Distinguishability of countable quantum states and von Neumann lattice
aa r X i v : . [ qu a n t - ph ] F e b Distinguishability of countable quantum states andvon Neumann lattice
Ryˆuitirˆo Kawakubo and Tatsuhiko Koike , Department of Physics, Keio University, Yokohama 223-8522, Japan REC for NS, Keio University, Yokohama 223-8521, JapanE-mail: [email protected] and [email protected]
18 September 2018
Abstract.
Condition for distinguishability of countably infinite number of purestates by a single measurement is given. Distinguishability is to be understoodas possibility of an unambiguous measurement. For finite number of states, it isknown that the necessary and sufficient condition of distinguishability is that thestates are linearly independent. For infinite number of states, several natural classesof distinguishability can be defined. We give a necessary and sufficient conditionfor a system of pure states to be distinguishable. It turns out that each levelof distinguishability naturally corresponds to one of the generalizations of linearindependence to families of infinite vectors. As an important example, we applythe general theory to von Neumann’s lattice, a subsystem of coherent states whichcorresponds to a lattice in the classical phase space. We prove that the condition fordistinguishability is that the area of the fundamental region of the lattice is greater thanthe Planck constant, and also find subtle behavior on the threshold. These facts revealthe measurement theoretical meaning of the Planck constant and give a justificationfor the interpretation that it is the smallest unit of area in the phase space. The casesof uncountably many states and of mixed states are also discussed.
Keywords : foundations of quantum mechanics, measurement theory, quantuminformation, unambiguous measurement, von Neumann lattice, Riesz-Fischer sequence
1. Introduction
Different states of a system are assumed to be distinguishable in classical mechanics.This fundamental assumption, however, is abandoned in quantum mechanics. Statescannot be distinguished without error unless they are orthogonal. One would then liketo consider a problem of distinguishing states in a given set, which is sometimes calledstate discrimination problem.Strategies for state discrimination can be classified into two types. In the firsttype, one makes a measurement with n outcome in order to distinguish n input states.If a certain outcome is detected, we presume that the system is in the input state istinguishability of countable quantum states and von Neumann lattice n + 1 outcomes. The extra outcome corresponds to the answer“do not know”. At the expense of this inconclusive outcome, it is possible, underrather weak conditions on the input states, to distinguish the inputs with certainty when the other n outcomes is obtained. A measurement with this property is calledan unambiguous measurement, which is the central subject of this paper. Note thatall measurements of the first type should be regarded as belonging to the second type,if we obliged to estimate the input even when we obtain the inconclusive outcome.Consequently, unambiguous measurements cannot make the average error probabilitysmaller than that in measurements of the first type. In other words, unambiguousmeasurements are less suitable for quantitative studies of state discrimination. On thecontrary, they are appropriate for qualitative studies, which we shall carry out in thispaper.Unambiguous measurements for two pure states were discussed by Ivanovic [2]for the first time, and later its theory was developed by Dieks [3], Peres [4], Jaegerand Shimony [5]. Chefles [6] obtain a necessary and sufficient condition that a familyconsists of finitely many pure states can be unambiguously measured. The conditionis the linear independence of the given family of states. Sun et al. [7] and Eldar [8]discuss optimal unambiguous measurements in relation to semidefinite programming. Anecessary and sufficient condition for mixed states to be unambiguously measured waspresented by Feng et al. [9], which is slightly complicated. Note that all the studiesabove concern a family consisting of a finite number of states.Another subject of this paper is a von Neumann lattice. A von Neumann lattice isa family of states which corresponds to the lattice on the phase space in the classicalmechanics. This family is investigated in several contexts. von Neumann [10] originallyexamines this family for simultaneous measurement of position and momentum.Gabor [11] discussed these families in the context of communication theory and electricalengineering, which is a pioneering work in time-frequency analysis. Interpolationproblem for entire functions also has relation to von Neumann lattices [12]. Propertiesof a von Neumann lattice depend on the area of its fundamental region in the phasespace. Von Neumann stated without proof that this family is complete when the areais roughly smaller than the Planck constant h . However, it was about 40 years laterthat Perelomov [13] and Bargmann et al. [14] gave the proof for this fact. Today, manyof the properties have been revealed, which potentially offer measurement theoreticalinterpretations.In this paper, we investigate unambiguous measurements on countably many states.First, we develop a general theory of the distinguishability of countably many states.We define a distinguishability of states as a possibility of unambiguous measurementson it. Then, we provide a necessary and sufficient condition for countable pure states tobe distinguishable. We also consider uniform distinguishability, and give the maximum istinguishability of countable quantum states and von Neumann lattice h , the smallest unit of area in the classical phase space. Dependingon whether the spaces between the states is larger or smaller than the Planck constant,distinguishability of states changes drastically.This paper is organized as follows. In Section 2, we define distinguishability of thestates using unambiguous measurements. Then we briefly review properties of vectorsin a Hilbert space, which can be considered as generalizations of liner independence,in Section 3. In Section 4, We show that distinguishability of countable pure states isequivalent to properties of vectors which we see in the previous section. Section 5 isdevoted to investigations of von Neumann lattices. Conclusions and discussions are givenin Section 6 and Section 7. We discuss the case of uncountable states in Appendix A.
