Quantum State Discrimination and Quantum Cloning: Optimization and Implementation

Transcription

1 City University of New York (CUNY) CUNY Academic Works Dissertations, Theses, and Castone Projects Graduate Center 5-05 Quantum State Discrimination and Quantum Cloning: Otimization and Imlementation Andi Shehu Graduate Center, City University of New York How does access to this work benefit you? Let us know! Follow this and additional works at: htts://academicworks.cuny.edu/gc_etds Part of the Physics Commons Recommended Citation Shehu, Andi, "Quantum State Discrimination and Quantum Cloning: Otimization and Imlementation" (05). CUNY Academic Works. htts://academicworks.cuny.edu/gc_etds/6 This Dissertation is brought to you by CUNY Academic Works. It has been acceted for inclusion in All Dissertations, Theses, and Castone Projects by an authorized administrator of CUNY Academic Works. For more information, lease contact

2 Quantum State Discrimination and Quantum Cloning: Otimization and Imlementation by Andi Shehu A dissertation submitted to the Graduate Faculty in Physics in artial fulfillment of the requirements for the degree of Doctor of Philosohy, The City University of New York 05

3 05 Andi Shehu Some rights reserved. This work is licensed under a Creative Commons Attribution 4.0 United States License. htt://creativecommons.org/licenses/by/4.0/ ii

4 iii This manuscrit has been read and acceted for the Graduate Faculty in Physics in satisfaction of the dissertation requirement for the degree of Doctor of Philosohy. Date Prof. János A. Bergou Chair of Examining Committee Date Prof. Igor L. Kuskovsky Executive Officer Suervisory Committee: Prof. Mark Hillery Prof. Christoher C. Gerry Prof. Ed Fieldman Prof. Neea T. Maitra THE CITY UNIVERSITY OF NEW YORK

5 Abstract iv Quantum State Discrimination and Quantum Cloning: Otimization and Imlementation by Andi Shehu Advisor: János A. Bergou In our work we exlore the field of quantum state discrimination and quantum cloning. Recently the roblem of otimal state discrimination with a Fixed Rate of Inconclusive Outcomes (FRIO strategy) has been solved for two ure quantum states and a few other highly symmetric cases. An otical imlementation to FRIO for ure states is rovided. The hysical imlementation can be carried out with the use of a six-ort interferometer constructed with otical fibers beam slitters, hase shifters and mirrors. The inut states are comosed of qubits which are realized as hotons in the dual-rail reresentation. The non-unitary measurements are carried out at the outut for the resence or absence of a hoton. The setu otimally interolates between minimum error and unambiguous state discrimination. We also extend the FRIO strategy to two mixed states, whose eigenvectors in their sectral reresentation form a Jordan basis. We derive the minimum error rate P E for a fixed inconclusive rate Q and, in articular, the otimal distribution of the total Q over the Jordan subsaces. As Q is varied between the two limits, 0 <Q<Q c,astructure with multile thresholds, Q (th) (= 0) <Q (th) <...<Q (th) N <Q c,emerges.wealsosolvethe roblem of state searation of two known ure states in the general case where the known

6 states have arbitrary rior robabilities. The solution emerges from a geometric formulation of the roblem. This formulation also reveals a deeer connection between cloning and state discrimination. The results are then alied in designing a scheme for hybrid cloning which interolates between aroximate and robabilistic exact cloning. It is shown that state searation and hybrid cloning are generalized schemes to well established state discrimination and cloning strategies. The relationshis between cloning, state searation and state discrimination are derived in several limints. v

7 Acknowledgments First and foremost I would like to thank my advisor Prof. János A. Bergou for guiding me throughout the years of the PhD rogram. It was due to his guidance, encouragement and ositive attitude I was able to comlete the rogram. Secial thanks go to the members of the research grou: Vadim Yerokhin, colleague and very good friend of many years for the work and the good times we shared in the office; Prof. Emilio Bagan for heling us solve roblems in ingenious ways which we at times believed to be imossible. In return we introduced him to the Knob Creek; Dr. Georgina Olivares for heling with the writing of the dissertation, several rehearsals of the defense over Skye and the words of encouragement; Dr. Ugur Guney for the IT suort, setting u the LYX software this dissertation is being written on and some of the grahics used in here; Prof. Mark Hillery for organizing the seminars and the helful discussions; Prof Emeritus. Ed Feldman for the work and the great stories he shared; Valeria Feherataky-Bergou for the many gifts of chocolate and wine and for being the mother of the grou. I would also like the thank my good friends and colleagues Armando Rua, Jorge Colon, Kay Hiranaka Tetiana Nosuch, Zhenmao Wan, Denis Sh., Mallory Gobet, Phil Stallworth, Marc Berman and Joel DeJesus for making the PhD life memorable. Many thanks to my arents Luan and Flutur, my siblings Elona, Evis and Kliton Shehu for the love, suort and guidance throughout my life and education. I did become a doctor as my arents thought I would, just not the one to call in case of a medical emergency. vi

8 Contents Acknowledgments vi List of Figures ix Chater. Quantum State Discrimination.. Unambiguous Discrimination 4.. Minimum Error Discrimination Chater. Otimal discrimination of a certain class of mixed states with a fixed rate of inconclusive outcome (FRIO) 8.. Review of the FRIO solution for two ure states 8.. FRIO discrimination of two Rank mixed states 3.3. FRIO discrimination of two Rank N mixed states, POVM regime Projective regime Summary and conclusion 36 Chater 3. Quantum Cloning No-Cloning Theorem Exact cloning with failure rate Exact Cloning then Unambiguous Discrimination State Searation Deterministic State Deendent Quantum Cloning Hybrid Cloning: Interolation between exact and aroximate cloning 69 Chater 4. Exerimental realization to FRIO Analytical Solution of Interolation 76 vii

9 CONTENTS viii 4.. Lagrange Multiliers Method Choosing the hysical imlementation Imlementation: equal riors Imlementation: Unequal riors 9 Aendix : Reck-Zeilinger Algorithm 97 Aendix : Lagrange Multiliers 0 Bibliograhy 03

