We present the optimal measurement strategy for distinguishing between three
quantum states exhibiting a mirror symmetry. The three states live in a
two-dimensional Hilbert space, and are thus overcomplete. By mirror symmetry we
understand that the transformation {|+> -> |+>, |-> -> -|->} leaves the set of
states invariant. The obtained measurement strategy minimizes the error
probability. An experimental realization for polarized photons, realizable with
current technology, is suggested.Comment: 4 pages, 2 figure

B. Huttner

Claire R. Gilson

Erika Andersson

H. P. Yuen

J. Mizuno

Kieran Hunter

M. A. Neumark

M. Ban

M. Sasaki

P. Hausladen

R. B. M. Clarke

S. M. Barnett

S. Virmani

Stephen M. Barnett

English

arXiv

We present the optimal measurement strategy for distinguishing between three quantum states exhibiting a mirror symmetry. The three states live in a two-dimensional Hilbert space, and are thus overcomplete. By mirror symmetry, we understand that the transformation {|+〉⃗|+〉,|-〉⃗-|-〉} leaves the set of states invariant. The obtained measurement strategy minimizes the error probability. An experimental realization for polarized photons, realizable with current technology, is suggested

Andersson, E.

Barnett, S.M.

Gilson, C.R.

Hunter, K.

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Minimum-error discrimination between three mirror-symmetric states

We present the optimal measurement strategy for distinguishing between three quantum states exhibiting a mirror symmetry. The three states live in a two-dimensional Hilbert space, and are thus overcomplete. By mirror symmetry, we understand that the transformation {|+〉→|+〉,|-〉→-|-〉} leaves the set of states invariant. The obtained measurement strategy minimizes the error probability. An experimental realization for polarized photons, realizable with current technology, is suggested

Name not available

We present the optimal measurement strategy for distinguishing between three quantum states exhibiting a mirror symmetry. The three states live in a two-dimensional Hilbert space, and are thus overcomplete. By mirror symmetry, we understand that the transformation {\+&gt;--&gt;\+&gt;,\-&gt;--&gt;-\-&gt;} leaves the set of states invariant. The obtained measurement strategy minimizes the error probability. An experimental realization for polarized photons, realizable with current technology, is suggested.</p

Andersson, Anna Erika Elisabeth

Barnett, Stephen M

Gilson, Claire R

Hunter, Kieran

Heriot Watt Pure

Andersson, Erika

Barnett, Stephen M.

Gilson, Claire R.

