Optimal Measurements for Symmetric QuantumStates with Applications to Optical Quantum

CommunicationHari Krovi

Raytheon BBN TechnologiesEmail: hkrovi@bbn.com

Saikat GuhaRaytheon BBN Technologies

Email: sguha@bbn.com

Zachary DuttonRaytheon BBN Technologies

Email: zdutton@bbn.com

Marcus P. da SilvaRaytheon BBN Technologies

Email: msilva@bbn.com

Abstract—Minimum probability of error (MPE) measurementsare quantum mechanical measurements that discriminate be-tween a set of candidate states and achieve the minimum errorallowed by quantum mechanics. Conditions for a measurement tobe an MPE measurement have been derived by Yuen, Kennedyand Lax. But finding explicit measurements that satisfy theseconditions is a hard problem in general and even in cases wherean explicit measurement is known, calculating the MPE of themeasurement is not easy. In some cases, MPE measurements havebeen found such as when the states form a single orbit undera group action i.e., there is a transitive group action on thestates. For such state sets, termed Geometrically Uniform (GU)in [1], it was shown that the ‘pretty good measurement’ (PGM)is optimal. However, calculating the MPE and other performancemetrics for the PGM involves inverting large matrices, and it istherefore not easy in general to evaluate. Our first contributionis a formula for the MPE and conditional probabilities of GUsets using group representation theory.

Next, we consider sets of pure states that have multiple orbitsunder the group action. Such states are termed compound geo-metrically uniform (CGU). CGU sets appear in many practicalproblems in quantum communication and imaging. For example,they include all linear codes formed using pure-state modulationconstellations, which are known to achieve the ultimate (Holevo)capacity of optical communication. Optimal (MPE) measurementfor general CGU sets are not known. In this paper, we show howour representation-theoretic description of optimal measurementsfor GU sets naturally generalizes to the CGU case. We show howto compute the optimal measurement for CGU sets by reducingthe problem to solving a few simultaneous equations. The numberof equations depends on the sizes of the multiplicity spaceof irreducible representations. For many group representations(such as those of several practical good linear codes), this is muchmore tractable than solving large semi-definite programs—whichis what is needed to solve for MPE measurements for an arbitraryset of pure states using the Yuen-Kennedy-Lax conditions [2].We show examples of the evaluation of optimal measurementsfor CGU states.

I. INTRODUCTION

Optimal discrimination of quantum states is central to alarge number of key problems in quantum information the-ory. Quantum state discrimination finds applications in: (i)sensing—for instance, in task-specific optical imaging [3],quantum reading [4], and pixelated image discrimination [5];

This material is based upon work supported by the Defense AdvancedResearch Projects Agency’s (DARPA) Information in a Photon (InPho)program, under Contract No. HR0011-10-C-0159.

(ii) communication—for instance, in decoding error correctingcodes for classical communication over a quantum channel [6],[7], [8], [9] and optimal M -ary phase discrimination undera photon budget constraint [10]; and (iii) computation—forinstance, in quantum algorithms for hidden subgroup prob-lems [11], and quantum state and process tomography [12],[13]. The problem of describing a quantum measurement tooptimally discriminate between a set of quantum states, i.e.,to optimize a given metric, was first considered by Yuen,Kennedy and Lax in [2], where they showed that the optimalmeasurement is one whose measurement operators satisfy aparticular semi-definite program, which is described later inthis paper 1. For arbitrary states however, finding the solutionsof the YKL semi-definite program can be computationallyhard. However, upper and lower bounds on the MPE have beenrecently obtained for the general problem [14]. Restrictingto pure states with a group symmetry makes the problem offinding the exact MPE more tractable. Helstrom considered theproblem of finding the optimal measurement and the exactMPE for states that have cyclic group symmetry [15]. Thiswas extended to arbitrary abelian groups with a transitiveaction by Forney and Eldar [16]. In fact, they showed thatthe pretty good measurement (PGM), also called the leastsquares measurement (LSM), as defined in [20] and [21], isoptimal in this case. Later, for any non-abelian group with atransitive action on the states, Eldar et al., [22] showed thatthe PGM was again optimal. However, an expression for theMPE was not known. In this paper, we fill this gap using grouprepresentation theory.

