Information leak and incompatibility of physical context: A modified approach
IInformation leak and incompatibility of physical context: A modified approach
Arindam Mitra , , ∗ Gautam Sharma , , † and Sibasish Ghosh , ‡ Optics and Quantum Information Group, The Institute of Mathematical Sciences,C. I. T. Campus, Taramani, Chennai 600113, India. Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India. (Dated: February 16, 2021)A beautiful idea about the incompatibility of Physical Context(IPC) was introduced in [Phys.Rev. A 102, 050201(R) (2020)]. Here, a context is defined as a set of a quantum state and twosharp rank-one measurements, and the incompatibility of physical context is defined as the leakageof information while implementing those two measurements successively in that quantum state. Inthis work, we show the limitations in their approach. The two primary limitations are that, (i) theirapproach is not generalized for POVM measurements and (ii), they restrict information theoreticagents Alice, Eve and Bob to specific quantum operations and do not consider most general quantumoperations i.e., quantum instruments and (iii), their measure of IPC can take negative values inspecific cases in a more general scenario which implies the limitation of their information measure.Thereby, we have introduced a generalization and modification to their approach in more generaland convenient way, such that this idea is well-defined for generic measurements, without theselimitations. We also present a comparison of the measure of the IPC through their and our method.Lastly, we show, how the IPC reduces in the presence of memory using our modification, whichfurther validates our approach.
I. INTRODUCTION
Measurement incompatibility is a key feature of Quan-tum theory which distinguishes it from the classicalworld[1]. A pair of observables are incompatible if theyare not measurable simultaneously,i.e., their outcomescan not be obtained jointly via a single joint measure-ment. Today, the connections among incompatibility,nonlocality and steering are well known [2, 3]. Nonclassi-cal features like Bell inequality violation as well as steer-ing can be demonstrated only using incompatible mea-surements [4, 5]. It is also well known that incompati-ble measurements provide an advantage over compatiblemeasurements in several information-theoretic tasks inquantum information theory [6, 7]. Measurement com-patibility can be characterized as the existence of a com-mon (i.e, constructed using same ancilla state and Hilbertspace) commuting Naimark extensions [8]. It has beenrecently shown there are several layers of classicality in-side the compatibility of measurement[9, 10].Recently, a novel idea was presented in ref.[18] toget a better understanding of non-classicality associatedwith incompatibility. The authors of [18] introduced theconcept of incomaptibility of the physical context(IPC),which is a function of a given context, where a contextcomprises of a quantum state and two measurements.In a way their measure of IPC captures the notion ofnon-classicality associated with the context, as it van-ishes when the state is a maximally mixed state or themeasurements are commuting with each other. It wasdefined as the difference between the information remain- ∗ [email protected] † [email protected] ‡ [email protected] ing in a quantum state after the first sharp measurementand after the second sharp measurement. Moreover, theIPC is also linked with the information leakage when aneavesdropper performs a measurement on the state beingtransferred in a QKD like game.However, as we will show later that their approachhas several limitations. Firstly, They restrict informationtheoretic agents Alice, Eve and Bob to specific quantumoperations and did not consider most general quantumoperations i.e., quantum instruments. Secondly, if we donot restrict Alice, Eve and Bob to specific quantum op-erations which they did, then their measure of IPC cantake negetive value, which implies that the state aftersecond measurement by eavesdrpper Eve has more in-formation than the state after first measurement whichphysically does to not make