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryAndersson, E. and Barnett, S.M. and Gilson, C.R. and Hunter, K. (2002) Minimum-errordiscrimination between three mirror-symmetric states. Physical Review A, 65 (5). 052308-1. ISSN1050-2947Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/arXiv:quant-ph/0201074 v1   17 Jan 2002Minimum-error discrimination between three mirror-symmetric statesErika Andersson1, Stephen M. Barnett1, Claire R. Gilson2, and Kieran Hunter11Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 0NG, Scotland2 Department of Mathematics, University of Glasgow, Glasgow G12 8QQ, Scotland(Dec 14, 2001)We present the optimal measurement strategy for distin-guishing between three quantum states exhibiting a mirrorsymmetry. The three states live in a two-dimensional Hilbertspace, and are thus overcomplete. By mirror symmetry weunderstand that the tranformation {|+〉 → |+〉, |−〉 → −|−〉}leaves the set of states invariant. The obtained measurementstrategy minimises the error probability. An experimental re-alization for polarized photons, realizable with current tech-nology, is suggested.03.67.-a,03.67.HkQuantum communication theory is concerned with thetransmission of information using quantum states andchannels. A sender encodes a message onto a set of signalstates, and the task for the receiving party is to decodethe message as well as possible. Suppose |ψi〉 is a setof M quantum states, each occurring with probabilitypi. This set, the “letters of the alphabet”, and the priorprobabilities are known also to the receiving party.A general measurement strategy can be described interms of a probability operator measure (POM), alsocalled a probability operator-valued measure [1–3]. Thedifferent measurement outcomes, labeled by j, are associ-ated with operators pˆij , called the elements of the POM.Given that the system to be measured is prepared in state|ψi〉, the probability to obtain result j isp(j|i) = 〈ψi|pˆij |ψi〉. (1)For a von Neumann measurement, the operators pˆij areprojectors onto the orthonormal eigenstates of the ob-servable to be measured. In general, however, pˆij are notorthogonal. Nevertheless, all the eigenvalues of pˆij haveto be positive or zero, and the POM elements sum to theidentity operator,∑j pˆij = 1ˆ. These conditions reflectthe facts that probabilities are non-negative and that themeasurement always yields a result, even if the resultmight not provide any information.The task is now to find an optimal measurement strat-egy based on the knowledge of the signal states and theirprior probabilities. One possibility is to choose the mea-surement strategy which minimises the probability of er-ror in assigning the correct signal state. If the prior prob-ability for signal state |ψi〉 is pi, the error probability isgiven byPerror = 1−M∑j=1〈ψj |pˆij |ψj〉pj . (2)The conditions which the minimum-error strategymust satisfy are known to be [1]pˆij(pj |ψj〉〈ψj | − pk|ψk〉〈ψk|)pˆik = 0 ∀ j, k, (3)M∑j=1pj |ψj〉〈ψj |pˆij − pk|ψk〉〈ψk| ≥ 0 ∀ k, (4)where the inequality in the second condition states thatall the eigenvalues of the operator on the left-hand sidemust be greater than or equal to zero. These conditionsare in general not transparent enough to be used for ob-taining the optimal solution. In fact, the optimal strategyis only known for some special cases, including the casewith only two signal states [1], symmetric states [1,4,5],and equiprobable states that are complete in the sensethat a weighted sum of projectors onto the states equalsthe identity operator [6]. For linearly independent states,the optimal measurement is known in some cases [7].We will consider the situation in which the signal statesare given by|ψ1〉 = cos θ|+〉+ sin θ|−〉|ψ2〉 = cos θ|+〉 − sin θ|−〉 (5)|ψ3〉 = |+〉,with prior probabilitites p1,2 = p and p3 = 1− 2p, where0 ≤ p ≤ 1/2. Due to symmetry it is enough to con-sider the range 0 ≤ θ ≤ pi/2. The states |+〉 and |−〉are orthonormal basis states. As a special case, the so-called trine states are obtained when θ = pi/3 and all theprobabilities are equal to 1/3 [8].The transformation {|+〉 → |+〉, |−〉 → −|−〉} leavesthe set of signal states unchanged. We expect that thePOM elements will exhibit the same symmetry as the setof signal states, and are led to the Ansatzpˆi1,2 = |φ1,2〉〈φ1,2|,pˆi3 = (1− a2)|+〉〈+| = |φ3〉〈φ3|, (6)where|φ1,2〉 = 1/√2(a|+〉 ± |−〉), (7)with “+” referring to |φ1〉, “-” to |φ2〉, and 0 ≤ a ≤ 1.If p is large enough (certainly if p = 1/2), we mightexpect that the optimal measurement strategy is theone which distinguishes optimally between |ψ1〉 and |ψ2〉.This is the case when a = 1 