase    10.1103/physreva.79.053843 [ * * ,   ( ) ] http://books.google.com/books?id=4htgaaaayaaj[__ ]  ( , ,  )
we propose quantum versions of the bell - ziv - zakai lower bounds on the error in multiparameter estimation . as an application we consider measurement of a time - varying optical phase signal with stationary gaussian prior statistics and a power law spectrum @xmath0 , with @xmath1 . with no other assumptions , 
 we show that the mean - square error has a lower bound scaling as @xmath2 , where @xmath3 is the time - averaged mean photon flux . 
 moreover , we show that this accuracy is achievable by sampling and interpolation , for any @xmath1 . 
 this bound is thus a rigorous generalization of the heisenberg limit , for measurement of a single unknown optical phase , to a stochastically varying optical phase .