PHYSICAL REVIEW A 90, 022116 (2014)

Negative conditional entropy of postselected states

Sina Salek,* Roman Schubert,† and Karoline Wiesner‡

School of Mathematics, University of Bristol, Bristol, England BS8 1TW, United Kingdom(Received 10 April 2014; published 25 August 2014)

We define a quantum entropy conditioned on postselection which has the von Neumann entropy of purestates as a special case. This conditional entropy can take negative values, which is consistent with part of aquantum system containing less information than the whole, which can be in a pure state. The definition isbased on generalized density operators for postselected ensembles. The corresponding density operators areconsistent with the quantum generalization of classical conditional probabilities following Dirac’s formalism ofquasiprobability distributions.

DOI: 10.1103/PhysRevA.90.022116 PACS number(s): 03.65.Ca, 03.65.Ta, 03.65.Wj, 03.67.−a

I. INTRODUCTION

Postselection refers to keeping the record of the outcome ofsome ensemble quantum measurement only for those parts ofthe ensemble which at a later point in time are in a desired, so-called postselected state and discarding the remaining results.Ensembles prepared in a state |ψ〉 and postselected in a state|φ〉 are described by a generalized density operator as [1,2]

ρψ |φ := |ψ〉〈φ|〈φ|ψ〉 . (1)

This generalization is appropriate when a weak measurementor no measurement has been performed between pre- andpostselection. For a proposal that incorporates strong mea-surements see [3]. Nevertheless, we restrict our attention to thecase where no strong measurement has been performed. Thesegeneralized density operators for postselected ensembles areused to obtain the so-called weak values �w [4] of an operator� as

�w = Tr[ρψ |φ�]. (2)

Experimentally, a weak value is obtained by weaklycoupling an ensemble of states to a measuring apparatus andpostselecting at a later time. To have an intuition about weakmeasurement on pre- and postselected ensembles, take thethree-box problem [5]. Here, at time t = 0 a state |ψ〉 isprepared in a superposition of states |A〉, |B〉, and |C〉 (thethree boxes), e.g., |ψ〉 = 1√

3(|A〉 + |B〉 + |C〉). At a later time

t = 1 the system is weakly measured in the basis {|A〉,|B〉,|C〉}and then postselected in some other state |φ〉 which isnot orthogonal to |ψ〉, e.g., |φ〉 = 1√

3(|A〉 + |B〉 − |C〉). The

resulting weak values of the operator projecting into the threeboxes at time t = 1 with postselection in state |φ〉 at timet = 2, calculated from Eq. (2), are, respectively,

Aw = 1, Bw = 1, Cw = −1. (3)

These results can also be mathematically studied in theframework of Dirac quasiprobabilities. In 1945, Paul Diracintroduced a complex phase-space distribution to make an

*sina.salek@bristol.ac.uk†roman.schubert@bristol.ac.uk‡k.wiesner@bristol.ac.uk

“Analogy Between Classical and Quantum Mechanics” [6],given by

Pr(am,bn) = Tr[ρ�m

A�nB

], (4)

where am and bn are the eigenvalues of the operators �A and�B , and �m

A and �nB are the projectors onto the corresponding

eigenstates.The Dirac distribution (4) satisfies all the conditions of

classical Kolmogorov probabilities, except that it is not apositive real function. It was shown that the negativity andcomplexness of this function are due to the noncommutativityof the quantum-mechanical observables [7] and, hence, canbe taken as a measure of quantumness in a similar way to thenegativity of the Wigner function (see [8,9]).

The Dirac distribution is normalized and gives correctmarginals,

∑

n

Pr(am,bn) =∑

n

Tr[ρ�m

A�nB

] = Tr[ρ�m

A

](5)

and∑

m

Pr(am,bn) =∑

m

Tr[ρ�m

A�nB

] = Tr[ρ�n

B

], (6)

it obeys the sum rule and the product rule, and it is compatiblewith Bayes’s law.

Note that Dirac distributions are not limited to phase space.In fact any two operators with nonvanishing overlap betweeneach of their eigenstates can be used to construct a Diracdecomposition,

ρ =∑

m,n

Pr(am,bn)|am〉〈bn|〈bn|am〉 , (7)

as long as the operators have the same Hilbert-space dimensionas the state ρ and their eigenvectors are mutually nonorthog-onal and none of those eigenvectors are orthogonal to thestate ρ. Hence a space spanned by any two such observableswould be sufficient to describe all the information availablefrom the state ρ. This is due to the fact that one can describeany quantum state of d-dimensional Hilbert space with d2 − 1elements.