2. Distinguishability
We shall discuss the problem of distinguishing a quantum state in a given family bya single measurement. We allow an answer “do not know” or “unknown”, but donot allow taking one state for another in the family. The problem is referred to asunambiguous measurement. We shall consider an arbitrary quantum system describedby a Hilbert space H . Let ( ρ i ) i ∈ I be a given family of countable states, where ρ i ’s aredensity operators on H . In our terminology, countable includes finite. We will oftendenote a family ( | ψ i ih ψ i | ) i ∈ I of pure states simply by ( ψ i ) i ∈ I in the following.A quantum measurement and the resulting probability density of the outcome isdescribed by a POVM (e.g. [15, § j ) j ∈ J on J , where J is acountable set, is a list of bounded operators Π j on H that satisfies the positivity, Π j > j ∈ J , and the normalization, P j ∈ J Π j = 1. The sum should be understood inthe sense of the weak operator topology. the conditional probability of obtaining anoutcome j ∈ J when the input was ρ i is given by q ji (Π) := tr[Π j ρ i ] . (1)When I ⊂ J , the success probability of obtaining the outcome i ∈ I ⊂ J for the input i ∈ I is given by q i (Π) := q ii (Π) = tr[Π i ρ i ] . (2)Though these quantities depend on ( ρ i ) i ∈ I , we omit them in the notation since we usuallyfix a family ( ρ i ) i ∈ I in our discussion.We shall define the distinguishability of each state in a given family of states. Definition 1 (Distinguishability) . Let ( ρ i ) i ∈ I be a countable family of states.(i) A POVM Π = (Π j ) j ∈ J distinguishes (the states in) ( ρ i ) i ∈ I if J = I ⊔ { ? } , a disjointunion of I and a set containing one element which we denote “?”, and the following istinguishability of countable quantum states and von Neumann lattice q ji (Π) = 0 for all i, j ∈ I, i = j, (3) q i (Π) > i ∈ I. (4)(ii) A POVM Π uniformly distinguishes the states in ( ρ i ) i ∈ I if Π distinguishes ( ρ i ) i ∈ I and has constant q i (Π). The constant is called the uniform success probability .(iii) A POVM Π perfectly distinguishes the states in ( ρ i ) i ∈ I if Π distinguishes ( ρ i ) i ∈ I and q i (Π) = 1 for all i ∈ I .The family ( ρ i ) i ∈ I of states is said distinguishable , uniformly distinguishable , or perfectlydistinguishable if there exists a POVM that distinguishes, uniformly distinguishes, orperfectly distinguishes, respectively, the states in ( ρ i ) i ∈ I .It is obvious by definition that perfect distinguishability implies uniformdistinguishability, and that uniform distinguishability implies distinguishability.Uniform distinguishability allows another characterization: (ii) ′ There exists aPOVM Π such that it distinguishes the states ( ρ i ) i ∈ I and inf i ∈ I q i (Π) > ′ satisfies the condition(ii) ′ . Let q ′ i := q i (Π ′ ) and q ′ := inf q ′ i >
0. Then the POVM Π = (Π j ) j ∈ I ⊔{ ? } , whereΠ j := q ′ q ′ i Π ′ j , j ∈ I, Π ? := Π ′ ? + X i ∈ I (cid:18) − q ′ q ′ i (cid:19) Π ′ i , satisfies the condition (ii), because its success probabilities q i (Π) = q ′ > i ∈ I .Distinguishability and uniform distinguishability are equivalent if the family ( ρ i ) i ∈ I consists of only finite number of states, because the infimum of finitely many positivenumbers is positive.The conditions equivalent to distinguishability and perfect distinguishability werediscussed for finite number of states in [6]. Uniform distinguishability, which is differentfrom the condition only when the family contains infinite number of states, is newlydefined in this paper. In the case that the number of states is countable, we will derivethe necessary and sufficient condition for each type of distinguishability, which is thetheme of this paper. The assumption of countability is not a restriction if the Hilbertspace is separable. See the Appendix A for details.It is worth noting that all conditions defined here are invariant under any unitarytransformation, in particular, under any unitary time evolution. In fact, if a POVMΠ = (Π j ) j ∈ I ⊔{ ? } distinguishes the states ( ρ i ) i ∈ I , then the POVM Π ′ = ( U Π j U ∗ ) j ∈ I ⊔{ ? } distinguishes the states ( U ρ i U ∗ ) i ∈ I , for any unitary operator U . istinguishability of countable quantum states and von Neumann lattice
3. Properties of a family of vectors in a Hilbert space
In this section, we shall review some properties of a family of vectors in a Hilbert space,which are related to the notion of linear independence. Careful discussion is necessaryif the number of vectors is infinite. They will turn out to have substantial relation todistinguishability of quantum states in the next section. Let span S be the minimallinear subspace containing S , where S is a subset of H . In other words, span S is theset of all linear combinations of finitely many elements of S . Let span S be the normclosure of span S . Definition 2 (e.g. [16, p.28], [17, Definition 3.1.2, p.135]) . Let ( ψ i ) i ∈ I be a family ofvectors in H .(n) The family ( ψ i ) i ∈ I is linearly independent if ψ i / ∈ span { ψ j ∈ H | j = i, j ∈ I } foreach i ∈ I .(i) The family ( ψ i ) i ∈ I is minimal if ψ i / ∈ span { ψ j ∈ H | j = i, j ∈ I } for each i ∈ I .(ii) The family ( ψ i ) i ∈ I is Riesz-Fischer if there exists
A > A X i ∈ I | α i | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I α i ψ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (5)holds for scalars ( α i ) i ∈ I with all but finitely many being zero. We call the positivenumber A a Riesz-Fischer bound .(iii) The family ( ψ i ) i ∈ I is orthonormal if h ψ j , ψ i i = δ ij holds for all i, j ∈ I .Orthonormality obviously implies the Riesz-Fischer property. The Riesz-Fischerproperty implies minimality, because otherwise there would be i ∈ I such that ψ i ∈ span { ψ j | j = i } , and one could make the norm of ψ i − P j = i α j ψ j smaller thanany given positive number by choosing α j ∈ C accordingly, thus violating (5) for any A >
0. Minimality implies linear independence by definition.In the particular case that I is a finite set, linear independence, minimalityand the Riesz-Fischer property are all equivalent. Indeed, if I is finite, then { ( α i ) ∈ C I | P i ∈ I | α i | = 1 } is compact, and we deduce Riesz-Fischer property fromlinear independence.In our discussion on distinguishability, the dual of a family ( ψ i ) i ∈ I of vectors in H play a crucial role. Two lists ( ψ i ) i ∈ I and ( φ i ) i ∈ I of vectors in H are said dual or biorthogonal to each other, if h φ j , ψ i i = δ ij for all i, j ∈ I . The condition for theexistence of a dual is given by the following proposition. For the proof, see [16, p.28],[17, Lemma 3.3.1]. We also attach a proof in Appendix B for the convenience of thereader. Proposition 1.
A family ( ψ i ) i ∈ I of vectors in H admits a biorthogonal family if andonly if it is minimal. In that case, the biorthogonal family is unique if ( ψ i ) i ∈ I is completei.e. span { ψ i | i ∈ I } = H . The Riesz-Fischer property has a dual notion. istinguishability of countable quantum states and von Neumann lattice Definition 3.