sense. Thirdly, it is not pos-sible to extend this idea to generic POVM measurementsthrough their approach and without introducting quan-tum instruments. Fourthly, in presence of memory theIPC can increase which is against the intuition that in-compatibility is non-increasing as we add memory.In this work, we have generalized their idea for POVMsand modified their information measure. Our measure ofthe IPC can never be negative and it is non-increasingon addition of memory. In this way our approach is hasa wider applicability.The rest of this paper is organised as follows: In sec-tion (II), we discuss the preliminary concepts necessaryfor this paper. Then, we discuss the limitations of theapproach given in [18] and discuss our main results insection (III). Further, in section (IV), we include thepresence of memory in our analysis. Finally, in section(V) we summarise our work and discuss future direction. a r X i v : . [ qu a n t - ph ] F e b II. PRELIMINARIESA. Observables and Channels
An observable A with outcome set Ω A in quantummechanics, is a collection of positive hermitian matri-ces { A ( x ) | x ∈ Ω A } such that (cid:80) x A ( x ) = I . A pair ofobservables ( A, B ) acting on same d dimensional Hilbertspace H and with outcome sets Ω A and Ω B respectively,is compatible if there exist a joint observable G actingon same Hilbert space H and outcome set Ω A × Ω B suchthat for all ρ ∈ S ( H ), x ∈ Ω A and y ∈ Ω B A ( x ) = (cid:88) y G ( x, y ); B ( y ) = (cid:88) x G ( x, y ) (1)where S ( H ) is the state space. Only for PVMs, compat-ibility implies commutativity. We denote the set of allobservables as O .On the other hand, a quantum channel is a CPTP mapfrom one state space S ( H ) to another state space S ( H ),where H and H are two Hilbert spaces. We denote theconcatenation of two quantum channels Λ and Λ as Λ ◦ Λ . Therefore, for all ρ ∈ S ( H ) (Λ ◦ Λ )( ρ ) = Λ (Λ ( ρ )).Consider two quantum channels Γ : S ( H ) → S ( H ) andΛ : S ( H ) → S ( H (cid:48) ). If there exist a quantum channelΘ : S ( H (cid:48) ) → S ( H ) such that Γ = Θ ◦ Λ holds, we denoteit as Γ (cid:22)
Λ. If both Γ (cid:22)
Λ and Λ (cid:22)
Γ hold, we denoteit as Γ (cid:39)
Λ and we call it as Γ and Λ are concatenationequivalent. We denote the set of all channels equivalentto channel Λ as [Λ].There exists a special type of channel known as com-pletely depolarising channel , which we will use in follow-ing section. A channel Σ is called completely depolarisingchannel if for all T ∈ L + ( H ), Σ( T ) = Tr( T ) η for somefixed η ∈ S ( H ), where L + ( H ) is set of positive linearoperators on Hilbert Space H . We denote the set of allchannels as C . B. Quantum Instruments and Measurementmodels
In quantum measurements, there are two equivalentconcepts, namely measurement models and quantum in-struments [11, 12]. Measurement models are descriptionsof measurement process, where as instruments are theconsise version of it. Consider a measured system S as-sociated with a Hilbert space H S and with density matrix ρ and a ancilla system associated with another Hilbertspace H a and with density matrix σ a . To perform a mea-surement on a measured system, at first a joint unitary U have to be applied on the composite system where, U is acting on Hilbert space H S ⊗ H a . Then, a pointer ob-servable A (cid:48) with outcome set Ω (cid:48) A have to be measured onthe ancilla system. Now in this process if the observable A with same outcome set as A (cid:48) has, is the measured on the system S then for all x ∈ Ω A and ρ ∈ H S we haveTr[ ρA ( x )] = Tr[ U ( ρ ⊗ σ a ) U † ( I ⊗ A (cid:48) ( x ))] . (2)The average post measurement state is given byΛ( ρ ) = Tr H a [ U ( ρ ⊗ σ a ) U † ] . (3)Here, Λ is a quantum channel. This measurement modelis specified by the quadruple ( H a , σ a , U, A (cid:48) ).A quantum instrument I through which the measure-ment of an observable A can be implemented, is a collec-tion of CP trace non-increasing maps { Φ x } such that forall x ∈ Ω A and ρ ∈ H S we haveTr[ ρA ( x )] = Tr[Φ x ( ρ )] (4)and (cid:88) x Φ x ( ρ ) = Λ( ρ ) (5)where Λ is a quantum channel. We call such an in-strument as A -compatible instrument. If