When the states are compound geometrically uniform(CGU), i.e., they have multiple orbits under the group action(restricting to group actions that are permutation representa-tions), very little is known about the structure of the optimalmeasurement. Solving the general CGU state discriminationproblem is particularly useful for the decoding of linear codesfor classical quantum (cq) channels, i.e., for sending classicaldata over a quantum (such as, an optical) channel. It was re-cently shown that a generalization of Arikan’s polar codes [17]can achieve the Holevo capacity of any cq channel [8].

1We use the terms optimal, minimum probability of error (MPE) and Yuen-Kennedy-Lax (YKL) measurement interchangeably.

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In particular, cq polar codes, which are linear codes, canachieve the Holevo capacity of optical communication usingcoherent-state (ideal laser light) modulation [18]. However,unlike Arikan’s polar codes, the quantum polar codes areconstructed using a Holevo-information criterion (as opposedto Shannon mutual information), and require the receiver tomake a succession of collective (joint-detection) measurementson the received codeword. This is why these codes can supporta higher rate (the Holevo capacity) on the physical opticalchannel, as opposed to Arikan’s classical polar code, whichcan support unto the Shannon capacity of the channel inducedby any of the conventional optical receivers. It is highlylikely that there exist other linear codes, which paired withjoint-detection measurements, would also achieve the Holevocapacity. Any linear code has an automorphism group andthe action of this group on the code is a permutation actionand hence is a CGU set. This group action carries over, ingeneral, to modulated code words. For example, for binarycodes used over a binary-phase-shift coherent-state alphabet(|α〉, | − α〉), the bit flip operation maps to the π phase-shiftoperation in the modulated domain, i.e., Uπ| ± α〉 = | ∓ α〉,Uπ = eiπa

†a. Therefore, finding the optimal decoder of aset of pure states with a CGU action will enable finding theoptimal performance of any coherent-state modulated linearcode. Finding the optimal measurement for CGU may alsolead to useful insights towards finding a structured design ofan optical receiver to implement the optimal measurement.Our examples in this paper are motivated by optical quantumcommunications with coherent-state codewords 2.

In this paper, we consider both the GU and the CGUaction on states that are linearly independent. In the caseof GU action, we show how one can use representationtheory to calculate the probability of error and conditionalprobabilities. Then we consider the CGU action and show thatone can reduce the problem to a set of simultaneous equations.The number of these equations depends on the sizes of themultiplicity spaces of the representation and the number oforbits. We present examples to show the usefulness of thismethod when the representations have small multiplicities andfew orbits.

This paper is organized as follows. In section II, we describethe problem of discriminating between quantum states whenthe states are pure and are linearly independent. We describethe gram matrix approach given in Helstrom [15] along witha caveat about this approach. In section III, we describethe optimal measurement for GU states. This descriptiongeneralizes the ones in [15], [16] for abelian groups to non-abelian groups using group representation theory. Then insection, IV, we describe how one can obtain the optimalmeasurement for CGU states. We show how to reduce thenumber of simultaneous equations based on the representationof the group. Then we give examples to illustrate this method.Finally, in sectionV, we present our conclusions.

2Coherent-state modulation is known to be sufficient to attain the Holevocapacity of communication on a pure-loss optical channel [19]

II. OPTIMAL MEASUREMENTS FOR PURE STATES

In this section, we describe two approaches to finding theoptimal measurement. Suppose we are given an ensemble{pi, |ψi〉} , 1 ≤ i ≤ n, of n linearly-independent pure statesand an associated prior distribution. It can be shown thatwhen distinguishing pure states, the optimal measurement isa projective measurement and is unique [15]. Therefore, letus assume that the optimal measurement is given by theorthonormal basis |wi〉. Now define two matrices: the matrixM whose columns are the states

√pi|ψi〉 and the matrix

X whose elements are xij =√pj〈wi|ψj〉. Since the states

are linearly independent, each state lies in an n dimensionalHilbert space and M is an invertible n×n matrix. The matrixX denotes the solution to the state discrimination problemsince all the information about the measurement vectors canbe obtained from X . It satisfies the equations

X†X = Γ, and (1)

xkmx∗mm = xkkx

∗mk , (2)

where Γ is the Gram matrix of the set of states, i.e., (Γ)ij =〈ψi|ψj〉. Although in [15], it is suggested that these equationslead to the solution, these two equations alone do not give aunique solution. In [2], it was shown that along with the abovetwo equations, an inequality must be satisfied. Only then doesone get a unique solution for pure states. However, in certaincases of interest, one can get a small set of solutions usingthe above two equations as we show later.