The list ( φ i ) i ∈ I of vectors in H is Bessel if there exists
B < ∞ , called a Bessel bound , such that the following equivalent conditions are fulfilled.1. For scalars ( α i ) i ∈ I with all but finitely many being zero, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I α i φ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B X i ∈ I | α i | (6)holds.2. For each vector ψ ∈ H , X | h ψ, φ i i | B k ψ k (7)holds.For the proof of equivalence of two conditions, see [16, Chapter 4, Section 2,Theorem 3].The Riesz-Fischer property allows characterizations by the dual ( [16, Chapter 4],see also [18, Proposition 2.3 (ii)]). Proposition 2.
For a family ( ψ i ) i ∈ I of vectors and A > , the following conditions areequivalent.
1. ( ψ i ) i ∈ I is Riesz-Fischer with bound A .
2. ( ψ i ) i ∈ I admits a biorthogonal family which is Bessel with bound A − . For each ( γ i ) i ∈ I ∈ C I with P i ∈ I | γ i | < ∞ , there exists φ ∈ H that satisfies h φ, ψ i i = γ i and k φ k A − P i ∈ I | γ i | . The moment problem is to find a vector φ ∈ H that satisfies h φ, ψ i i = γ i , i ∈ I , fora given family ( ψ i ) i ∈ I and a numerical sequence ( γ i ) i ∈ I , In that context, the equivalenceof (i) and (iii) states that the Riesz-Fischer property guarantees existence of a solutionof the moment problem for each square summable ( γ i ) i ∈ I .
4. Distinguishability of general family of pure states: the first main result
Now we shall state the condition for each type of distinguishability for a family ( ψ i ) i ∈ I of pure states. Theorem 1.
Let ( ψ i ) i ∈ I be a family of countably many pure states. (i) The states in ( ψ i ) i ∈ I are distinguishable if and only if ( ψ i ) i ∈ I is minimal. (ii) The states in ( ψ i ) i ∈ I are uniformly distinguishable with uniform success probability q if and only if ( ψ i ) i ∈ I is Riesz-Fischer with bound q , (iii) The states in ( ψ i ) i ∈ I are perfectly distinguishable if and only if ( ψ i ) i ∈ I is anorthonormal family.istinguishability of countable quantum states and von Neumann lattice if part of (i) needs the assumption of countability.It should be noted that the statement (ii) of Theorem 1 reveals the significanceof the Riesz-Fischer bound in quantum measurement theory: success probability ofuniform distinction.We remark that in the particular case of finite family ( ψ i ) i ∈ I , the statements (i) and(ii) of Theorem 1 are identical, which reproduces the results obtained by Chefles [6].As was discussed in the previous sections, linear independence, minimality and theRiesz-Fischer property are equivalent, for finitely many vectors.We begin the proof of Theorem 1 with Lemma 1.
If the POVM
Π = (Π j ) j ∈ I ⊔{ ? } distinguishes the pure states in ( ψ i ) i ∈ I , thenthe following holds for i, j, k ∈ I .
1. Π j ψ i = 0 for i = j . q i (Π) = 1 if and only if Π ? ψ i = 0 . h ψ i , Π k ψ j i = q i (Π) δ ik δ jk .Proof. The condition tr[ ρE ] = 0 for ρ = | ψ ih ψ | and E > Eρ = 0. This facttogether with the assumption of the lemma imply the first and second claims. The thirdfollows directly from the first. Proof of Theorem 1.
We must prove the six claims below.1 (Distinguishability implies minimality). Suppose Π distinguishes ( ψ i ) i ∈ I . Then itfollows from Lemma 1 that ( φ i ) i ∈ I defined by φ i := Π i ψ i h ψ i , Π i ψ i i is biorthogonal to ( ψ i ) i ∈ I . Thus ( ψ i ) i ∈ I is minimal by Proposition 1.2 (Minimality implies distinguishability). Suppose ( ψ i ) i ∈ I is minimal. Then byProposition 1 there exits a family ( φ i ) i ∈ I biorthogonal to ( ψ i ) i ∈ I . DefineΠ i := p i | φ i ih φ i |k | φ i ih φ i | k , Π ? := 1 − X i ∈ I Π i , where P i ∈ I p i = 1 and p i > i ∈ I . This is possible since I is countable.The operators Π j constitutes a POVM Π = (Π j ) j ∈ I ⊔{ ? } because positivity of Π ? isguaranteed by an inequality for the operator norm, (cid:13)(cid:13)P i ∈ I Π i (cid:13)(cid:13) P p i = 1. It followsfrom the biorthogonality that Π distinguishes ( ψ i ) i ∈ I .3 (Uniform distinguishability implies the Riesz-Fischer property). Suppose Πuniformly distinguishes ( ψ i ) i ∈ I . Let ψ = P i ∈ I α i ψ i , with all α i but finitely many beingzero. One has (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i α i ψ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = h ψ, ψ i = X k ∈ I h ψ, Π k ψ i + h ψ, Π ? ψ i > X i,j,k ∈ I α i h ψ i , Π k ψ j i α j + 0 = q (Π) X k | α k | , istinguishability of countable quantum states and von Neumann lattice ? and the last equality from Lemma 1.Thus ( ψ i ) i ∈ I is Riesz-Fischer with bound q (Π).4 (The Riesz-Fischer property implies uniform distinguishability). Suppose ( ψ i ) i ∈ I is Riesz-Fischer with bound A . By Proposition 2, there exists a biorthogonal family( φ i ) i ∈ I which is Bessel with bound A − . It follows from (7) that for any ξ the sum P i ∈ I | h φ i , ξ i | converges. This in particular implies that only countably many h φ i , ξ i are nonzero. With an appropriate order, the sequence P i ≤ k | h φ i , ξ i | becomes Cauchy.From this fact and (6), one can show that X := X i ∈ I | φ i ih φ i | converges in the strong operator topology and defines a bounded liner map with bound k X k A − . We define the POVM Π = (Π j ) j ∈ I ⊔{ ? } byΠ i := | φ i ih φ i |k X k , where positivity of Π ? is verified byΠ ? := 1 − X i ∈ I Π i = 1 − X k X k > . This POVM uniformly distinguishes ( ψ i ) i ∈ I with uniform success probability q (Π) =1 / k X k > /A − = A .5 (Perfect distinguishability implies orthonormality). Let Π distinguish ( ψ i ) i ∈ I .Then Lemma 1 implies that h ψ i , ψ j i = X k ∈ I h ψ i , Π k ψ j i + h ψ i , Π ? ψ j i = δ ij q j (Π) + 0 = δ ij . Thus ( ψ i ) i ∈ I is an orthonormal family.6 (Orthonormality implies perfect distinguishability). If ( ψ i ) i ∈ I is an orthonormalfamily, the POVM Π defined by Π i := | ψ i ih ψ i | and Π ? := 1 − P i ∈ I Π i perfectlydistinguishes ( ψ i ) i ∈ I .If a family ( ψ i ) i ∈ I is given, one interested not only whether or not it is uniformlydistinguishable but also the value of the largest possible success probability. Thefollowing theorem gives the value and the construction of the measurement. Theorem 2.