10 List of Figures.. UD Detectors 7.. von Neumann UD for i 9..3 von Neumann UD for i 0.. Min Error 4.. Q c and Q b vs...! Q ot vs. Q and! Q ot vs. Q. 8..! P ot e, vs. Q and! P ot e, vs. Q. 9.3.! i Q ot i vs. Q 33.3.! i P ot e,i vs. Q 33.4.! i Q ot i vs. Q P ot e,i vs. Q Unitarity curves Unitarity curves for s Q min vs State searation s 0 vs. s Six ort interferometer NxN interferometer 00 ix

11 CHAPTER Quantum State Discrimination An integral art of quantum information and quantum rocessing is measurement theory []. It is the robabilistic nature of quantum mechanics that one cannot simly obtain information encoded in states [], the state is not an observable in quantum mechanics [3]. When a quantum circuit or rocessor has acted on the inut states to erform a task, the outut needs to be read out. Thus after the rocessing occurs the task is to determine the state of the system. If the inut states are orthogonal the rocess is trivial. Simly setting u detectors along the orthogonal directions and a click in those detectors will determine the state of the system. On the other hand discriminating among non orthogonal quantum states is not trivial. Since quantum mechanics does not allow for erfect discrimination of non orthogonal states the task becomes that of a measurement otimization roblem. Not being able to erfectly discriminate quantum states is key to various quantum crytograhic schemes and quantum comuting. The origin of the state discrimination field is attributed to the works of Helstrom [4] andholevo[5]. The field however gained momentum in the 90 s as quantum information theory became very active rimarily due to the factorization work of Peter Schor [6] andquantumkeydistributionrotocolssuchasb9[7]. Various otimum state discrimination measurement strategies have been develoed with resect to some figure of merit. Two of those methods which we focus on are otimum Unambiguous Discrimination (UD) and Minimum Error (ME). In UD strategy, first suggested by Ivanovic [8], the observer Bob, is not allowed to make an error. Whenever he is handed a state i i he cannot conclude that he was given j i. We will show that this cannot be done with 00% success rate and that the observer must allow for inconclusive results and find an otimum measurement strategy which minimizes the average rate of inconclusive results. In the Minimum Error strategy the observer is not allowed to have inconclusive results. Thus

12 . QUANTUM STATE DISCRIMINATION errors are allowed and the task is to find otimum measurements that minimize the average error rate. It has been shown that ME and UD are secial cases of a more general scheme of otimum state discrimination measurement which can be aroached by relaxing the conditions at either end [9]. In the ME scheme the otimal error rate can be further reduced by allowing for some rate of inconclusive results. Thus the otimal average error rate, P E, becomes a function of a given rate of allowed inconclusive results Q, P E (Q). On the other hand, in UD, the otimal rate of the average inconclusive outcomes, Q, may be reduced by allowing for some error rate P E. The failure rate becomes a function of a given error rate Q(P E ). In our work we use various quantum measurements schemes to read out information out of aquantumsystem. Foramorethoroughunderstandingofquantumtheoryofmeasurements we go along the lines of the review aer by J.A Bergou [0]. Starting with the standard quantum measurement theory due essentially to von Neumann the generalized measurements (Positive Oerator Valued Measures, POVMs) are introduced as more useful measurement schemes in otimization roblems. Using Neumark s theorem the POVMs can be realized exerimentally..0.. Standard Quantum Measurements. We start with the ostulates of standard or rojective quantum measurements introduced by von Neumann [] analyzingamodel for the couling of the system with the meter or ancilla and generalizing the redictions of the model. The ostulates are: () Observables in quantum mechanics have a Hermintian oerator which has a sectral reresentation = P N j j jihj, where the eigenvalues are real and assuming non-degeneracy for simlicity. The eigenvectors { ji} form a comlete orthonormal basis set. () The Hilbert sace is sanned by the rojectors P j = jihj, suchthat P j P j =. (3) The eigenvalues of the rojectors are 0 or due to the orthogonality of the states P i P j = P i ij.

13 . QUANTUM STATE DISCRIMINATION 3 (4) Any measurement of the will yield one of the eigenvalues j. (5) If j is obtained in a measurements, the state of the system collases onto: ji = P j i if the system was initially in a ure state, j = P j P j h Pj i Tr(P j ) initially in a mixed state. if the system was (6) The robability of obtaining ji is j = P j = P j = h P j i. The robability of obtaining j is j = Tr(P j P j )=Tr(P j ) =Tr(P j ). (7) If a measurement is erformed but the result is not recorded the ost-measurement state collases onto: = P j P j ih P j if the system was initially in a ure state, = P j j j = P j P j P j if the system was initially in a mixed state..0.. POVMs. Due to the orthogonality condition of the rojective measurements one cannot have more orthogonal rojections than the dimensionality, hence the ossible outcomes cannot exceed the number of the dimensionality. Sometimes we would like to allow for more outcomes than the dimensionality, as in the case of otimal UD measurements where we have three outcomes in a two dimensional roblem. Next we introduce a ositive oerator j 0 as a generalization of P j and the robability of obtaining state j becomes j = Tr( j j ). To normalize the robabilities we require that the ositive oerators j are a decomosition of the identity P j j = I. This is decomosition is called a Positive Oerator Valued Measure (POVM) and j the elements of the POVM. The generalization of the ostulates of quantum mechanics in terms of the POVM can be exressed as: () The decomosition of the identity in terms of ositive oerators, j 0, P j j = I is called a POVM. () The elements of the POVM can be exressed in terms of the detection oerators j = A j A j where the oerators satisfy the requirements P j A j A j = I but they need not be Hermitian. (3) A detection yields an element on POVM.