It is useful to view this in terms of the polar decomposition.The left and right polar decomposition of the matrix M isgiven by

M = U√M†M =

√MM†U .

In the above equation, M†M is just the Gram matrix Γ ofthe set of states {|ψ1〉, . . . , |ψn〉} and U is a unitary matrix.Denote

√M†M by P . It is known that if M is invertible, then

P and U are unique, with P being a positive semi-definitematrix. Clearly, P satisfies P †P = Γ as does any matrix ofthe form V P , where V is unitary. Since V P always satisfiesEq. 1 for any unitary V , it is chosen so that V P satisfies Eq. 2as well. Therefore, the matrix X is in general of the form V Pand the measurement vectors |wi〉 are columns of the matrixUV †. Finally, note that if the solution X turns out to be suchthat pkxkk = pmxmm, then Eq. 2 becomes xkm = x∗mk forall k and m i.e., X is Hermitian. In the next section, we willshow that the above condition is satisfied for GU states.

III. GEOMETRICALLY UNIFORM STATES AND THE PRETTYGOOD MEASUREMENT

We say that a set of states are geometrically uniform ifthere is a group G acting transitively on them i.e., for everytwo states |ψi〉 and |ψj〉 there exists a group element g suchthat R(g)|ψi〉 = |ψj〉, where R is some representation ofthe group G. This implies that all the elements of the setare obtained from a single element, say |ψ1〉 by the actionof the group. If the states are linearly independent, then the

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representation of the group on the space spanned by the statesis the induced representation of the trivial representation of thestabilizer subgroup of |ψ1〉. For a state discrimination problemto be GU, one usually assumes that the priors are the samefor all states.

Here we show that for geometrically uniform states, thePGM or LSM is the optimal measurement. This was provedfor cyclic groups in [15], for abelian groups in [16] andfor non-abelian groups in [22]. The pretty good measure-ment has been defined in [20] and [21] as a measurementto discriminate between the states ρi with priors pi. Themeasurement operators of the PGM are given by Πi =piρ−1/2ρiρ

−1/2, where ρ =∑i piρi. If the states are pure

(|ψi〉) and linearly independent, this measurement becomes aprojective measurement. Consider the polar decomposition ofthe matrix M (defined above). Observe that ρ = MM† andso U = ρ−1/2M . The columns of U form the measurementbasis of the PGM. From the left polar decomposition, noticethat the columns of U are also the measurement basis ifthe solution matrix X coincides with P =

√M†M . Since

P is Hermitian, it would be the solution if in addition allthe diagonal elements of P are equal (since the priors areequal). To see that this is true for geometrically uniformstates, observe that ρ commutes with the representation R andxkk = 〈wk|ψk〉 = 〈w1|R(g)−1R(g)|ψ1〉 = 〈w1|ψ1〉 = x11.

Now we describe the measurement using non-abelian grouprepresentation theory along the lines of [15], [16] where itwas done for abelian groups. In accordance with the actionof the group, we have |ψi〉 = U(gi)|ψ1〉 for any i. Finally,we assume that the priors pi are all the same. Any transitivepermutation action on a linearly independent set is an inducedrepresentation. We pick a base point, say |ψ1〉 and with respectto this point, there is a subgroup G0 of G which stabilizes |ψ1〉.The representation on the vector space spanned by |ψi〉 is theinduced representation of the trivial representation of G0 toG. If the set of states is S, then we have that |S| = |G|/|G0|.