Suppose that a family ( ψ i ) i ∈ I of pure states is uniformly distinguishable.By Proposition 1 and Theorem 1, it admits a unique biorthogonal family ( φ i ) i ∈ I containedin span { ψ i | i ∈ I } . Then among all POVMs which uniformly distinguish ( ψ i ) i ∈ I , thePOVM Π = (Π j ) j ∈ I ⊔{ ? } defined by Π j := | φ j ih φ j | (cid:13)(cid:13) P k ∈ I | φ k ih φ k | (cid:13)(cid:13) , j ∈ I ,1 − X k ∈ I Π k , j =?, (8) istinguishability of countable quantum states and von Neumann lattice attains the maximum uniform success probability q (Π) = 1 (cid:13)(cid:13) P k ∈ I | φ k ih φ k | (cid:13)(cid:13) . (9)Applying Theorem 2 to the case of two pure states ( ψ i ) i =1 , , we can immediatelyobtain the maximum uniform success probability q (Π) = 1 − |h ψ | ψ i| , (10)by calculating the smallest nonzero eigenvalue of the operator P i =1 , | ψ i ih ψ i | . This isessentially the result of Dieks [3], though he gave a prior uniform probability density( p i ) i =1 , = (1 / i =1 , to ( ψ i ) i =1 , while we do not. For finitely many states, uniform priorprobability density is canonical in the sense of having a maximum entropy. However, inthe case of infinity many states, uniform probability density does not exist. This is thereason why we do not employ the prior probability in our discussion. Our definition ofuniform distinguishability is meaningful even when the cardinality (number of elements)of I is infinite.Prior to the proof for the theorem, we show a lemma which states that anyunambiguous measurement of a given family of pure states is essentially the projectionto its biorthogonal family. The lemma corresponds to equation (2.9) in [6], extended tothe case of countable families ( ψ i ) i ∈ I . Lemma 2.
Assume that a POVM Π distinguishes a family ( ψ i ) i ∈ I of pure states so that,by Theorem 1 and Proposition 1, ( ψ i ) i ∈ I admits a unique biorthogonal family ( φ i ) i ∈ I contained in span { ψ i | i ∈ I } . Then Π satisfies P Π i P ∗ = q i (Π) | φ i ih φ i | , where P is the orthogonal projection of H onto span { ψ i | i ∈ I } .Proof. Assume for a while span { ψ i | i ∈ I } = H and therefore P = 1. For any ψ = P α i ψ i ∈ span { ψ i | i ∈ I } , it follows from Lemma 1 that h ψ, (Π i − q i (Π) | φ i ih φ i | ) ψ i = 0holds for all i ∈ I . Thus, by continuity of the inner product, Π i , and | φ i ih φ i | ,the equation above holds for all ψ ∈ span { ψ i | i ∈ I } = H . One therefore hasΠ i = q i (Π) | φ i ih φ i | . This proves the lemma in the case span { ψ i | i ∈ I } = H . Whenspan { ψ i | i ∈ I } 6 = H , define Π ′ byΠ ′ j := ( P Π j P ∗ , j ∈ I , P Π ? P + (1 − P ) , j =?.Then Π ′ is a POVM on the Hilbert space P H . The claim reduces to the casespan { ψ i | i ∈ I } = H . Proof of Theorem 2.
As was shown in the proof of Theorem 1, k P i ∈ I | φ i ih φ i | k is finiteand the POVM Π in the theorem is well-defined. Let Π ′ be an arbitrary POVM which istinguishability of countable quantum states and von Neumann lattice ψ i ) i ∈ I uniformly and let P be the orthogonal projection from H ontospan { ψ i | i ∈ I } . It follows from from Lemma 2 and Π ′ ? > P = P P ∗ > X i ∈ I P Π ′ i P ∗ = q (Π ′ ) X i ∈ I | φ i ih φ i | . Thus one has 1 > k P k > q (Π ′ ) (cid:13)(cid:13)P i ∈ I | φ i ih φ i | (cid:13)(cid:13) = q (Π ′ ) /q (Π), i.e., q (Π) > q (Π ′ ).
5. Distinguishability of von Neumann lattices: the second main result
In this section, we shall discuss distinguishability of coherent states represented by alattice in the complex plane, which is called the von Neumann lattice. The coherentstates may be defined for a particle in one dimension or any quantum system whichallows a harmonic oscillator, so that they can represent photons, phonons, or otherbosonic particles. We do not specify the physical system here and treat them in general,though we may sometimes use the terminology for a particle.Let H be the Hilbert space which represents the states of a quantum system. Let a be the annihilation operator on H that satisfies aa ∗ − a ∗ a = 1. Let | i be a state whichsatisfies a | i = 0. The state | i is unique up to phase factor and is called the vacuum.The coherent state | z i , where z ∈ C , is defined by [19, 20] | z i := exp[ za ∗ − za ] | i . (11)They are minimum uncertainty states for the position operator Q = 2 − / ( a + a ∗ ) and themomentum operator P = (2 i ) − / ( a − a ∗ ). This allows one to regard a coherent state | z i as the quantum state that corresponds to the classical state represented by a single point z = 2 − / ( q + ip ) ∈ C in the phase space. It is easily verified by the equation a | z i = z | z i that coherent states ( | z i ) z ∈ C is linearly independent. It follows that mutually differentfinite number of states ( | z i i ) i =1 , ,...,n are uniformly distinguishable.In the context of simultaneous measurement of position and momentum,von Neumann considered the following family of coherent states, which correspondsto a lattice on the phase space. Definition 4.