14 .. UNAMBIGUOUS DISCRIMINATION 4 (4) The state of the system collases onto: ji = A j i q if the system was initially h A j A j i in a ure state, j = mixed state. A j A j Tr(A j A j ) = A j A j = A j A j Tr(A j A j ) Tr( j ) if the system was initially in a (5) The robability of obtaining j is j = Tr(A j j A j )=Tr(A j A j j )=Tr( j j ). (6) If a measurement is erformed but the result is not recorded the ost-measurement state collases onto: = P j j j = P j A j A j. It is these generalized measurements we will use in our otimization work. In the following sections POVM elements are used to otimize the unambiguous discrimination and minimum error schemes... Unambiguous Discrimination In this section we give a review of the existing schemes of Unambiguous Discrimination (UD). Particularly that of two ure states as it is directly related with our work. When erforming UD the detectors are not allowed to make an error but can admit inconclusive outcomes. We first show by contradiction that it is not ossible to succeed at unambiguously discriminating quantum states with 00% success rate. Then we show that in order to erform UD a third detector must be added which accounts for inconclusive results. The task is to minimize this rate of inconclusive outcomes. In Subsection (..) the roblem is solved via the POVM strategy. In the following Subsection (..) we show how the solution can be imlemented via the Neumark theorem.... Unambiguous Discrimination: Two ure states via POVM. An ensemble of quantum states is reared with two ossible ure states i or i. Each state is reared with an a riori robability or,suchthat + =. The observer has full knowledge of the states and their riors. The rearer, Alice, icks u a state and hands it over to the observer, Bob. Bob s task it to determine which state he is given by erforming a single a POVM on the individual system he is given. As it was stated earlier, the observer is not allowed to make an error when erforming a measurement. Let us assume Bob can indeed discriminate the given states with 00%

15 .. UNAMBIGUOUS DISCRIMINATION 5 success rate. Let and be detectors which cover the full Hilbert sace sanned by the states i and i, + = I (..) In the UD strategy the detector i identifies only the state i i and never clicks for j i, such that i j i =0. Multilying Equation (..) by i from the right and h from the left results in = h i =, which is the robability of successfully identifying i. Similarly it can be shown that the state i can be detected with a robability one, = h i =. Seems as if one can indeed discriminate two non-orthogonal quantum states with a 00% success rate. However multilying Equation. (..) with h from the left and i from the right it follows that h i =0, where we use i j i =0. This means that the inut states are orthogonal to begin with, which is a contradiction because we started with nonorthogonal quantum states. Thus one cannot discriminate non-orthogonal quantum state with 00% success rate (orthogonal states can indeed be discriminated with no error rate, they corresond to classical states). One can still erform Unambiguous Discrimination but with a modified scheme. Equation (..) is modified by adding a third detector 0 which can click for both states i and i : = I (..) The clicks from 0 are all inconclusive, i.e we gain no information from 0. Defining individual failure rates q = h 0 i and q = h 0 i as the failure robabilities, the task becomes that of minimizing the overall failure rate, Q = q + q. (..3) Equivalently otimizing the success rate

16 .. UNAMBIGUOUS DISCRIMINATION 6 P E = +, such that P E + Q =., Let us now exlicitly determine the POVM oerators to be used in the otimization of Q, seefigure(..).firstdefinethestatestobeinatwodimensionallane, i = cos 0i +sin i, i = cos 0i sin i. The detectors must be orthogonal with the states for which they should not identify, i.e i j i =0, = c? ih?, = c? ih?, where? i =sin 0i +cos i and? i = sin 0i +cos i. The coefficients c i 0 are yet to be determined based on the otimum strategies. Using the definition of success robabilities i = h i i i i the constants c i can be relaced, = = h? i? ih?, h? i? ih?. (..4) To determine the failure oerator, insert (..4) into (..): 0 = I = I h? i? ih? h? i? ih?. (..5)

17 .. UNAMBIGUOUS DISCRIMINATION 7 Figure... POVM setu that unambiguously discriminates between i and i otimally. The detector D = is setu along?, detector D = is setu along? and the failure oerator is setu symmetrically between i and i for = =.WhenaclickintheD i detector occurs we know for certain that i i was reared (i =, ) as the inut state since it is the only one that has a comonent along this direction. A click in the D 0 detector is considered inconclusive as both states have a comonent along this direction direction After writing everything exlicitly, the ositivity constraint of the eigenvalues of 0 gives the condition q q h i, (..6) where we used q i = i. Using the condition (..6) and taking the equality sign, the total failure rate in (..3) can be exressed in terms of a single constraint. Define the overla s h i and relacing q = s /q into (..3), Q = s q + q, the otimization follows = s q.

18 .. UNAMBIGUOUS DISCRIMINATION 8 q q This leads to individual failure rates q = s and q = s. Inserting them back into Equation (..3) gives the otimal Q which it will be defined as Q 0, Q 0 = s. (..7) Let us now check the conditions where this result holds. The individual error rates must q be smaller or equal to one, q i ale. Hence q = s ale gives the lower bound on the a-rior robabilities, s +s. Similarly the condition that q ale gives the uer bound on the riors ale +s. Putting the two conditions together the POVM regime is valid in the range: s +s ale ale +s. (..8) Outside of this range it is interesting to see that the measurement strategy merges into the rojective measurement. If one of the incoming states is reared with a much higher robability, say, we design an exeriment where we have only two detection oerators. One of them, D 0,the failure oerator, simly rojects onto state i, the detector D rojects onto? i, thus it never clicks for i, so that a click on D is associated with the state i. Aclickalong D is failure. The setu for the detectors which roduce failure rate Q is shown in Figure (..). The total failure rate is: Q = h i +. (..9) Similarly for, the corresonding setu with detectors yielding Q is shown in Figure (..3) Q = + h i. (..0) Putting the ieces together, the minimum value of Q for the three different regimes can be written as:

19 .. UNAMBIGUOUS DISCRIMINATION 9 Figure... AvonNeumannmeasurementthatdiscriminates i unambiguously. The failure detector 0 = P is set u along the i direction and the second detector = P? is set u orthogonal to i therefore never clicks for i. When a click in the 0 detector occurs we learn nothing as both states have an overla along P. 8 s s if ale +s >< ale Q = h i + if >, +s >: + h i if < s +s. +s, (..) It is very interesting that the POVM gives the minimum Q when it is valid. Outside the boundaries it merges with the von Neumann rojective measurement.... Unambiguous Discrimination: Two ure states via Neumark s Theorem. Theoretically the roblem of minimizing the average failure rate for two ure states has been solved in the revious section. However to be able to imlement those schemes we resort to Neumark s theorem which states that any POVM oerator can be realized by generalized measurements [?]. The system where the incoming states live is embedded in a