I = { Φ x } isan A -compatible instrument, then another instrumentΘ ◦ I = { Θ ◦ Φ x } is also an A -compatible instrument[13], where (Θ ◦ Φ x )( ρ ) = Tr[Φ x ( ρ )]Θ (cid:16) Φ x ( ρ )Tr[Φ x ( ρ )] (cid:17) . Wedenote the set of all A -compatible instruments as J A .Therefore, given a measurement model ( H a , σ a , U, A (cid:48) ),one can associate a quantum instrument I such that forall x ∈ Ω A and ρ ∈ H S we haveTr[Φ x ( ρ )] = Tr[ U ( ρ ⊗ σ a ) U † ( I ⊗ A (cid:48) ( x ))] . (6)Similarly, given a quantum instrument it is possible tofind out a measurement model such that equation (6)holds [14]. This implies that these two concepts areequivalent. C. Observable-Channel compatibility
A quantum channel Λ is compatible with an observable A if there exists a quantum instrument I = { Φ x } suchthat equations (4) and (5) together hold. Otherwise, theyare incompatible. If a channel Λ and an observable A are compatible, we denote it as Λ ◦◦ A [15]. We call Λ as A -compatible channel. It is well known that completelydepolarising channels are compatible with any observable [12]. For a quantum channel Λ ∈ C and an observable A ∈ O , following sets are introduced in [15]: τ c (Λ) = { X ∈ O | Λ ◦◦ X } ; (7) σ c ( A ) = { Γ ∈ C | Γ ◦◦ A } . (8)Let us now write down the following theorem which wasoriginally proved in ref.[13]: Theorem 1.
Suppose A ∈ O be an observable and ( V, K , ˆ A ) be its Naimark extension, i.e, K is a Hilbertspace, V is an isometry and ˆ A = { ˆ A ( x ) } is a PVM suchthat V † ˆ A ( x ) V = A ( x ) for all x ∈ Ω A . Then, σ c ( A ) = { Λ ∈ C | Λ (cid:22) Λ A } (9) where for any state ρ , Λ A ( ρ ) = (cid:80) x ˆ A ( x ) V ρV † ˆ A ( x ) . We call Λ A as parent channel of σ c ( A )and we also thecall corresponding A -compatible instrument I A a parentinstrument in J A . Clearly, Λ A depends on the choice ofthe Naimark extension. But any two parent channels areconcatenation equivalent. Therefore, we have freedom tochoose it. D. Holevo Bound
The Holevo bound captures the maximum classicalinformation that can be extracted from a ensemble ofquantum states[16]. Suppose, we have an ensemble E = { p X ( x ) , ρ x } , and our task is to determine the classicalindex x by doing some measurements. The density ma-trix operator corresponding to this ensemble has the form ρ = (cid:80) x p X ( x ) ρ x . Now, we can do a measurement Λ y , sothat the information gain after doing the measurementis given by the mutual information I = I ( X ; Y ) afterthe measurement, where Y is the random variable cor-responding to the outcome of measurement. It is knownthat the maximum value of this mutual information isgiven by the Holevo bound[16, 17], given by χ ( E ) = S ( ρ ) − (cid:88) x p X ( x ) S ( ρ x ) (10)where S ( ρ ) is the Von-Neumann entropy of the state ρ .It is interesting to note that the holevo bound χ is alsothe mutual information of a classical-quantum state ofthe form ρ CQ = (cid:80) x p X ( x ) | x (cid:105) (cid:104) x | ⊗ ρ x . Under the actionof a channel Λ the ensemble transforms as E → E (cid:48) = { p X ( x ) , ρ (cid:48) x } . But we know that the mutual informationis non-increasing under the action of channels[17], whichimplies that the Holevo information is also non-increasingunder the action of quantum channels, i.e., χ ( E ) ≥ χ ( E (cid:48) ) . (11) E. Incompatibility of physical context
In a recent work [18], the concept of IPC was intro-duced which was further used to show quantum resourcecovariance[19]. To define this idea, we need the notionof context. A context is defined as C = { ρ, X, Y } , where ρ is an arbitrary quantum state. Also, X = { X i } and Y = { Y j } are two observables, with X i and Y j as therespective eigen projectors. Other than the definition of context, we also need agame using which we define the incompatibility of a con-text C . The game goes like this. Alice prepares thequantum state ρ , and of course it has some informationcontent which can be quantified by using any known mea-sure. The authors in ref.[18] quantify the information of ρ using the following I ( ρ ) = ln d − S ( ρ ) , (12)where S ( ρ )=- T r ( ρ ln ρ ) is the von Neumann entropy of ρ and d is the dimension of the Hilbert space. This infor-mation is non-negative, i.e., I ( ρ ) ≥