Let ω , ω ∈ C be such that Im( ω /ω ) >
0. Let L ( ω , ω ) := { n ω + n ω ∈ C | n , n ∈ Z } ⊂ C , (12) vNL ( ω , ω ) := {| Ω i | Ω ∈ L ( ω , ω ) } ⊂ H. (13)A family vNL ( ω , ω ) is called a von Neumann lattice. The set { t ω + t ω | t , t ∈ [0 , } ⊂ C is called the fundamental region of L ( ω , ω ) or of vNL ( ω , ω ). The area of the fun-damental region is usually denoted by S .Though in some literature the name of von Neumann lattice is used only for thecase S = π , we use the term for all S . Von Neumann lattices are called Weyl-Heisenbergsystems in the field of time-frequency analysis or in the Gabor analysis [11, 21].Von Neumann stated without proof that a von Neumann lattices is complete when S . / istinguishability of countable quantum states and von Neumann lattice S < π vNL ( ω , ω ) Yes No No vNL ( n ) ( ω , ω ) , n < ∞ Yes No No S = π vNL ( ω , ω ) Yes No No vNL (1) ( ω , ω ) Yes Yes No † vNL ( n ) ( ω , ω ) , n < ∞ No Yes No ‡ S > π vNL ( ω , ω ) No Yes Yes Table 1.
Properties of von Neumann lattices : vNL ( n ) denotes the set obtained byremoving arbitrary n elements from vNL . Facts marked by † and ‡ can be deducedfrom (38) in [13]. However, they have not been stated explicitly. Combining the properties of von Neumann lattices and Theorem 1, we arrive atour second main result.
Theorem 3.
Let S be the area of the fundamental region of vNL ( ω , ω ) , Then, thefollowings hold. (i) When
S < π , vNL ( ω , ω ) is not distinguishable. (ii) When S = π , vNL ( ω , ω ) is not distinguishable. However, the set vNL ( ω , ω ) with more than one element removed is distinguishable, but isnot uniformly distinguishable. (iii) When
S > π , vNL ( ω , ω ) is uniformly distinguishable, but is not perfectlydistinguishable. A finite subset of coherent states is always uniformly distinguishable, whereasvon Neumann’s lattice, which is infinite, behaves quite different. The result shows thateach level of distinguishability is directly related to the area S = Im ( ω ω ) and does notdepend on ω and ω separately. Furthermore, it can be shown that distinguishabilityof von Neumann’s lattice is determined solely by the density of points in the complexplane, 1 /S , and is robust to deformation of the lattice (See proofs in [13] and [12]).The threshold S = π corresponds to the area of the Planck constant h in the classicalphase space [note that d z = ( h/ π ) − (2 − / dq )(2 − / dp )]. Physically, the area h is theminimum unit of area of the phase space, which appeared e.g. in the Bohr-Sommerfeldquantum condition [23, (48.2)]. Theorem 3 reveals the measurement theoretical meaningof the Planck constant. istinguishability of countable quantum states and von Neumann lattice Theorem 4. | ω | , | ω | → ∞ with sin arg( ω ω ) > ε for some ε > forces maximaluniform success probability q (Π) of vNL ( ω , ω ) approaches . More specifically, when ω , ω ∈ C satisfies S = Im( ω ω ) > π , there exists a POVM Π which distinguishes vNL ( ω , ω ) uniformly with uniform success probability q (Π) > − (cid:18) √ π sin arg( ω ω ) min i | ω i | (cid:19) . Proof.
Let Ω nm = nω + mω ∈ L ( ω , ω ). For any numerical sequence ( α n,m ) ( n,m ) ∈ Z with only finitely many being nonzero, one has (cid:13)(cid:13)(cid:13)X α m,n | Ω m,n i (cid:13)(cid:13)(cid:13) = X m,n X k,ℓ α k,ℓ α k + m,ℓ + n h Ω k,ℓ | Ω k + m,ℓ + n i = X k,ℓ | α k,ℓ | + X ( m,n ) =(0 , X k,ℓ α k,ℓ α k + m,ℓ + n h Ω k,ℓ | Ω k + m,ℓ + n i . The second sum in the last line can be estimated as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( m,n ) =(0 , X k,ℓ α k,ℓ α k + m,ℓ + n h Ω k,ℓ | Ω k + m,ℓ + n i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( m,n ) =(0 , e −| Ω m,n | / X k,ℓ | α k,ℓ α k + m,ℓ + n | X ( m,n ) =(0 , e −| Ω m,n | / X k,ℓ | α k,ℓ | , where we have used the formula | h z | w i | = e −| z − w | and the triangle inequality in thefirst line and the Cauchy-Schwarz inequality in the second. Therefore, we have (cid:13)(cid:13)(cid:13)X α m,n | Ω m,n i (cid:13)(cid:13)(cid:13) > A X k,ℓ | α k,ℓ | , A := 1 − X ( m,n ) =(0 , e −| Ω m,n | / . In the case
A > vNL ( ω , ω ) is Riesz-Fischer with bound A . So, by the Theorem 1, vNL ( ω , ω ) is uniformly distinguishable with uniform success probability at least A .Since (cid:12)(cid:12) λλ + λν (cid:12)(cid:12) > (cid:12)(cid:12) Im (cid:0) λλ + λν (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) Im (cid:0) λν (cid:1)(cid:12)(cid:12) , we have | Ω m,n | = 1 | ω m | | ω m ω m + ω m ω n | > | ω m | | Im { ω m ω n }| = 1 | ω m | | mn | S > S max i | ω i | | n | when m = 0, and | Ω m,n | > S max i | ω i | n istinguishability of countable quantum states and von Neumann lattice m, n ∈ Z . Hence1 − A X ( m,n ) =(0 , exp (cid:20) − S i | ω i | ( m + n ) (cid:21) = − X k ∈ Z exp (cid:20) − S i | ω i | k (cid:21)! − (cid:18) Z ∞−∞ exp (cid:20) − S i | ω i | x (cid:21) dx (cid:19) = − (cid:18) √ π max i | ω i | S (cid:19) . Since S = | ω | | ω | sin arg ( ω ω ), the desired estimate follows.Theorem 4 proves a part of Theorem 3 directly. Theorem 4 justifies the intuitionthat, as the lattice becomes large, the von Neumann lattice approaches an orthogonalfamily.It should be noted however that the uniform success probability cannot be estimatedby S only. In particular, the condition S → ∞ is not sufficient for the uniform successprobability q (Π) to approach unity. It can be seen easily as follows. The uniform successprobability q (Π) cannot be greater than the maximal uniform success probability ofdistinguishing two states {| i , | Ω i} in vNL ( ω , ω ), where Ω is either ω or ω . Onetherefore has, by (10), q (Π) − | h | Ω i | = 1 − exp (cid:20) − | Ω | (cid:21) (14)for Ω = ω , ω . This proves that q (Π) cannot be estimated solely by S .