20 .. UNAMBIGUOUS DISCRIMINATION 0 Figure..3. AvonNeumannmeasurementthatdiscriminates i unambiguously. The failure detector 0 = P is set u along the i direction and the second detector = P? is set u orthogonal to i therefore never clicks for i. When a click in the 0 detector occurs, we learn nothing as both states have an overla along P. larger Hilbert sace called ancilla. Then a unitary oerator entangles the degrees of freedom of the system with those of the ancilla. After this interaction rojective measurements are erformed within this larger system in the ancilla. These measurements will also transform the system states in the original Hilbert sace because of the entanglement. To show the ower of Neumark s theorem we will re-derive the otimal failure rate of two nonorthogonal states. The incoming states { i s, i s } which live in the state Hilbert sace H S are embedded with the ancilla ii a which live in the ancilla Hilbert sace H A. Now the system and the ancilla live in the larger Hilbert sace H = H S H A. The incoming states in this larger Hilbert sace can be written in the roduct form { i s ii a, i s ii a }, where ii a is the initial state of the ancilla. The unitary oerator does the following:

21 .. MINIMUM ERROR DISCRIMINATION U i s ii a = 0 i s i a + q i s 0i a, U i s ii a = 0 i s i a + q i s 0i a, (..) where i is the robability of successfully identifying the state i i s, q i is the robability of failing to identify i i s, and i + q i =. The unitary oerator aims to take the two incoming states and make them orthogonal. When there is a click on the ancilla i a the inut states have been searated and outut states ii 0 s are orthogonal and therefore fully distinguishable. If there is a click along the ancilla 0i a the incoming states have been collased into a single state which carries no information about the system. That is why the choice on the setu of having the failed state i s be the same, there should be absolutely no information left in the failed state, otherwise it is not otimal. Taking the inner roduct of the two equations in (..) gives the constraint to the otimization s = q q, (..3) where s was defined to be the overla of the inut states s h i. In just one line Neumark s setu has roduced the constrain and the rest of the derivation, otimizing (..3), is the same as in the POVM section and we do not need to reeat here. Imlementation methods have been derived and we will show an examle in Chater 4... Minimum Error Discrimination In the Minimum Error (ME) strategy one is not allowed to abstain from identifying an incoming state, i.e for every incoming state the observer must say which state he was given. Since it was shown that erfect discrimination is not ossible the detectors inevitably will make errors. A click in a detector can only identify a state with some robability of success and misidentify the state with some robability of error.

22 .. MINIMUM ERROR DISCRIMINATION... Minimum Error: Two mixed states via POVM. Given an ensemble of two mixed states {, } reared with different a riori robabilities {, } the task is to minimize the rate for which the detectors misidentify a state. The minimum error roblem for two ure or mixed states was first solved by Helstrom [4]. We show an alternative derivation to ME of two ure states develoed by Herzog [] and Fuchs[3]. When the detector i clicks for state j it is an error, r i = Tr( j i ),aclinkforstate i is success i = Tr( i i ). Thus for two states we want to minimize the following exression. P E = Tr( )+ Tr( ). (..) Using the relation + =and + = I, Equation (..) can be rewritten as: P E = Tr( (I )) + Tr( ), = + Tr[( ), = Tr[( ). Let = P E = + Tr( )= Tr( ). (..) To minimize P E, should roject onto the eigenvectors of the negative eigenvalues of, on the other hand should roject onto the ositive eigenvectors. Let us write into its sectral decomosition. dx = = i iih i. (..3) To imlement the rojection of the POVM oerators onto the ositive (or negative) i= eigenvectors the eigenvalues negative, ositive and zero: i can be slit into three categories without any loss of generality:

23 .. MINIMUM ERROR DISCRIMINATION 3 i < 0 for ale i<i o, i > 0 for i o ale i<d, i = 0 for d ale i<d s. (..4) Then from the sectral decomosition we can rewrite (..6) in terms of the otimal POVM. P E = + ix o i= ih i ii = where = P i o i= i iih i and = P d s i=i o i iih i. Xd s i=i o ih i ii, (..5) The POVMs need to satisfy the condition 0 aleh i j ii ale which comes from the definition of the normalized robabilities r i = Tr( i j ). These POVMs are basically von Neumann rojectors onto the corresonding eigenvectors. If we now relace the detection oerators by the otimal detectors the minimum error can be exressed just in terms of the eigenvalues of. ix o Xd s P E = i = i, i= i= i= = d s [ X i ] = [ Tr ], = [ Tr ] (..6) When the states to be discriminated are ure, { i, i}, the minimum error can be reduced to P E = [ 4 h i ]. (..7)

24 .. MINIMUM ERROR DISCRIMINATION 4 Figure... A von Neumann measurement which minimizes the error rate of two ure states reared with equal riors. The detectors are laced symmetrically along the states { i, i} for = =.... Minimum Error: Two ure states via Neumark s theorem. Just as we did in the UD case, we will solve the ME roblem of discriminating two ure states via the Neumark setu because of it lends itself into an otical imlementation. Although in this case the solution is not as straightforward. The incoming states { i s, i s } which live in the state Hilbert sace H S are embedded with the ancilla ii a which live in the ancilla Hilbert sace H A. Now the system and the ancilla live in the larger Hilbert sace H = H S H A. The incoming states in this larger Hilbert sace can be written in the roduct form { i s ii a, i s ii a }. The unitary oerator does the following: U i s ii a = i i + r i i, (..8) U i s ii a = i i + r i i, (..9) where i is the robability of having successfully identified the state i i and r i is the robability of misidentifying the state i i for j i.

25 .. MINIMUM ERROR DISCRIMINATION 5 Taking the inner roduct of the two equations in (..9) gives the constraint: s = r + r. (..0) The quantity we are looking to minimize in the average error rate: P E = r + r, (..) subject to the constraint in (..0). Adding the constraint to.. with one Lagrange multilier and using i + r i = F E = r + r + h s ( r )r ( r )r i. (..) Differentiating with resect to r E = + aler r r r r E = + ale r r r r r r =0. Rearranging the above two equations so that the left hand side is only a function of r i and the right hand sides turn out to be equivalent, r ( r )= r r ( r )( r ), (..3) r ( r )= r r ( r )( r ). (..4) The right hand sides of Eq.(..3) and (..4) can be set to a constant i ri ( r i ) C, which can later be determined from the unitarity constraint..0,

26 .. MINIMUM ERROR DISCRIMINATION 6 r i = ± s C i! = s i!, (..5) where A i q r i = [ A i], (..6) i and C. The smaller r i is icked (lower sign in..5 ) as this reresents error rate, which is to be minimized. Now relace r i into the constraint (..0) and solve for : s = ( r )r + ( r )r, (s s = ( + A )( A )+ ( A )( + A ), q s = A A + ( A )( A ), s = A A + ) = After some tedious but trivial algebra:, 4 +. Now substitute the value of = 4s ( s ) 4 s. (..7) from (..7) into (..5) to get the exlicit form of the individual error rates, r i = " # i s 4 s (..8) Inserting r and r into (..) Helstrom bound is retrieved [4]

27 .. MINIMUM ERROR DISCRIMINATION 7 P E = " # s s, 4 s 4 s P E = h 4 s i. (..9) As we mentioned above the advantage of solving the ME roblem via Neumark is that we now have exlicit exressions for the individual error rates, r and r. In the Imlementation chater, it is shown that the unitary oerator which carries out this oeration can be written in terms of r i and then be decomosed into beam slitters and hase shifters using otical interferometers.