The Yuen, Kennedy, Lax conditions are

Υ− piψi ≥ 0,

(Υ− piψi)Πi = 0, and

Υ =∑i

piψiΠi =∑i

piΠiψi , (3)

where ψi = |ψi〉〈ψi|. Since the optimal measurement basisfor a GU set is also GU, it is easy to see that we only needthe equations where in the first two i = 1.

Let the optimal measurement basis be given by {|wi〉}which are also GU under the G action (and let Πi = |wi〉〈wi|).Therefore we have

Υ =1

|G|∑g∈G

U(g)ψ1Π1U(g−1).

Now let this representation consist of irreducible represen-tations λ with multiplicity mλ. Consider the Fourier basis|λ,m, k〉 where λ labels the irreducible representation, m itsmultiplicity and k its representation space whose dimension

is denoted dλ. The matrices U(g) are block diagonal in thisbasis and therefore the operator Υ is also block diagonal bySchur’s lemma. In order to find the probability of error, weneed to access to an arbitrary matrix element of Υ inside theblocks.

〈λ,m, k|Υ|λ′,m′, k′〉 =

1

|G|∑g

〈λ,m, k|U(g)ψ1Π1U(g−1)|λ′,m′, k′〉 . (4)

We denote ψ1 and Π1 as ψ and Π respectively. The action ofany U(g) on the state |λ,m, k〉 is given as follows

U(g)|λ,m, k〉 =∑k′

λ(g)k′,k|λ,m, k′〉 ,

where λ(g)k′,k is the k′, k matrix entry of the irreduciblerepresentation λ. Using this we get

〈λ,m, k|Υ|λ′,m′, k′〉 =

1

|G|∑g,l,l′

λ∗(g−1)k,lλ′(g−1)k′,l′〈λ,m, l|ψΠ|λ′,m′, l′〉 . (5)

Using the orthogonality relations among matrix entries ofirreducible representations, we obtain

〈λ,m, k|Υ|λ′,m′, k′〉 =1

dλδλ,λ′δk,k′

∑l

〈λ,m, l|ψΠ|λ,m′, l〉.

This shows that Υ is block diagonal with the blocks given bymultiplicity places. Let φ = (1/|S|)ψ. Now using the YKLequations, inside these invariant spaces, we see that

〈λ,m, k|(ΥΠ− φΠ)|λ′,m′, k′〉 = 0

= Υλ,mγλ,m,kγ∗λ′,m′,k′ − xλ,m,kγ∗λ′,m′,k′〈φ|w〉 = 0 , (6)

where xλ,m,k = 〈λ,m, k|φ〉 and γλ,m,k = 〈λ,m, k|w〉. Inorder to find the optimal measurement, we need to solve forγ. Suppose that γλ,m,k = xλ,m,k/cλ,m, where we need tosolve for cλ,m. We have from Eq. 6 Υλ,m = cλ,m〈φ|w〉, ifγλ,m,k 6= 0. We have that

Υλ,m =|S|dλ

∑k

xλ,m,kγ∗λ,m,k〈φ|w〉

=|S|dλ

∑k |xλ,m,k|2

cλ,m〈φ|w〉 . (7)

Using the above two equations for Υλ,m we get that

|S|∑k |xλ,m,k|2

dλ(cλ,m)2= 1 .

Notice that if

cλ,m =

√∑k |S||xλ,m,k|2

dλ,

then it satisfies the above equation. We also need Υ − φ ≥0. Let |µ〉 be an arbitrary normalized state. Then the above

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equation becomes 〈µ|(Υ− φ)|µ〉 ≥ 0. The left hand side canbe written in the Fourier basis as

∑λ,m,k

|µλ,m,k|2Υλ,m −

∣∣∣∣∣∣∑λ,m,k

xλ,m,kµ∗λ,m,k

∣∣∣∣∣∣2

=∑λ,m,k

|µλ,m,k|2cλ,m∑

λ′,m′,k′

|xλ′,m′,k′ |2

cλ′,m′−

∣∣∣∣∣∣∑λ,m,k

xλ,m,kµ∗λ,m,k

∣∣∣∣∣∣2

. (8)