0, is ensured because S ( ρ ) ≤ ln d . After state preparation, Alice performs anoisy measurement with X on the prepared state, so that ρ transforms as ρ → N X ( ρ ) = d (cid:88) i =1 X i ρX i . (13)So, after this operation the information content in thestate is I = I ( N X ( ρ )). This state N X ( ρ ) is then deliv-ered to Bob, who verifies the information content of thestate. In case Bob finds that there is no loss of informa-tion, Alice and Bob will agree that the channel is freefrom information leakage.But it might happen that there is an eavesdropper,Eve, who performs a noisy measurement Y on the state N X ( ρ ), before it is delivered to Bob. The state is thentransformed as N X ( ρ ) → ( N Y ◦ N X )( ρ ) = N Y X ( ρ ) = d (cid:88) j =1 Y j N X ( ρ ) Y j . (14)Thus, the remaining information content in the state N XY ( ρ ) is I = I ( N Y X ( ρ )). And therefore, the leakagein the information content is given by I C = I − I = I ( N X ( ρ )) − I ( N Y X ( ρ )) , = S ( N Y X ( ρ )) − S ( N Y X ( ρ )) . (15)Hence, only if I C >
0, Alice and Bob will know thatthere is information leakage from the channel. Noticethat I C = 0 in two kind of scenarios: 1) If X and Y commute with each other and 2) if ρ is a maximally mixedstate. In the first scenario N X ( ρ ) = N Y X ( ρ ) becausethe two operators are compatible with each other. Andin the second type of scenarios I = 0 and there is noinformation to loose which results in I = I . Thus, werequire the incompatibility I C to be non-zero for Bob todetect any leakage of informationHence, the concept of IPC can be defined as Definition 1.
Context incompatibility is the resourceencoded in a context C = { ρ, X, Y } that allows one totest the safety of a communication channel against in-formation leakage. It is quantified as I C = I − I = I ( N Y X ( ρ )) − I ( N X ( ρ )) . It is operationally related to theamount of information lost from the system under an ex-ternal measurement. III. MAIN RESULTSA. Limitations of incompatibility of physicalcontext
In this section, we discuss the limitations of approachgiven in [18].1. First of all, according to the approach given in [18],the post-measurement states after measuring sharpobservable X ∈ O on a quantum state ρ is N X ( ρ ).Therefore, to measure an observable X , Alice andEve both are restricted to use a particular channel N X ∈ C , or equivalently they are restricted to usea particular quantum instrument I X = { Φ X ( x ) } such that (cid:80) x Φ X ( x ) = N X . Since, we have nocontrol atleast over eavesdropper Eve, there is noreason to assume such a restriction.2. Second, to generalize it, suppose we remove such re-striction, i.e., to measure an observable, now Aliceand Eve can use all possible instruments that arecompatible with that observable. Then to measurethe observable X if Alice uses an arbitrary instru-ment I (cid:48) X = { Φ (cid:48) X ( x ) } such that Λ (cid:48) = (cid:80) x Φ (cid:48) X ( x )and to measure Y Eve uses a special instrument I depoY = { Φ depoY ( y ) } such that for all ρ ∈ S ( H ) anda fix pure state η , Λ depoη ( ρ ) = (cid:80) y Φ Y ( y )( ρ ) = η is acompletely depolarising channel. Now, as S ( η ) = 0,from equation (15) we have I C = I (Λ (cid:48) ( ρ )) − I ((Λ depoη ◦ Λ (cid:48) )( ρ ))= − S (Λ (cid:48) ( ρ )) ≤ . (16)The negativity of IPC implies that the post-measurment state of Eve has more informationthan the post-measurement state of Alice, whichdoes not make sense. Such a problem is occur-ing because Von Neumann entropy is not mono-tonically non-increasing under action of a quantumchannel. Therefore, in this general context, theirinformation measure is not a proper informationmeasure.3. Thirdly, as we know that for any POVM, post mea-surement state depends on the quantum instrumentused to implement that POVM, their results cannot be generalised for POVMs without introduc-ing quantum instruments or equivalently withoutintroducing measurement models! Therefore,in our attempt to generalize the idea of IPCfor POVMs, we need to modify idea and present it in adifferent way which we describe in next section.