28 CHAPTER Otimal discrimination of a certain class of mixed states with a fixed rate of inconclusive outcome (FRIO) In this chater we will derive the otimal strategy with a Fixed Rate of Inconclusive Outcomes (FRIO) that otimally interolates between the two well known limits, Helstrom bound for minimum error and IDP for unambiguous discrimination. In articular, as the main finding of our aer, we will show that the otimal distribution of the fixed rate of inconclusive outcomes, Q, amongthe-dimensionalsubsacessannedbytheairofjordan basis vectors is highly non-trivial and an interesting threshold-like structure emerges: As we start increasing Q from Q =0,firstonlyonesubsacereceivestheentireinconclusiverate. Then, as we increase Q further, at a certain threshold a second subsace starts sharing the inconclusive rate. If we increase Q further, at another threshold a third subsace also starts sharing Q,andsoon,untilabovealastthresholdallsubsacessharetheavailable inconclusive rate... Review of the FRIO solution for two ure states We first resent a brief review of the method develoed in [4] forthetwourestate otimal FRIO roblem since the rest of the aer relies heavily on this method. We derive the maximum robability of success or, equivalently, the minimum robability of error in identifying the states, when a certain fixed rate of inconclusive outcomes is allowed. By varying the inconclusive rate, the scheme otimally interolates between Unambiguous and Minimum Error discrimination (UD and ME). In all of these scenarios (UD, ME or FRIO) one is given a system which is romised to be reared in one of two known ure states, i or i, but we don t know which. The ure states are reared with rior robabilities and,resectively,suchthat + =. 8

29 .. REVIEW OF THE FRIO SOLUTION FOR TWO PURE STATES 9 It is well known that two ure states can be discriminated both unambiguously and with minimum error. In Section (.) we showed the solution of the otimal average inconclusive rate, Q c for UD: 8 + cos, if < >< Q c = + cos, if > >: cos +cos (l), +cos (r), cos Q 0, if (l) ale ale (r), where h i cos is the overla of the inut states. (..) In Section (.) we also showed the otimal average error rate for ME: PE ME = 4 cos. (..) It has long been suggested [5] thattheabovestatediscriminationointsareartof a more general scheme. One which interolates between otimal ME and UD. The FRIO strategy achieves that goal by minimizing the error rate while allowing for some rate of inconclusive results. Hence the strategy has three measurement outcomes, one that identifies with the first state, one that identifies with the second state and one that does not identify with a state at all, corresonding to the inconclusive outcome. The authors in [4] solvethe roblem via the POVM method. Three POVM elements are needed such that: = I, (..3) Aclickin is identified with state i, aclickin is identified with the second state i and any clicks in the oerator 0 corresonds to inconclusive outcomes. However the oerators i can also click for j i. We wish to minimize the average error rate, P E = Tr[h i]+ Tr[h i], = tr [ ih ]+ tr [ ih ], = tr [ ]+ tr [ ],

30 .. REVIEW OF THE FRIO SOLUTION FOR TWO PURE STATES 0 where = ih and = ih are the corresonding ure state density matrices (Equivalently we wish to maximize the average success rate P s = tr [ ]+ tr [ ]), for a fixed rate of inconclusive results, Q = tr [ 0 ]+ tr [ 0 ]. (..4) The solution to the roblem involves a neat trick which transforms the three element POVM defined in Eq.(..3) into a two element POVM, namely, + = I 0, / ( + ) / = / (I 0 ) /, + = I, (..5) where I 0 and i / i /. Should be noted that / =(I 0 ) / exists unless 0 has a unit value, in which case the roblem is treated searately. It was imortant to notice that for otimal FRIO, 0 must be rank one oerator. That means that 0 mas both incoming non-orthogonal states onto a single state which is then discarded. On the other hand the other two POVM elements ma the inut states onto two different states. The FRIO roblem has essentially been transformed into a new arametrized otimization scheme with two POVM elements, and. The error robability in the new arametrized form, P E =( Q) P E,becomes P e = Tr( )+Tr( ), (..6) where the normalized states i and normalized a riori robabilities i are i = / i / Tr( i )), i = itr( i ) Q, (..7)

31 .. REVIEW OF THE FRIO SOLUTION FOR TWO PURE STATES Equation (..6) and + = I define a ME discrimination roblem for the transformed states and riors given in Eq. (..7). The otimal solution to this ME discrimination roblem immediately follows by using the tilde quantities in (..), with ii = / i i/ h i i i being the roerly normalized transformed states. We can immediately write down the solution in arametrized form P ME E = r 4 D E! Writing the inut states as, i i = c i 0i + s i i, where c i cos i, s i sin i,weobtain the transformed states and riors from Eq. (..7). Since the otimal 0 is a ositive rank one oerator it can be written as 0 = 0ih0, where is its eigenvalue, 0 ale ale, and the eigenstate belonging to is 0i and the orthogonal state is i. In this basis = ( ) 0ih0 + ih. The tilde error rate now becomes: P e ME = n 4 (cos c c ) /( Q) o, (..8) where. It follows from (..4) that = Q. (..9) c + c Hence Eq. (..8) deends only on one arameter, say, which determines the orientation of 0 relative to that of the two ure states. The minimization over simlifies considerably, using Eq. (..9) and defining c / ( c + c ) / cos ', andc / ( c + c ) / sin '. The resulting exression is minimum for ' = /4, yielding Pe min = Q q ( Q) (Q 0 Q), (..0) for all Q ale Q 0 s. This is the otimal error rate for an intermediate range of the rior robabilities, a rather nice looking formula for a somewhat comlicated roblem. We can check that is reroduces the Helstrom bound for zero failure rate Q =0,