6. Conclusion
We examined distinguishability of countable pure states. We defined distinguishabilityof countable states as the possibility of unambiguous measurements on these states.There we classified the distinguishability into three, namely, distinguishability, uniformdistinguishability, and perfect distinguishability. Distinguishability and uniformdistinguishability, which are equivalent when the number of states is finite, split when thenumber becomes infinite. We then proved a criterion of distinguishability for countablepure states in Theorem 1. The theorem establishes a relation between operationaldefinitions of distinguishability and intrinsic properties of a family of state vectors inthe Hilbert space. In addition, we gave the maximal uniform success probability and aPOVM which attain it in Theorem 2.After developing a general criterion of distinguishability, we discussed distinguisha-bility of von Neumann’s lattice, which is a family of states corresponds to a lattice of thephase space in the classical mechanics. Besides its own interest in measurement theory,it serves as an excellent example for the general theory in the sense that all the subtleties istinguishability of countable quantum states and von Neumann lattice h . Theresult is robust to deformation of the lattice.The Planck constant h is without doubt the most fundamental constant in quantumphysics. It appears in the canonical commutation relation [ Q, P ] = ih/ (2 π ) andcharacterizes quantum physics in many ways. One is the uncertainty relation of physicalquantities. A simple, well-known inequality is Kennard’s inequality [24, (27)], whichgives a bound for the standard deviations σ of the observables Q and P , σ ( Q ) σ ( P ) > h π . Modern interpretations of Heisenberg’s noise-disturbance uncertainty relation andcorresponding rigorous inequalities are also found in [25] (See also [26, Section 6].) and[27]. Another important aspect of h is the Bohr-Sommerfeld quantum condition [23,(48.2)] I p dq = h (cid:18) n + 12 (cid:19) n = 0 , , , . . . . It not only stimulated the discovery of quantum mechanics but also can be shown inquantum mechanics with the Wentzel-Kramers-Brillouin (WKB) approximation. Thiscondition explains the familiar fact in statistical mechanics that a single quantumstate occupies an area of h in the classical phase. Our analysis of von Neumannlattices revealed the significance of the Planck constant h through the context of statediscrimination, thereby giving another measurement-theoretic meaning of h , and givesa rigorous version of justification for h to be the unit of the phase space.
7. Discussion
We have not discussed criterions of distinguishability for mixed states. For finitely manymixed states, Feng et al. [9] obtained a condition of distinguishability. We generalizetheir result to the case of countably many states and present it in a slightly differentmanner.
Proposition 3.
Let ( ρ i ) i ∈ I be a countable family of density operators on a Hilbert space H . Then, the following are equivalent. (i) ( ρ i ) i ∈ I is distinguishable. (ii) For all i ∈ I , \ k = i ker ρ k ! \ ker ρ i = ∅ . (iii) For all i ∈ I , \ k ∈ I ker ρ k ! ( \ k = i ker ρ k ! . istinguishability of countable quantum states and von Neumann lattice For all i ∈ I , span [ k = i im ρ k ! ( span [ k ∈ I im ρ k ! . Here, L denotes the norm closure of L ⊂ H , ker ρ = { ξ ∈ H | ρξ = 0 } and im ρ = { ρξ | ξ ∈ H } for a bounded operator ρ on H .Proof. Assume (i). Then, for distinct elements i and j of I , one has 0 = tr[ ρ i Π j ] =tr[Π / j ρ i Π / j ] hence ρ i Π / j = 0. Therefore, for each i ∈ I , there exists ψ i such that ρ k (Π / i ψ i ) = 0 for all k = i and ρ i (Π / i ψ i ) = 0. This ensures (ii). That (ii) implies(i) is as in the proof of Theorem 1. That (ii) is equivalent to (iii) is seen by a trivialset-theoretical identity X \ ( X ∩ Y ) = X \ Y . The equivalence of (iii) and (iv) is due toim ρ i = (ker ρ i ) ⊥ and span (cid:16)S j K ⊥ j (cid:17) = (cid:16)T j K j (cid:17) ⊥ , where each K j is a closed subspaceof H , and ⊥ denotes the orthogonal complement.When ( ρ i ) i ∈ I is finite family, (iv) reduces tospan [ k = i im ρ k ! ( span [ k ∈ I im ρ k ! for all i ∈ I . This is the condition that Feng et al. [9] presented.The criterion enables us to investigate the time evolution of distinguishability i.e.the relation between distinguishability of ( ρ i ) i ∈ I and that of ( ρ ′ i ) i ∈ I . Here we assumeeach state ρ i , at time t , evolves to ρ ′ i at t ′ > t . We already note in the remark belowthe Definition 1 that distinguishability is invariant under unitary evolutions. Thus weshould concern non-unitary evolutions of the system, which changes a pure state into amixed state in general. Hence we need a criterion for mixed states.Distinction of coherent states is not only a subject of theoretical interest but also ofa practical problem. A coherent state of light is easy to handle and is often used in opticalcommunication. Let us consider the following simple example. The sender generatesseveral coherent states and sends one of them, which travels through optical fibers. Thereceiver detects it and determine which coherent state was sent. The simplest case is todistinguish two states, the vacuum | i and another coherent state | ω i . A slightly morecomplicated problem is to distinguish nine states, i.e., the vacuum and the eight statesenclosing the vacuum C = { | Ω i | Ω = 0 , ± ω , ± ω , ± ω ± ω } . A still more complicatedone is distinction of 25 states in the set C , doubly surrounding the vacuum. In thismanner, we consider distinction of the states in the set C n ( n − which is a finite subsetof a von Neumann lattice. It approaches the whole von Neumann lattice as n → ∞ . Wedenote by S the area of fundamental region of lattice corresponding to C n ( n − (weassume S > C n ( n − is linearly independent so that by Theorem 1 itis uniformly distinguishable with uniform success probability q n >
0. The q n satisfies q > q > · · · >
0. Thus there exists a finite limit lim n →∞ q n > q . Here, q is theuniform success probability for the whole lattice which is positive when S > π and istinguishability of countable quantum states and von Neumann lattice q n for smaller n may be of practical interest, andthe asymptotic behavior for large n may be of theoretical interest. Acknowledgments
This work was supported by JSPS Grant-in-Aid for Scientific Research No. 24540282and by MEXT-Supported Program for the Strategic Research Foundation at PrivateUniversities Topological Science.