Now consider the second half of the above expression∣∣∣∣∣∣∑λ,m,k

xλ,m,kµ∗λ,m,k

∣∣∣∣∣∣2

=

∣∣∣∣∣∣∑λ,m,k

√cλ,mγλ,m,k

√cλ,mµ

∗λ,m,k

∣∣∣∣∣∣2

≤∑λ,m,k

|µλ,m,k|2cλ,m∑

λ′,m′,k′

|xλ′,m′,k′ |2

cλ′,m′, (9)

where the last line was obtained through Cauchy-Schwartz.This shows that the γ are the solutions. We can assumethat the basis of the multiplicity space is picked in such away that xλ,m,k is non-zero for only one m. Then cλ,m =√|S|Tr(Pλψ)/dλ and

|w〉 =∑λ,m,k

|λ,m, k〉xλ,m,kcλ,m

=∑λ

√dλ|S|

Pλ|ψ〉√〈ψ|Pλ|ψ〉

,

where Pλ is the projector onto the isotypic space λ.We now calculate the probability of success using this ex-

pression. The probability of success is given by Ps = |〈w|ψ〉|2.This can be written as

Ps =

∣∣∣∣∣∑λ

√dλ|S|√〈ψ|Pλ|ψ〉

∣∣∣∣∣2

.

For any group with a representation U , an expression for Pλis given by (for a character χλ)

Pλ =dλ|G|

∑g

χλ(g−1)U(g).

One can simplify the expression 〈ψ|Pλ|ψ〉 as follows.

〈ψ|Pλ|ψ〉 =dλ|G|

∑g∈G

χλ(g−1)〈ψ|U(g)|ψ〉 . (10)

But we have 〈ψ|U(g)|ψ〉 = 〈ψ|U(g1gg2)|ψ〉 for all g1, g2 ∈G0, where G0 is the stabilizer group of |ψ〉 i.e., the subgroupof G such that U(g)|ψ〉 = |ψ〉, ∀g ∈ G0. This means

〈ψ|Pλ|ψ〉 =dλ|G|

∑i,g∈Ci

χλ(g−1)〈ψ|U(g)|ψ〉 , (11)

where i is a sum over (G0, G0) double coset representativesand Ci is the double coset. We evaluate this for the case whenthe candidate states are optical coherent states. Each state hasn modes with vacuum in n−2 modes and two equal intensity

coherent states some two modes. The symmetric group Sn actson this set of states and in this representation of Sn, there areonly three irreps. The trivial, standard and the a third irrepwhose Young diagram has two rows with two boxes in thesecond row (label them a, b, c respectively). In this case, G0

is S2×Sn−2 and the double coset representatives are e, (1, 3)and (1, 3)(2, 4).

IV. OPTIMAL MEASUREMENT FOR CGU SETS

In this section, we describe how to obtain the optimal mea-surement for CGU state sets. We use the Helstrom descriptionof the problem of finding the optimal measurement for purestates i.e., by viewing it as the solution of a set of simultaneousequations. However, as we pointed out earlier, one need notobtain a unique solution. With every obtained solution, wehave to check the third condition to find the right one. Weagain resort to representation theory to simplify the equationsand obtain far fewer equations (in many practical cases ofinterest). We begin by recalling the set of simultaneous equa-tions which give the solution given above in Eqs. 1 and 2.X†X = Γ, xkmx

∗mm = xkkx

∗mk where Γ is the Gram

matrix of the set of states. Since we have CGU symmetryin the problem, the Gram matrix is symmetric about a groupG and its representation U(g). Suppose this representationdecomposes into irreducible spaces λ of dimension dλ withmultiplicity mλ (as before). Then the Gram matrix is blockdiagonal in this basis with a block corresponding to eachirreducible representation. Denote these blocks as Γ′λ. Theseblocks also have a special structure where they are identity inthe representation space i.e., Γ′λ = Γλ⊗ Idλ . We note that thesolution X would have the same block diagonal decompositionsince it commutes with the same representation of G. Thenwe have that inside an isotopic space X†λXλ = Γλ. Noticethat these matrices are of dimension mλ ×mλ. Therefore, ifthe multiplicity spaces are small, then this task is easy. Now,solving for such an Xλ can be done only up to a unitary sincefor any solution Xλ, UλXλ is also a solution, where Uλ is anarbitrary unitary operator. In order to find a solution, we needto use the set of equations in Eq. 2.