B. Modified measure of Information leakage
In this section we present a generalization of the gamepresented in Sec.II E. Now, in the game, after the statepreparation of ρ , instead of only doing a sharp measure-ment we allow Alice to perform a more generic measure-ment. Now, Alice performs her measurement with thePOVM, A on the quantum state ρ ∈ S ( H ) using the A -compatible instrument I (cid:48) A = { Φ A,x } such that Λ (cid:48) A = (cid:80) x Φ A,x and generates the ensemble E A = { p x , ρ x } ,where p x = Tr[Φ A,x ( ρ )] and ρ x = Φ A,x ( ρ )Tr[Φ A,x ( ρ )] . Here, Λ (cid:48) A is the quantum channel such that Λ (cid:48) A : S ( H ) → S ( K ).Furthermore, we quantify the information content of theensemble E A via the Holevo bound as: χ ( ρ, I (cid:48) A ) = S (Λ (cid:48) A ( ρ )) − (cid:88) x p x S ( ρ x ) . This measure of information has been previously usedto quantify information gain in ref.[20]. Similarly, theeavesdropper Eve performs the POVM measurement B on the quantum state Λ (cid:48) A ( ρ ) ∈ S ( K ) using the B -compatible instrument I (cid:48) B = { Φ B,y } such that Λ (cid:48) B = (cid:80) y Φ B,y and generates the ensemble E B = { p x , Λ (cid:48) B ( ρ x ) } .Here, Λ (cid:48) B is the quantum channel such that Λ (cid:48) B : S ( K ) →S ( K (cid:48) ). It should be noted that, Alice and Bob do nothave access to Eve’s measurement outcomes, her mea-surement can be represented using a channel. Now, theinformation remaining in the state (Λ (cid:48) B ◦ Λ (cid:48) A )( ρ ) is givenby its Holevo bound, i.e., χ ( ρ, I (cid:48) A , I (cid:48) B ) = S ((Λ (cid:48) B ◦ Λ (cid:48) A )( ρ )) − (cid:88) x p x S (Λ (cid:48) B ( ρ x )) . Therefore, Bob who was expecting to receive an ensem-ble with information χ ( ρ, I (cid:48) A ), would receive a differentensemble with information content χ ( ρ, I (cid:48) A , I (cid:48) B ). Thus,the new form of information leakage of the channel is I Hc ( ρ, I (cid:48) A , I (cid:48) B ) = χ ( ρ, I (cid:48) A ) − χ ( ρ, I (cid:48) A , I (cid:48) B )= S (Λ (cid:48) A ( ρ )) − S ((Λ (cid:48) B ◦ Λ (cid:48) A )( ρ ))+ (cid:88) x p x S (Λ (cid:48) B ( ρ x )) − (cid:88) x p x S ( ρ x ) . (17)As Holevo bound is monotonically non-increasing un-der the action of quantum channels, I Hc ( ρ, I (cid:48) A , I (cid:48) B ) ≥ I Hc ( ρ, I (cid:48) A , I (cid:48) B ) >
0, Alice and Bob will be able todetect the information leakage in the channel.Now, if Eve is rational, her goal will be to minimizeleakage along with collecting information. Therefore,to measure B she will choose an instrument such that I Hc ( ρ, I (cid:48) A , I (cid:48) B ) takes the minimum value. Now, let Λ B be a parent channel in σ c ( B ) and corresponding B -compatible instrument be I B . Then, as for