Appendix A. Countability of state family
In this appendix, we briefly discuss the case that states are not countable. We shallshow that uncountably many states cannot be distinguishable when H is separable.Note that in the standard formulation of quantum mechanics (e.g. [10, II-1, PostulateE]) the Hilbert space is assumed to be separable.We begin by extending the definition of distinguishability to the case of uncountablestates. We cannot define POVM with P j ∈ J Π j = 1 when the set J of outcomes isuncountable since the sum exceeds countable additivity. We therefore go back to themeasure-theoretical definition of the POVM. Let B ( H ) be the set of bounded operatorson H .Let ( J, J ) denote a measurable space, where J is a set and J a σ -algebra on J . The map Π : J → B ( H ) is called a POVM on ( J, J ) if it satisfies the followingconditions [15, § J ) = 1 and Π(∆) > ∈ J .(ii) Π ( S k ∆ k ) = P k Π(∆ k ) in the sense of weak operator topology for all disjointcountable collection { ∆ k } ⊂ J .We shall redefine distinguishability of states, or extend Definition 1(i), to includefamilies of uncountable states. Definition 5.
A POVM Π on ( J, J ) distinguishes the family ( ρ i ) i ∈ I of states if thefollowing conditions hold.(i) J = I ⊔ { ? } , and { i } ∈ J holds for all i ∈ I .(ii) For all i, j ∈ I , tr[Π( { i } ) ρ j ] is positive if i = j and vanishes if i = j .The states ( ρ i ) i ∈ I are distinguishable if there exists a POVM Π which distinguishesthem.We show a general relation between dim H and the number of input states thatis necessary for distinguishability. Here, dim H is defined as the cardinality of anorthonormal basis of H , which is countable if and only if H is separable. Theorem 5.
Let ( ρ i ) i ∈ I be a state family of a Hilbert space H . If ( ρ i ) i ∈ I isdistinguishable in the sense of Definition 5, then dim H > | I | , where | I | denote thecardinality of the set I .istinguishability of countable quantum states and von Neumann lattice Proof.
The proof is divided into two steps.1 (Case that all ρ i are pure). Let ρ i = | ψ i ih ψ i | , ψ i ∈ H . Then ( ψ i ) i ∈ I is minimal,which can be shown in a similar manner to the proof of Theorem 1. For J ⊂ I ,let K J := span { ψ i | i ∈ J } . Minimality of ( ψ i ) i ∈ I implies that for k / ∈ J , there is anormalized vector e k which is orthogonal to K J and ψ k ∈ C e k + K J . Using transfiniteinduction on I , we can construct a orthonormal family ( e i ) i ∈ I . Therefore, dim H > | I | .2 (General case). Assume a POVM Π distinguishes ( ρ i ) i ∈ I . Let Π j denote Π( { j } ).As in the proof of Proposition 4, Π / j ρ i = 0 holds for all i, j ∈ I with i = j and Π / i ρ i = 0for all i ∈ I . Hence, for each i ∈ I , there exists φ i ∈ H such that Π / i ρ i φ i = 0. Define ψ i = ρ i φ i / k ρ i φ i k . Sincetr [Π j | ψ i ih ψ i | ] = 1 k ρ i ψ i k (cid:13)(cid:13)(cid:13) Π / j ρ i φ i (cid:13)(cid:13)(cid:13) for all i, j ∈ I , Π distinguishes ( | ψ i ih ψ i | ) i ∈ I . Therefore, the claim follows from the firststep.Theorem 5 shows that separability of H forces any distinguishable family of statesto be countable. On the other hand, there exist distinguishable family of uncountablestates in a non-separable Hilbert space. A simple example is the following. Example 1.
Let H be a non-separable Hilbert space, and ( e i ) i ∈ I be a completeorthonormal system of H , which is uncountable. Let ( I, I ) be a measurable space,where 2 I := { ∆ | ∆ ⊂ I } . Define Π : 2 J −→ B ( H ) byΠ(∆) := X i ∈ ∆ | e i ih e i | , ∆ ∈ I . The sum converges in the strong operator topology. Then Π is a well-defined POVMon ( I, I ) and (perfectly) distinguishes ( e i ) i ∈ I .In the example above, Π satisfies “uncountable additivity” P j ∈ J Π( { j } ) = 1 =Π( J ). However, when we define a measure µ ψ : 2 J −→ [0 , ∞ ) ⊂ R for a vector ψ ∈ H as µ ψ (∆) = h ψ, Π(∆) ψ i , then “uncountable additivity” of µ ψ reduces to a trivial matterbecause P h ψ, Π( { i } ) ψ i = k ψ k < ∞ and { j ∈ J | µ ψ ( { j } ) = 0 } is countable. Appendix B. Proof of Proposition 1
We give a proof of Proposition 1 for convenience of the reader. We do so by showinga slightly generalized proposition below. For a normed space X , let X ∗ denote thetopological dual of X , which consists of continuous functionals. In this case, we say( ψ i ) i ∈ I of X and ( φ i ) i ∈ I of X ∗ are biorthogonal when φ i ψ j = δ i,j for all i, j ∈ I . Proposition 4.
Let X be a normed space and ( ψ i ) i ∈ I be a family of vectors in X . ( ψ i ) i ∈ I is minimal if and only if ( ψ i ) i ∈ I admits a biorthogonal family. If the conditionholds and span { ψ i | i ∈ I } = X , then the biorthogonal family is unique.istinguishability of countable quantum states and von Neumann lattice Proof.