For theoretical considerations on the Dirac distribution seethe work by Johansen [7] and Hofmann [10]. An experimentalprocedure for measuring the Dirac distribution of a generalquantum state has been given by Lundeen and Bamber [11].

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SINA SALEK, ROMAN SCHUBERT, AND KAROLINE WIESNER PHYSICAL REVIEW A 90, 022116 (2014)

We will now describe how weak values can be understoodin the framework of Dirac distributions.

Note that we can rewrite the weak value, Eq. (2), as

�w = 〈φ|�|ψ〉〈φ | ψ〉 . (8)

Now we can interpret �w as a conditional Dirac distributionof an ensemble which is pre- and postselected in states |ψ〉and |φ〉, respectively. Using Bayes’s law and defining ρψ :=|�〉〈�| and ρφ := |�〉〈�|, the weak value can be written as

�w = Pr(�|φ) = Tr[ρψρφ�]

Tr[ρψρφ]. (9)

In this last equation the interpretation as a conditionalquasiprobability follows from the use of Bayes’s law.

This close connection between weak values and Diracquasiprobabilities gives an operational meaning to the complexvalues of the Dirac distribution as a result of weak measure-ments. In the case where the measurement was performedby coupling the momentum of the measurement pointer tothe quantum system, the real part of the weak value refersto the shift in the position of the measurement pointer,while the imaginary part refers to the shift in the momentumof the measurement pointer [12]. The same interpretation canbe given to the real and complex part of the Dirac distribution.Given this operational meaning of the Dirac distribution, weproceed to use this formalism to make an analogy betweenclassical conditional probabilities and quantum conditionalstates.

In the following, we establish first that Eq. (1) is actuallya form of a conditional state by using the framework ofDirac distributions. This allows us then to define a corre-sponding conditional entropy of postselected quantum states.As mentioned earlier, due to noncommutativity of quantum-mechanical operators, Dirac probabilities of nonclassical statestake values outside the unit interval [0,1]. We shall see in thispaper that this quantumness is quantified by the (negative)values of the entropy we define.

II. QUANTUM CONDITIONAL STATES ANDCONDITIONAL ENTROPY

A. Quantum states, conditioned on postselection

Using the conditional Dirac distribution obtained inEq. (9), we now construct the corresponding conditional Diracdecomposition ρψ |φ in analogy to Eq. (7), given postselectionin some state |φ〉.

The summation here runs over the eigenstates of aprojection operator � measured weakly in between thetimes of preselection and postselection. We define � :=|h〉〈h|, where |h〉 is one out of a complete set of ba-sis states {|h〉} which are mutually nonorthogonal to thestate |φ〉 [16].

The conditional Dirac decomposition ρψ |φ we obtain is

ρψ |φ :=∑

h

Pr(h|φ)|h〉〈φ|〈φ|h〉 . (10)

ρψ |φ is the density operator of all paths leading from state |ψ〉to postselected state |φ〉. One may note that the operator ρψ |φ

is not Hermitian. It has been argued in the weak measurementliterature why this should not be a cause of concern [11].ρψ |φ thus defined is a trace-one operator and can indeed,by construction, be determined by weak measurements. Inthe case of the three-box problem, the eigenstates |h〉 of theprojector � would be |A〉, |B〉, and |C〉, representing thesystem in being in one of the three boxes A, B, and C attime t = 1.

By writing

Pr(h|φ) = 〈φ|h〉〈h|ψ〉〈φ|ψ〉 (11)

and inserting it into Eq. (10) we obtain

ρψ |φ =∑

h

|h〉〈h|ψ〉〈φ|〈φ|ψ〉

= |ψ〉〈φ|〈φ|ψ〉 , (12)

which is indeed the generalized density operator for post-selected ensembles, defined in Eq. (1). This links the in-terpretation Eq. (10) as conditional quantum states to thegeneralized density operator for postselected ensembles asin Eq. (1).