any otherchannel Λ (cid:48) B ∈ σ B , Λ (cid:48) B (cid:22) Λ B holds and Holevo boundis monotonically decreasing under action of a quantumchannel, χ ( ρ, I (cid:48) A , I (cid:48) B ) ≤ χ ( ρ, I (cid:48) A , I B ) ∀I B . (18)Therefore, implementation of a parent instrument keepsmaximum amount of accessible information or equiva-lently maximum Holevo bound! Therefore, for a giveninstrument of Alice the minimum leakage of informationis I Hc ( ρ, I (cid:48) A , B ) = min I (cid:48) B I Hc ( ρ, I (cid:48) A , I (cid:48) B )= χ ( ρ, I (cid:48) A ) − max I (cid:48) B χ ( ρ, I (cid:48) A , I (cid:48) B )= χ ( ρ, I (cid:48) A ) − χ ( ρ, I (cid:48) A , I B )= I Hc ( ρ, I (cid:48) A , I B ) . (19)Note that the choice of B depends on output state space S ( K ) of the quantum channel Λ (cid:48) A and in that sense arbi-trary.Now, if Alice is also rational and she does not knowthe presence of Eve, she will try to create an ensem-ble with most accessible information such that the re-ceiver i.e, Bob can get best amount of information, orequivalently, she will use an A -compatible instrument forwhich χ ( ρ, I (cid:48) A ) is maximum. Let, Λ A be a parent channelin σ c ( A ) and corresponding A -compatible parent instru-ment be I A . Then, using arguments as above χ ( ρ, I (cid:48) A ) ≤ χ ( ρ, I A ) ∀I (cid:48) A . (20)Therefore, if Alice uses the instrument I A , in this casethe information leakage will be minimum when Alice usesa parent channel from σ c ( A ), is I Hc ( ρ, A, B ) = I Hc ( ρ, I A , I B ) . (21)Therefore, assuming both Alice and Eve to be ratio-nal, I Hc ( ρ, A, B ) is the appropriate amount of informa-tion leak . C. Incompatibility of physical context: A modifiedversion
First of all we modify the notion of context so that, C = { ρ, X , Y } , where X and Y are POVM measurementsacting on S ( H ) and S ( H (cid:48) ) respectively. Since, X and Y are given, to define IPC, we restrict Alice’s instru-ment I (cid:48) , H (cid:48) X = { Φ (cid:48) , H (cid:48) X ,x } such that Λ (cid:48) X = (cid:80) x Φ (cid:48) , H (cid:48) X ,x andΛ (cid:48) X : S ( H ) → S ( H (cid:48) ). We denote the set of all such X -compatible instruments as J H (cid:48) X . With this restric-tion also being rational, Alice’s goal will be to maximize χ ( ρ, I (cid:48) , H (cid:48) X ). Let, for some I H (cid:48) X ,max ∈ J H (cid:48) X ,max I H(cid:48) X χ ( ρ, I H (cid:48) X ) = χ ( ρ, I H (cid:48) X ,max ) . (22)Therefore, similar to the previous subsection, in thiscase the appropriate amount of information leak is I ( C ) = I Hc ( ρ, I H (cid:48) X ,max , I Y ) . (23)For the special case of, H (cid:48) = H , we have I ( C ) = I Hc ( ρ, I H X ,max , I Y ) . (24)Therefore, we can define the generalized version ofIPC as: Definition 2.