When X = 0, the statement is trivial. We assume X = 0 in the following.Let Y = span { ψ i | i ∈ I } , Y i = span { ψ j | i = j ∈ I } and these norm closure Y , Y i ,respectably.First, suppose ( ψ i ) i ∈ I is minimal. Because ( ψ i ) i ∈ I is linearly independent, one candefine for each i ∈ I a linear functional φ ′ i : Y −→ C by the condition φ ′ i ψ j = δ i,j , where i, j ∈ I . We shall show φ ′ i is continuous on Y . One has k φ ′ i k := sup ψ ∈ Y \{ } | φ ′ i ψ |k ψ k = 1inf ψ ∈ ψ i + Y i k ψ k . The denominator is positive because ( ψ i ) i ∈ I is minimal. Therefore, k φ ′ i k < ∞ and thelinear functional φ ′ i is continuous on Y . Due to the Hahn-Banach theorem, φ ′ i admitsa continuous extension to the whole space X . The extension, which we denote by φ i ,belongs to X ∗ . By construction, ( φ i ) i ∈ I is a biorthogonal family for ( ψ i ) i ∈ I .Second, let ( φ i ) i ∈ I be a biorthogonal family for ( ψ i ) i ∈ I . Suppose that ( ψ i ) i ∈ I werenot minimal. Then there would exist i ∈ I such that ψ i ∈ Y i , i.e., there would exist asequence ( ξ n ) n ∈ N in Y i such that ξ n → ψ i . Since φ i ∈ X ∗ is continuous and φ i Y i = 0,one would have 1 = φ i ψ i = φ i lim n ξ n = lim n φ i ξ n = lim n , a contradiction.For the proof of the uniqueness part of the proposition, assume Y = X and ( ψ i ) i ∈ I has two biorthogonal families ( φ i ) i ∈ I and ( φ ′ i ) i ∈ I . Because the functional φ i − φ ′ i iscontinuous on X and vanishes on Y , one has φ i = φ ′ i on Y = X .When X = H is a Hilbert space, the Riesz theorem establishes a conjugateisomorphism H ∗ ≃ H so that a biorthogonal family ( φ i ) i ∈ I of H ∗ can be regardedas a family in H . References [1] C. W. Helstrom. Detection theory and quantum mechanics.
Information and Control , 10(3):254–291, 1967.[2] I. D. Ivanovic. How to differentiate between non-orthogonal states.
Phys. Lett. A , 123(6):257–259,1987.[3] D. Dieks. Overlap and distinguishability of quantum states.
Phys. Lett. A , 126(5–6):303–306,1988.[4] A. Peres. How to differentiate between non-orthogonal states.
Phys. Lett. A , 128(1–2):19, 1988.[5] G. Jaeger and A. Shimony. Optimal distinction between two non-orthogonal quantum states.
Phys. Lett. A , 197(2):83 – 87, 1995.[6] A. Chefles. Unambiguous discrimination between linearly independent quantum states.
Phys.Lett. A , 239(6):339–347, 1998.[7] X. Sun, S. Zhang, Y. Feng, and M. Ying. Mathematical nature of and a family of lower boundsfor the success probability of unambiguous discrimination.
Phys. Rev. A , 65:044306, 2002.[8] Y. C. Eldar. A semidefinite programming approach to optimal unambiguous discrimination ofquantum states.
IEEE transactions on information theory , 49(2):446–456, 2003.[9] Y. Feng, R. Duan, and M. Ying. Unambiguous discrimination between mixed quantum states.
Phys. Rev. A , 70:012308, 2004.[10] J. von Neumann.
Mathematische Grundlagen der Quantenmechanik . Springer, 1932.[11] D. Gabor. Theory of communication. part 1: The analysis of information. j.inst elect. eng - PartIII: Radio and Communication Engineering , 93(26):429–441, 1946. istinguishability of countable quantum states and von Neumann lattice [12] K. Seip. Density theorems for sampling and interpolation in the Bargmann-Fock space. Bull.Amer. Math. Soc. (N.S.) , 26:322–328, 1992.[13] A. M. Perelomov. On the completeness of a system of coherent states.
Theor. and Math. Phys. ,6(2):156–164, 1971.[14] V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder. On the completeness of the coherentstates.
Rep. Math. Phys. , 2(4):221–228, 1971.[15] E. B. Davies.
Quantum theory of open systems . Academic Press, 1976.[16] R. M. Young.
An Introduction to Nonharmonic Fourier Series . Pure and Applied Mathematics.Academic Press, 1980.[17] O. Christensen.
An Introduction to Frames and Riesz Bases . Applied and Numerical HarmonicAnalysis. Birkh¨auser, 2003.[18] P. Casazza, O. Christensen, S. Li, and A. M. Lindner. Riesz-Fischer sequences and lower framebounds.
Z. Anal. Anwend. , 21(2):305–314, 2002.[19] E. Schr¨odinger. Der stetige ¨ubergang von der mikro- zur makromechanik.
Naturwissenschaften ,14(28):664–666, 1926.[20] R. J. Glauber. Coherent and incoherent states of the radiation field.
Phys. Rev. , 131:2766–2788,1963.[21] C. Heil. History and evolution of the density theorem for gabor frames.
J. Fourier Anal. Appl. ,13(2):113–166, 2007.[22] V. Bargmann. On a hilbert space of analytic functions and an associated integral transform parti.
Commun. Pure Appl. Math. , 14(3):187–214, 1961.[23] L. D. Landau and E. M. Lifshits.
Quantum Mechanics: Non-relativistic Theory . Butterworth-Heinemann, 1977.[24] E. H. Kennard. Zur quantenmechanik einfacher bewegungstypen.
Z. Phys. , 44(4-5):326–352, 1927.[25] M. Ozawa. Universally valid reformulation of the heisenberg uncertainty principle on noise anddisturbance in measurement.
Phys. Rev. A , 67:042105, 2003.[26] M. Ozawa. Uncertainty relations for noise and disturbance in generalized quantum measurements.
Annal. Phys. , 311(2):350 – 416, 2004.[27] Y. Watanabe, T. Sagawa, and M. Ueda. Uncertainty relation revisited from quantum estimationtheory.