and1pˆi1 =12(|+〉+ |−〉)(〈+|+ 〈−|),pˆi2 =12(|+〉 − |−〉)(〈+| − 〈−|), (8)pˆi3 = 0.It is convenient to start by investigating when thisstrategy is optimal. The POM elements (8) can bechecked to satisfy condition (3). The inequality condi-tion (4) is seen to hold for k = 1, 2. For k = 3, usingmatrix notation,|+〉 ≡(10); |−〉 ≡(01), (9)it can be writtenp(cos θ + sin θ)(cos θ 00 sin θ)− (1− 2p)(1 00 0)≥ 0,(10)which is satisfied ifp ≥ 12 + cos θ(cos θ + sin θ). (11)Hence the measurement strategy given by Eq. (8), whichdistinguishes optimally between |ψ1〉 and |ψ2〉, is the bestchoice provided p1,2 = p is large enough.If p is too small to satisfy the inequality (11), we needto consider a three-element POM, as given by the originalAnsatz in Eq. (6). The equality condition (3) holdstrivially for j = 1 and k = 2. For j = 1, 2 and k = 3 itleads to the requirementa =p cos θ sin θ1− p(2 + cos2 θ) . (12)With this value of a, the inequality condition (4) fork = 3 is satisfied, the smaller eigenvalue being zero. Totest the inequality condition for k = 1, 2, it is convenientto make use of the fact that a 2 × 2 Hermitian matrixwith elements a11, a12, a21 and a22 is positive semidefiniteif and only if a11, a22 and the determinant of the matrixare all greater that or equal to zero. Upon evaluatingthe l.h.s. of condition (4) in matrix form, one finds thatits elements a11, a22 are both greater than zero. Thedeterminant turns out to be zero when the required valueof a is used.It follows that this is indeed the measurement strategywhich minimises the error probability. As shown in Fig.1, there are two regimes depending on p and θ. If p1,2 = pis large enough to satisfy the inequality (11), the strategywith pˆi3 = 0, which distinguishes optimally between |ψ1〉and |ψ2〉, is the solution. If p is smaller than this, thefull three-element measurement is needed, with a givenby Eq. (12). The crossover between these two regimescorresponds to p = (2+ cos θ(cos θ+sin θ))−1, where thetwo strategies will coincide.P success1000.5p0pi/2θTwo POM elementsThree POM elementspi/4FIG. 1. The success probability Psuccess as a function ofp and θ. The optimal measurement strategy will have eithertwo or three elements.It is straightforward to calculate the success probabil-ity, that is, the probability of correctly identifying thesignal state, of the devised optimal measurement strat-egy. For p ≥ 1/ [2 + cos θ(cos θ + sin θ)], we findPsuccess = 1− Perror = p(1 + sin 2θ). (13)As a special case, for p = 1/2 with 0 ≤ θ ≤ pi/2, thisreduces to the probability to distinguish with minimalerror between two non-orthogonal states. For p = 1/2and θ = pi/4, the success probability is equal to one,corresponding to the situation where the two states areorthogonal.For p ≤ 1/ [2 + cos θ(cos θ + sin θ)], the success proba-bility isPsuccess =(1− 2p) [p sin2 θ + 1− 2p− p cos2 θ]1− 2p− p cos2 θ , (14)which reduces to 2/3 for equal probabilities (p1,2 = p3 =1/3), as long as pi/4 ≤ θ ≤ pi/2.In many of the measurement situations where the opti-mal measurement strategy is known, the optimal strategyis the square-root or “pretty good” measurement [4,7,9]with POM elementspii = ρˆ−1/2pi|Ψi〉〈Ψi|ρˆ−1/2, (15)whereρˆ =3∑i=1pi|Ψi〉〈Ψi|. (16)It is interesting to note that the square root measure-ment strategy will coincide with the optimal strategy forsome nontrivial p and θ, but is in general not the optimalstrategy.2In a way similar to recent optical realisations of gen-eralised measurements [8,10–12], it is possible to realisethe devised measurement strategy for photon polariza-tion. The states |+〉 and |−〉 will now correspond toorthogonal polarisation states of a single photon, for ex-ample horizontal and vertical linear polarisation |H〉 and|V 〉. The realisation relies on the fact that any gener-alised measurement can be extended to a projective (vonNeumann) measurement in a higher-dimensional Hilbertspace, the so-called Naimark extension [1–3,13]. Physi-cally, this can be achieved by coupling the system to anancilla, or for example as in the optical realisations, bythe introduction of additional input ports to the system,thus extending the Hilbert space.An optical network with beam splitters and waveplates can be used to couple the different polarisationstates, and the measurement result is obtained by de-tecting where the photon exits, as shown in Fig. 2. Thesignal state is incident on one of the input ports of thepolarising beam splitter PBS1. In the case we are con-sidering, one auxiliary degree of freedom is needed, andthis is provided by the second input port of the beamsplitter. Only vacuum is incident through this