32 h P E = Q 0=.. REVIEW OF THE FRIO SOLUTION FOR TWO PURE STATES i h i Q 0 = s and the IDP bound for zero error rate q ( Q) (Q 0 Q) ) Q = Q 0 s. For the validity of (..0), ale must hold. The definitions of cos ' and sin ' after Eq. (..9) give c = c for ' = /4 which, in turn, leads to c = c = sin /( Q 0 ), and Eq. (..9) yields =( Q 0 )Q/( sin ). Setting = defines the boundary Q b, between the rojective and POVM regimes, Q b sin /( Q 0 ). (..) Hence ale if Q ale Q b and =if Q>Q b. In Fig... we lot Q c and Q b vs. together for a fixed overla, cos =0.5 ( = /3). The two curves intersect at = (l) and = (r), the same oints as in Eq. (..). The interval 0 ale ale is thus divided into three regions. In regions I and III, we have Q b <Q c and the solution (..0) is valid for 0 ale Q<Q b only. In Region II, (l) ale ale (r),we have Q c = Q 0 <Q b and the solution (..0) is valid for the entire 0 ale Q ale Q c range. Figure... Q c (dashed line, Eq. (..)) and Q b (solid line, Eq. (..)) vs. for = /3. Measurements can be otimized in the area under the dashed line, Q c. Measurements in the area above Q c are subotimal. In the shaded areas between Q c and Q b (regions I, left, and III, right) the otimal FRIO measurement is a rojective measurement, in the unshaded area below Q c (region II) the otimal measurement is a POVM. In the shaded arts of regions I and III one has Q b ale Q ale Q c and, necessarily, =. Hence, 0 = 0ih0 and = ih are rojectors. Therefore, / does not exist in these areas and the case needs secial consideration.

33 .. FRIO DISCRIMINATION OF TWO RANK MIXED STATES 3 The calculation of the error robability is most easily erformed by realizing that and become degenerate, both must be roortional to d = ih. The three-element POVM becomes a standard two element rojective measurement, { d = ih, 0 = 0ih0 }. We identify a click in d with ( )if ( ), so P e(s) = s, P s(e) = s, with Q = P e P s. These equations comletely determine the solution. There is nothing to otimize here, so we dro the suerscrit min in what follows. = immediately gives Q(P e ) as Q= P e () s P e () cos ± s P e () sin!. (..) Inverting this equation gives P e (Q) yields cos ( Q Q ) ( )(Q Q ) sin cos Q( P e = 4 sin Q) sin (..3), but the resulting exression is not articularly insightful. However, we note that for P e =0 (UD limit) one has Q = Q, given by the second line in Eq. (..), and for Q = Q b, P e reduces to (..0), as it should... FRIO discrimination of two Rank mixed states In our work we extend the FRIO discrimination scheme to a articular case of mixed states. We solve the FRIO roblem for two mixed states which exhibit a Jordan structure. The two states, with their resective rior robabilities and, can be written in the sectral decomosition form as, = = NX r i r i ihr i, i= NX s i s i ihs i. (..) i= In the Jordan structure these states satisfy the following conditions:

34 .. FRIO DISCRIMINATION OF TWO RANK MIXED STATES 4 hr i r j i = ij hs i s j i = ij hr i s j i = ij cos i (..) Due to the Jordan structure, instead of one N dimensional roblem we have N mutually orthogonal -dimensional subsaces. The i th subsace is sanned by r i i and s i i with rior robabilities r i and s i, resectively. Thus the discrimination of the two mixed states and can be reduced into that of ure state discrimination in each subsace, by discriminating r i i and s i i in each subsace. In this section we solve the case where N =,thereis-dimensionalsubsacesforeach density oerator. The density oerator in Eq. (..) can be exressed as = r r ihr + r r ihr, = s s ihs + s s ihs. (..3) The overall error rate is P E = tr ( )+ tr ( ), = (r tr ( s, r ihr )+r tr ( s, r ihr )) + (..4) (s tr ( r, s ihs )+s tr ( r, s ihs )) (..5) In this case the first subsace is sanned by { r i, s i} and the second subsace by { r i, s i}. The task becomes that of erforming FRIO discrimination of { r i i and s i i} within subsace i. The error rate in each subsace is:

35 .. FRIO DISCRIMINATION OF TWO RANK MIXED STATES 5 P e, = r tr ( s, r ihr )+ s tr ( r, s ihs ) P e, = r tr ( s, r ihr )+ s tr ( r, s ihs ) (..6) The rior robability of r i i is r i while the rior robability of s i i is s i. We define their normalized robabilities as,i,i r i, r i + s i s i, (..7) r i + s i such that,i +,i =. The error rate in each subsace becomes P e, =, tr ( s, r ihr )+, tr ( r, s ihs ) P e, =, tr ( s, r ihr )+, tr ( r, s ihs ) (..8) where P e,i = P e,i ( r i + s i ). We note that this reduces the roblem to the FRIO discrimination of two ure states in subsace i, with rior robabilities given above. It follows immediately that the solution is given by (..0) in the POVM regime of Q i, Q i ale Q c,i,q th,i,and(..)intherojective regime of Q i, Q th,i <Q i ale Q c,i, again with the above substitutions. In the following we will focus mainly on the POVM regime where the solution in each subsace is given exlicitly by Eq. (..0), P e, = ( Q P e, = ( Q q ( Q ) (Q 0, Q ) ), (..9) q ( Q ) (Q 0, Q ) ) (..0) Where we introduced a fixed rate of inconclusive outcomes for each subsace i, Q i,such that 0 ale Q i ale Q c,i where Q c,i is given by Eq. (..) with the obvious substitutions!,i,