We observe that the application of Bayes’s law in thedefinition (10) results in a density operator which is theextension of classical conditional probabilities. This analogycan be further clarified by multiplying ρψ |φ by probabilityPr(φ) of the system ending in state |φ〉 and sum Eq. (10) overa complete basis {|φ〉}. Thus, we retrieve the density operatorof Eq. (7) which contains the full information about thesystem,

ρ =∑

φ

Pr(φ)ρψ |φ, (13)

with Pr(φ) = Tr[ρ|φ〉〈φ|] being the probability of thestate being in the state |φ〉 at the time of postselection.This is what is expected from the analogy with classicalprobabilities.

B. Entropies, conditioned on postselection

In classical information theory, the entropy of ran-dom variable X conditioned on selection of a particu-lar instance y of random variable Y is H (X|Y = y) =−∑

x Pr(x|y) ln Pr(x|y), where Pr(x|y) is the conditionalprobability of x given the occurrence of y. Note thatthis function is the average 〈− ln Pr(x|y)〉. The classicalconditional entropy is, again, an average, i.e., H (X|Y ) =∑

y Pr(y)H (X|Y = y).In analogy to the classical conditional entropy, we define

the conditional entropy of a state |ψ〉, given postselection inone out of a possible set of states |φ〉, to be

Sc(ψ |� = φ) = − 12 Tr[ρψ |φ ln(ρ†

ψ |φρψ |φ)], (14)

where ρ†ψ |φ is the conjugate transpose of ρψ |φ . We note that this

particular form of the entropy is an average of the logarithmof the density operator of the system:

Sc(ψ |� = φ) = 〈− ln((ρ†ψ |φρψ |φ)1/2)〉. (15)

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NEGATIVE CONDITIONAL ENTROPY OF POSTSELECTED . . . PHYSICAL REVIEW A 90, 022116 (2014)

One could choose other functional forms for the conditionalentropy here. For instance, one could choose the function− 1

2 Tr[ρ†ψ |φρψ |φ ln(ρ†

ψ |φρψ |φ)] instead. However, this would notcorrespond to an average of the logarithm. This makes ourchoice for the conditional entropy function a natural one.Note that the argument of the logarithm gives the singularvalues of the operator ρψ |φ . The singular value decompositionis a generalization of diagonalization needed here due to theparticular choice of basis in the definition of the two-statedensity operator [Eq. (10)]. For Hermitian operators, suchas ordinary density operators, the singular values and theeigenvalues are the same.

Finally, we define the general quantum conditional entropyas an average of the entropy conditioned on a particular choiceof postselection as

SC(ψ |�) =∑

φ

Pr(φ)Sc(ψ |� = φ). (16)

Equations (14)–(16) can be simplified. By inserting Eq. (1)into Eq. (14) we obtain

Sc(ψ |� = φ) = −1

2

〈φ|ln(ρ†ψ |φρψ |φ)|ψ〉〈φ|ψ〉

= ln|〈φ|ψ〉|. (17)

Inserting this into Eq. (16) we obtain the simplifiedexpression

SC(ψ |�) =∑

φ

|〈φ|ψ〉|2ln|〈φ|ψ〉|. (18)

This new expression is very useful for finding upper and lowerbounds of the conditional entropy. We can see that SC has anegative lower bound of 1

dln 1√

d, where d is the dimension of

the Hilbert Space. This lower bound is reached when all states|φ〉 have the same magnitude in overlap with the state |ψ〉. SC

is bounded from above by the von Neumann entropy of thesystem without postselection.

The conditional entropy SC can be understood as theamount of information contained in a system conditionedon a particular postselection. The case discussed here isthe special case of preselection in a pure state which,by definition, has zero von Neumann entropy. The upperbound of the conditional entropy is zero in this case andthus the conditional entropy of a pure preselected state|ψ〉 can be negative. This is a necessary consequence fromchoosing a subset of the entire (pure) system for postselectionwhich necessarily decreases the entropy below zero. Thus,a negative conditional entropy of postselected ensemblesbecomes intuitive as a subset contains less information than thewhole. The concept of negative conditional entropy appearsin other contexts in quantum information. For instance, fora bipartite system ρAB in a pure, i.e., maximally entangledstate, the joint entropy is zero. On the other hand, locallythe subsystems ρA and ρB are in maximally mixed states.Hence, the entropy of one subsystem, conditioned on themeasurement of the other, S(ρA|ρB) = S(ρAB) − S(ρB ) < 0,takes on a negative value. This conditional entropy was alsocalculated in the form of S(ρA|ρB) = −Tr[ρAB lnρA|B], whereρA|B = limn→+∞[ρ1/n