We illustrate this with an example coming from opticalquantum communications. Suppose that we want to discrimi-nate between the following n fold tensor product states. Then states with a coherent state α in any of the n modes (andvacuum in the others), another n states with an β in any ofthe n modes. This set of states is CGU under the cyclic groupaction. Therefore, in the Fourier basis, the Gram matrix has n2× 2 blocks. The first block is(

1 + (n− 1)e−|α|2

C

C 1 + (n− 1)e−|β|2

),

where C = exp(αβ∗ − |α|2+|β|22 ) + (n − 1) exp( |α|

2+|β|22 ).

The other blocks are(1− n(n− 1) exp(−|α|2) D

D 1− n(n− 1) exp(−|β|2)

),

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where D = exp(αβ∗ − |α|2+|β|22 )− n(n− 1) exp( |α|

2+|β|22 ).

Suppose that X has entries x11, . . . x22 for the first block andy’s for the other 2× 2 blocks, then we obtain the equations

p2(x23 + (n− 1)y12)(x∗33 + (n− 1)y∗22) =

p1(x22 + (n− 1)y11)(x∗32 + (n− 1)y∗21) (12)p2(x23 − y12)(x∗33 − y∗22) = p1(x22 − y11)(x∗32 − y∗21) (13)

For many such orbits, these equations can be generalizedeasily. After solving these equations, one obtains the followingprobability of success(√

1 + (n− 1)e−|α|2 + e−12 |α−β|2 + (n− 1)e−

12 (|α|2+|β|2)

+

√1 + (n− 1)e−|α|2 − e− 1

2 |α−β|2 − (n− 1)e−12 (|α|2+|β|2)

+ (n− 1)

√1− e−|α|2 + e−

12 |α−β|2 − e− 1

2 (|α|2+|β|2)

+ (n− 1)

√1− e−|α|2 − e− 1

2 |α−β|2 + e−12 (|α|2+|β|2)

)2 1

4n2

V. CONCLUSIONS

We have developed a new and compact interpretation—using group representation theory—of the minimum proba-bility of error (MPE) measurement for distinguishing a set ofgeometrically-uniform pure quantum states—states that form asingle orbit under the group action i.e., a transitive action. Wedetermine the probability of error for the pretty good measure-ment, or equivalently the least squares measurement, whichis the optimal measurement for GU states. Using the sameframework, we extended our analysis to construct optimalmeasurements for compound geometrically uniform (CGU)state sets, which have multiple orbits under the group action.CGU sets appear in many practical problems, particularly intransmitting classical data over a (quantum) optical channel.All linear codes formed using pure-state modulation constel-lations, which are known to achieve the ultimate capacity ofoptical communication, are CGU. We showed how to computethe optimal measurement for CGU sets by reducing theproblem to solving a few simultaneous equations. The numberof equations depends on the sizes of the multiplicity spacesof irreducible representations. For many group representations(such as those of several practical good linear codes), this isa lot more tractable than solving large semi-definite programsto solve by brute force the Yuen-Kennedy-Lax conditions [2]for determining optimal measurements for an arbitrary set ofpure states. We showed some examples of the evaluation ofoptimal measurements for CGU states.

We know that linear codes achieve the Holevo capacity ofoptical communication [18]. However, there is a gap betweenthe Holevo capacity and the Shannon capacity of the opticalchannel attainable by a direct-detection receiver, and the gapwidens in the low photon number regime [23]. In continuingwork, we are investigating explicit finite blocklength codefamilies, specifically constant-composition codes (which bydefinition meet the Holevo capacity) with good symmetryproperties, whose capacity performance with the respectiveoptimal measurements, bridge the aforesaid capacity gap. We

are also working on trying to develop rigorous foundationsfor translating our CGU results to an algorithmic design ofstructured optical receivers built using a small universal setof known optical components that can implement the optimalmeasurement on any given linear code.

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