36 .. FRIO DISCRIMINATION OF TWO RANK MIXED STATES 6!,i and cos! cos i = hr i s i i)q 0,i =,i,i hr i s i i. Q 0 was introduced in Eq. (..) for the single subsace (two ure states) case, Q 0,i is its generalization for the case of two (or many) subsaces. We introduce the weight! i of subsace i as r i + s i =! i, (..) where, obviously,! +! =. (..) The total error rate in Eq. (..4) can be exressed as weighted sum of the error rates of the individual subsaces P e =! P e, +! P e,, =! ale ( Q )! ale ( Q ) q ( Q ) (Q 0, Q ) + q ( Q ) (Q 0, Q ) (..3) The task it to determine the otimal distribution of Q among the two subsaces, the distribution that minimizes the error rate for a fixed amount of inconclusive results. We can write the total inconclusive rate as a weighted of the inconclusive rates of the individual subsaces, Q =! Q +! Q. (..4) Since the total failure rate Q is fixed, then only one of the Q i s is an indeendent variable. Thus the total error rate can be exressed in terms of only one variable and be otimized in that variable. Inserting Q =(Q! Q ) /! into Eq. (..3) it becomes a function of the indeendent variable Q and the otimization with resect to this variable is straightforward:

37 .. FRIO DISCRIMINATION OF TWO RANK MIXED STATES 7 P e =!! ale ( Q ) 4 q ( Q ) (Q 0, Q ) + Q +! Q!! s Q +! Q!! Q 0, 3 Q +! Q 5.!! Simly setting =0and solving for Q, = 4!! ( Q 0, ) q 6 4! r ( Q 0, ) ( Q ) (Q 0, Q ) = ( Q 0, ) " ( Q ) (Q 0, Q )! ( Q 0, ) Q! +! Q! Q 0, Q +! Q!! Q 0, Q! +! Q! Q +! # Q!! In order to exress the otimal failure rates in comact form it will be useful to introduce at this oint the following convention. Without loss of generality in what follows we assume the hierarchy We also introduce the notation Then result of the otimization can be written as Q 0, Q 0,. (..5) Q () th 0 Q() th! (Q 0, Q 0, ) Q 0,. (..6) Q ot = 8 < : Q! if Q () th Q 0, Q 0 (Q Q () th Q() th )+ ale Q ale Q() th,! if Q () th <QaleQ 0, (..7)

38 .. FRIO DISCRIMINATION OF TWO RANK MIXED STATES 8 and 8 < Q ot = : 0 if 0 ale Q ale Q () th, Q 0, Q 0 (Q Q () th ) if Q() th <QaleQ 0. (..8) Q () th The threshold structure is the interesting feature of this roblem. In the region 0 ale Q ale only one of the subsaces accommodates the fixed failure rate Q, itisthesubsace with the larger Q 0,i. The other subsace allows for no inconclusive rate at all, Q =0,and oerates in the Minimum Error regime. Above this threshold Q () th <Qale Q 0,bothsubsaces share the failure rate but at different values. If! Q 0, >! Q 0,,thefirstsubsacealways accommodates more inconclusive rate than the second, otherwise above some value of the total Q the second subsace will accommodate more inconclusive rate than the first. The situation is deicted in Fig..., for some secific values of the arameters. (a) (b) Figure...! Q ot vs. Q (solid line) and! Q ot vs. Q (dashed line). For the figure we used = =/ and the following arameter values: (a) = /4, = /7, r = 3/4, r = /4, s = 3/4 and s = /4; (b) cos =/4 3, cos =/4, r =3/4, r =/4, s =3/4 and s =/4.

39 .. FRIO DISCRIMINATION OF TWO RANK MIXED STATES 9 To get the exression of the total error rate as a function of FRIO, insert (..7) and (..8) into (..3), 8 >< P E = >: ( Q) ( Q) q(! Q) (! Q 0, Q) q (! ) (! Q 0, ) if Q () th ale Q ale Q() th, q ( Q) (Q 0 Q) if Q () th <QaleQ 0, We will show that this is valid in all regions of the arameter Q. For Q =0it reduces to the otimal Minimum Error exression for two subsaces [6], as it should. On the other hand for P e =0it reduces to the otimal Unambiguous Discrimination of two subsaces [7, 8] WeclosethissectionbydislayingP e,i vs. Q in Fig..., for some secific values of the arameters. (a) (b) Figure...! P ot e, vs. Q (solid line) and! P ot e, vs. Q (dashed line). For the figure we used the same arameter values as in Fig.... The insert in (b) shows that the solid and dashed lines intersect for a very small value of Q.

40 .3. FRIO DISCRIMINATION OF TWO RANK N MIXED STATES, POVM REGIME FRIO discrimination of two Rank N mixed states, POVM regime In this section we resent the more general solution to the two rank N density matrices, where N is some arbitrary but fixed integer. The two density matrices exhibit the Jordan basis as described in (..) and (..). Solving the roblem is very similar to the method for two rank mixed states in the receding section. The definitions and roerties resented in Eqs. (..) (..) remain in effect but the art starting with Eq. (..) has to be modified accordingly.the weights of the subsaces, introduced in (..) now satisfy NX! i =. (.3.) i= The total inconclusive rate can again be written as a weighted sum of the inconclusive rates of the individual subsaces, NX Q =! i Q i. (.3.) Q is fixed, with the fixed value satisfying 0 ale Q ale Q 0, where now i= Q 0 NX! i Q 0,i, (.3.3) i= is the maximal inconclusive rate that the N subsaces can accommodate. Similarly, the total error rate is a weighted sum of the error rates of the individual subsaces, NX P e =! i P e,i. (.3.4) i= The remaining task is to determine the otimal distribution of Q between the N subsaces, the distribution that minimizes the total error rate P e,undertheconstraintthatq in Eq. (.3.) is fixed, i.e., to determine the otimal values of Q i as a function of the fixed Q. In order to erform the otimization we emloy the Lagrange multilier method because it leads to symmetric and easily tractable equations. Adding the Lagrange multilier times