input.PD1PD2PD3PBS1PBS2PBS3WP2WP1M2M1      | 0        2signal stateauxiliarydegree offreedom | Ψ   1iFIG. 2. Optical network for implementing the min-imum-error measurement on the three mirror-symmetricstates. The state to be measured is incident on PBS1. PBS1,PBS2 and PBS3 are polarising beam splitters, transmittinghorisontally polarised light and reflecting vertically polarisedlight. WP1 and WP2 are waveplates, and M1 and M2 aremirrors.In matrix notation, for correct settings of the wave-plates [14], the optical network effects the unitary trans-formationU =1√2 a 1√1− a2a −1 √1− a2√2(1− a2) 0 −√2a , (17)using|H〉1 ≡ 100 , |V 〉1 ≡ 010 , |V 〉2 ≡ 001 . (18)Here |H〉1 and |V 〉1 are used to encode the signal state,|V 〉2 being an auxiliary degree of freedom provided by thesecond input port of PBS1. The mode |H〉2 is not coupledto the signal state modes, as horizontally polarized lightincident through the second input port of PBS1 will betransmitted by both beam splitters, always resulting indetection at PD3. The rows of the matrix U are simplythe POM element vectors |φ1,2,3〉, but extended in thethird auxiliary dimension so that the matrix row vectorsare orthogonal.The measurement at the output then distinguishes be-tween horisontally and vertically polarised photons inmode 1 (PD1 and PD2, outcome 1 and 2), and (verti-cally polarised) photons in mode 2 (PD3, outcome 3). Ifthe incident state is |ψi〉, the state after the network isU |ψi〉 =3∑j=1|j〉〈φj |ψi〉, (19)where |1〉 ≡ |H〉1, |2〉 ≡ |V 〉1 and |3〉 ≡ |V 〉2. It is easyto confirm thatp(j|i) = 〈ψi|pˆij |ψi〉= 〈j|U |ψi〉〈ψj |U †|j〉, (20)so that the setup indeed realises the intended measure-ment strategy.We want to stress that an optical setup like this isstraightforward to realise. Indeed, a slightly different,but essentially equivalent network was used in the ex-periments of Clarke et al. [8,12]. Another equivalent net-work was suggested, for a different measurement task, bySasaki et al. [15], and has recently been realized exper-imentally [16]. In summary, we have obtained the mea-surement strategy which minimises the error probabil-ity when distinguishing between three mirror-symmetricstates. This example is noteworthy in that the number ofnon-zero POM elements, for the minimum-error strategy,depends on the parameters chosen for the set of states. Itwill be interesting to see if this unusual property persistsfor other measures of detection strategy such as mutualinformation [15] and fidelity [17].We would like to thank P. O¨hberg for his assistancein preparing the manuscript. This work was supportedby the European Commission Marie Curie Fellowshipscheme, the Royal Society of Edinburgh, the ScottishExecutive Education and Lifelong Learning Department,3the International Centre for Theoretical Physics and theUK Engineering and Physical Sciences Research Council.[1] C. W. Helstrom, Quantum detection and estimation the-ory (Academic, New York, 1976).[2] K. Kraus, States, Effects and Operations: FundamentalNotions of Quantum Theory (Springer, Berlin, 1983).[3] A. Peres, Quantum Theory: Concepts and Methods,(Kluwer Academic Publishers, Dordrecht 1993).[4] M. Ban, Y. Kamazaki, R. Momose, and O. Hirota, Int.J. Theor. Phys. 36, 1269 (1997).[5] S. M. Barnett, Phys. Rev. A 64, 030303(R) (2001).[6] H. P. Yuen, R. S. Kennedy, and M. Lax, IEEE Trans.Inf. Theory IT-21, 125 (1975).[7] M. Sasaki, K. Kato, M. Izutsu, and O. Hirota, Phys. Rev.A 58, 146 (1998).[8] R. B. M. Clarke, V.M. Kendon, A. Chefles, S. M. Barnett,E. Riis, and M. Sasaki, Phys. Rev. A 64, 012303 (2001).[9] P. Hausladen and W. K. Wootters, J. Mod. Opt. 41, 2385(1994).[10] B. Huttner, A. Mu¨ller, J. D. Gautier, H. Zbinden, andN. Gisin, Phys. Rev. A 54, 3783 (1996).[11] S. M. Barnett and E. Riis, J. Mod. Opt. 44, 1061 (1997).[12] R. B. M. Clarke, A. Chefles, S. M. Barnett, and E. Riis,Phys. Rev. A 63, 040305 (2001).[13] M. A. Neumark, Izv, Akad. Nauk. SSSR, Ser. Mat. 4,227 (1940).[14] The wave plates rotate the polarisation according toΛˆ(φ1,2/2) with φ1 = arccos a and φ2 = pi/4, whereΛˆ(φ2) =(cosφ sinφsinφ − cosφ); |H〉 ≡(10), |V 〉 ≡(01).[15] M. Sasaki, S. M. Barnett, R. Jozsa, M. Osaki, and O.Hirota, Phys. Rev. A 59, 3325 (1999).[16] J. Mizuno, M. Fujiwara, M. Sasaki, M. Akiba, T. Kawan-ishi, and S. M. Barnett, Phys. Rev. A. 65, 012315 (2002).[17] S. M. Barnett, C. R. Gilson, and M. Sasaki, J. Phys.A: Math. Gen. 34, 6755 (2001); C. Fuchs and M. Sasaki(private communication).4

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Minimum-error discrimination between three mirror-symmetric states

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