AB (1A ⊗ ρB)−1/n]n [13], which is closer

in outlook to the construction introduced in Eq. (16). Aswe have seen it is not obvious how to extend conditionalprobabilities to quantum conditional states. For spatiallyseparated states with strong measurement there have beenseveral proposals, e.g., Leifer [14] and Cerf and Adami[13]. One could take conditional entropies as a criterionfor choosing a definition of conditional state, as done byCerf and Adami [13]. Our work is similar in spirit to that,however with two main differences. While their definitions areconcerned with spatially separated states, the state separationin our work is temporal. This allows us to assume weakmeasurement between pre- and postselection. Other forms ofconditional state for postselected ensembles are conceivable.For instance, Silva et al. have defined a density operatorfor postselected ensembles by tensoring the two-time vectorwith its conjugate transpose [3]. Although one might beinclined to interpret the resulting density operator as a quantumextension of conditional probabilities, such interpretationcannot be valid since the mathematical description representseach pre- and postselection as a complete state and, hence,their conditional entropy [Eq. (16)] would be in the samerange as the von Neumann entropy of a quantum state, i.e.,positive.

C. Example: three-box postselection

We will now revisit the three-box problem [5] fromthe beginning and calculate the conditional entropies justdefined.

Choosing pre- and postselection as before, i.e., |ψ〉 =1/

√3(|A〉 + |B〉 + |C〉) and |φ〉 = 1/

√3(|A〉 + |B〉 − |C〉),

one obtains for the conditional entropy Sc of the system,Eq. (14), Sc(ψ |� = φ) = − ln 3. In order to calculate theconditional entropy SC , Eq. (16), given that the Hilbert-spacedimension of the system is 3, one needs to perform the samecalculation as above for two other postselected states, |φ′〉 and|φ′′〉. The only restrictions we put on the choice of these twopostselected states, for the sake of this example, are for themto be mutually nonorthogonal to the state of each box, i.e.,to |A〉, |B〉, and |C〉, and to the initial state |ψ〉, and to span,together with our original state |φ〉, the Hilbert space of thestate |ψ〉. One such choice of states is

|φ′〉 = 1√3|A〉 + −3 − √

3

6|B〉 + −3 + √

3

6|C〉 (19)

and

|φ′′〉 = 1√3|A〉 + 3 − √

3

6|B〉 + 3 + √

3

6|C〉. (20)

Calculating Sc of Eq. (14) for |φ′〉 and |φ′′〉 gives Sc(ψ |� =φ′) = − log3 4.10 and Sc(ψ |� = φ) = − log3 1.10. Withprobabilities Pr(φ) = 0.11, Pr(φ′) = 0.06, and Pr(φ′′) = 0.83(given by |〈φ|ψ〉|2 and correspondingly for φ′ and φ′′), theconditional entropy Eq. (16) becomes SC(ψ |�) = −0.26, withthe logarithm being calculated in base 3 for convenience.Since our entropy is a measure of correlation of the pre- andpostselection one should expect that for different choices of|φ′〉 and |φ′′〉 the value of the entropy would be different.

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SINA SALEK, ROMAN SCHUBERT, AND KAROLINE WIESNER PHYSICAL REVIEW A 90, 022116 (2014)

The three-box problem illustrates the negativity of theconditional entropy where each path through a box containspartial information of the system as a whole.

III. CONCLUSION

In this work we have defined a quantum conditional entropyfor postselected ensembles. To this end, we have adaptedthe formalism of Dirac quasiprobabilities and showed thatthe two-state density operator of postselected ensembles isa generalization of the classical conditional probability, giventhat a particular final state is selected. We have found upper andlower bounds on such entropies and interpreted the values asthe amount of information contained in a system conditioned

on a particular postselection. Furthermore, we showed thatthese states have properties beyond their classical counterparts.Most notably, conditional quantum entropies as defined in thiswork can have negative values. This is consistent with ourinterpretation of this entropy measuring quantumness. Thisentropy function should open up avenues for studying theproperties of weak measurement of more general states thandiscussed in this paper, such as mixed postselected states.

ACKNOWLEDGMENTS

We thank Jeff Lundeen, Holger Hofmann, Sandu Popescu,and Ralph Silva for their helpful comments and stimulatingdiscussions. K.W. thanks Engineering and Physical SciencesResearch Council for financial support.

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