41 .3. FRIO DISCRIMINATION OF TWO RANK N MIXED STATES, POVM REGIME 3 the constraint, Q P N i= Q i =0,to(.3.4)yieldsthefunction F = NX! i P e,i + (Q i= NX! i Q i ), (.3.5) i= where P e,i is inserted from Eq. (..9), so it is also a function of Q i.nextwevaryf treating the variables Q,Q,...,Q N as indeendent. Solving the resulting equations together with the constraint (.3.) determines the value of the Lagrange multilier which in return otimizes.3.4. Before we resent the results it will rove useful to introduce a hierarchy of the subsaces. So, in what follows we will assume Q 0, Q 0,... Q 0,N, (.3.6) which generalizes the ordering used in the case of two subsaces in the revious section. Then the otimal Q i can be written as 8 Q ot i = >< >: P k i=! i if Q (k) th 0 Q 0,i if i>k. P k i=! iq 0,i Q + Q 0,i ale Q ale Q(k+) th and i ale k, P k i=! P k i i=! iq 0,i P k i=! P k i i=! iq 0,i (.3.7) Here k =,,...,N and we introduced the notation and also Q (N+) th Q (k) th = P k i=! iq 0,i Q 0,k P k i=! i Q 0,k. (.3.8) = Q 0 = Q max,cf. (.3.3). Obviously,fork =, these results reroduce the results for two subsaces, Eqs. (..6) (..8). Inserting the otimal failure rates into the subsace-error rates P e,i,(..9),givesthe otimal error rates for the subsaces, P ot e,i. Then using these otimal subsace-error rates

42 .3. FRIO DISCRIMINATION OF TWO RANK N MIXED STATES, POVM REGIME 3 in (.3.4) gives the total otimal error rate P E = P N i= P ot e,i, 8 >< P E = >: " Q ale Q r Pk i=! i Q Pk i=! iq 0,i Q P q # N i=k+ (! i ) (! i Q 0,i ) if Q (k) th q and k<n ( Q) (Q 0 Q) if Q (N) th <Q ale Q which is valid in all regions of the arameter Q. For N =,(.3.9)reducestothetwosubsaces solution, (??). For the maximum allowable inconclusive rate, Q = Q 0,thesecond line in Eq. (.3.9) holds and it reduces to P E =0,corresondingtootimalUnambiguous Discrimination of the subsaces, while for Q =0the first line holds and it reduces to the Minimum Error exression for N subsaces, as exected. Again, the most interesting asect of the otimal solution is that a structure with multile thresholds emerges. For Q () th = 0 ale Q < Q () th the total available inconclusive rate is accommodated by the first subsace only and all others oerate at the Minimum Error level. According to the hierarchy introduced in Eq. the largest Q 0,i. Then between Q () th ale Q<Q(3) th (.3.6), the first subsace is the one with the second subsace, the one with the second largest Q 0,i will also articiate in sharing the available inconclusive rate, while the remaining N the interval Q (k) th and the remaining N Q (N) th rate. subsaces continue to oerate at the minimum error level. In general, in ale Q<Q(k+) th the first k subsaces share the available inconclusive rate k subsaces remain at the minimum error level. Finally, in the range ale Q ale Q 0 = Q max all N subsaces articiate in sharing the available inconclusive It is easy to show that the exressions are continuous at the threshold, i.e. the exressions valid below the threshold and the ones valid above the threshold tend to the same values at the threshold, although their sloes are, in general, different below and above the threshold. Furthermore, if! i Q 0,i >! i+ Q 0,i+,thei th subsace always accommodates more inconclusive rate than the i + st for all i =,,...,N, otherwise above some value of the total Q the

43 .3. FRIO DISCRIMINATION OF TWO RANK N MIXED STATES, POVM REGIME 33 i + st subsace will accommodate more inconclusive rate than the i th subsace, the Q ot i (Q) curves will intersect (see art (b) of Fig... for an examle). We illustrate these results on the examle of N =3.InFig..3.,welottheotimal failure rates! i Q ot i and in Fig..3., we lot the otimal error rates! i P ot e,i for the three subsaces as a function of the total failure rate Q for some secific values of the arameters. Figure.3.. Otimal subsace failure rates! i Q ot i vs. Q from Eq. (.3.7), for three subsaces (i =,, 3).! Q ot : solid line.! Q ot : dashed line.! 3 Q ot 3 : dotted line. For the figure we used = =/ and the following arameter values: = /4, = /3, 3 = /3, r =5/8, r =/4, r 3 =/8, s =3/8, s =/4 and s 3 =3/8. Figure.3.. Otimal subsace error rates! i P ot e,i vs. Q from Eq. (..9) with (.3.7), for three subsaces (i =,, 3).! P ot e, : solid line.! P ot e, : dashed line.! 3 P ot e,3 : dotted line. For the lots we used the same arameter values as for Fig..3.. The results resented so far are valid if the arameters are such that in all subsaces we are in the POVM regime, i.e., Q i is in the unshaded region (region II) of Fig. for all i. When the arameters are such that in some subsaces we are in the rojective regime, i.e., Q i falls in the shaded regions of Fig. (regions I and III) for some i we have to modify the

44 .4. PROJECTIVE REGIME 34 treatment to account for the fact that the error exression for the corresonding subsace, P e,i, is no longer given by Eq. (..9) but by (..3). We will study this case in the next section..4. Projective regime We have seen for the single subsace case that the POVM solution is valid if the inconclusive rate is smaller than a boundary value, Q ale Q b, where Q b is given by Eq. (..). With an obvious generalization, the POVM solution holds in subsace i if in that subsace Q i ale Q b,i holds where the subsace boundary value is given by Q b,i,i,i sin i /( Q 0,i ). (.4.) In the region Q b,i ale Q i ale Q c,i the otimal measurement is a standard rojective quantum measurement (SQM). The curves Q b,i and Q c,i intersect at,i = (l),i and,i = (r),i,the same oints as in Eq. (..). The interval 0 ale,i ale is thus divided into three regions. In regions I and III, we have Q b,i <Q c,i and the solution (..0) is valid for 0 ale Q i <Q b only. In Region II, (l),i ale,i ale (r),i,wehaveq c,i = Q 0,i <Q b,i and the solution (..0) is valid for the entire 0 ale Q i ale Q c,i range. Thus, Fig... is valid in every subsace, with the obvious change of axis labels to Q i and,i.so,q i >Q b,i occurs in regions I and III and in the shaded areas the otimal FRIO measurement is an SQM while in the unshaded area it is a POVM. We now illustrate the case when the FRIO measurement is a POVM in one and an SQM in the other subsace on an examle. The otimal distribution of the total available inconclusive rate Q between the two subsaces, Q ot and Q ot such that their sum satisfies Q ot +Q ot = Q,canbefoundbyotimizing the total error rate with resect to the failure rate of the subsaces. We now have to use the error exression (..3) in subsace for Q>Q b,. This leads to a numerical otimization roblem, the result of which is shown in Fig..4..