Context incompatibility is the resource en-coded in a context C = { ρ, X , Y } that allows one to testthe safety of the channel against information leakage.This resource is quantified via I ( C ) = I Hc ( ρ, I H X ,max , I Y ) ,where I H X ,max is the X -compatible instrument that max-imizes χ ( ρ, I (cid:48) , H (cid:48) X ) . Operationally, it is the proper infor-mation leakage in the channel caused by an external mea-surement on the state. Moreover, if Alice performs a sharp measurement X = { X i } , from theorem (1), choosing V = I or equivalentlychoosing H = K and X i = ˆ X i we get a parent channelΛ A = N ( X ) : S ( H ) → S ( H ). Let I X = { Φ X } be cor-responding X -compatible parent instrument. As, imple-mentation of a parent channel keeps maximum amount ofaccessible information or, equivalently maximum Holevobound, we have I H X ,max = I X . Then the proper informa-tion leakage will have the following form I ( C ) = χ ( ρ, N X ( ρ )) − χ ( ρ, N X , Λ Y ( ρ )) , = S ( N X ( ρ )) − S ((Λ Y ◦ N X )( ρ ))+ (cid:88) x p x S (Λ Y ( ρ x )) − (cid:88) x p x S ( ρ x ) . (25)where Λ Y is the Y -compatible parent channel correspond-ing to the Y -compatible parent instrument I Y . D. Relation between two definitions
Our generalization of the measure of IPC, gives asimplified form when we demand that both Alice andEve perform rank-1 sharp measurements X and Y us-ing parent instruments I X ∈ J X and I Y ∈ J Y , where N X ∈ σ c ( X ) and N Y ∈ σ c ( Y ) are corresponding chan-nels respectively. In this case I ( C ) reads as I ( C ) = (cid:88) x p x S ( N Y ( ρ x )) − I C . (26)The above equation relates our generalized measure ofIPC with the measure of IPC I C defined in [18]. Tocompare the two measures of the IPC, we remind thereader that I C is zero when 1) X and Y commute or 2) ρ is a maximally mixed state (see secII E). Coming to thenew measure of IPC we find that I ( C ) = 0 whenever X and Y commute, because then N Y ( ρ x ) are pure states.However, I ( C ) is not necessarliy equal to zero when ρ isa maximally mixed state(as in the Eq. (26), I C is zerobut S ( N Y ( ρ x ))’s are not zero).This implies that our measure captures the incompat-ibility of a context even when the state(belonging to thecontext) is a maximally mixed state. This is unlike theprevious measure of IPC I C given in ref [18], which saysthat the context is compatible if the state is a maximallymixed state. IV. INCOMPATIBILITY OF PHYSICALCONTEXT IN THE PRESENCE OF MEMORY
Motivated by the work in [21], where it was shown thatin the presence of memory the total uncertainty of twomeasurements gets reduced, we ask the question, howthe IPC will change in the presence of memory? To ac-comodate the presence of memory, we modify our gameslightly, for the scenario where we perform only rank-oneprojective measurements X and Y .In the modified game, our initial state σ in is the sub-system of the bipartite state σ in,M , where M acts as thememory. After the X measurement on the subsystem σ in , Alice produces the bipartite ensemble (cid:80) x p x ρ xAM ,where ρ xM = Tr A [ ρ xAM ] acts as the memory and Bobreceives the subsystem A prepared in the state ρ A =Tr M [ ρ AM ]. On this ensemble, if we use the approachin ref.[18], the information content of ρ A conditioned onmemory ρ M is given by I mem = ln d − S ( A | M ) = ln d − S ( ρ AM ) + S ( ρ M ) , where S ( A | M ) = S ( ρ AM ) − S ( ρ M ) is the conditionalentropy [17]. After the Y measurement by Eve on ρ A ,the ensemble transforms as (cid:80) x p x ρ xAM → (cid:80) x p x ( N Y ⊗ I )( ρ xAM ) = (cid:80) x p x ρ xA (cid:48) M , so that the remaining informa-tion content of the state ρ A (cid:48) is I mem = ln d − S ( A (cid:48) | M ) = ln d − S ( ρ A (cid:48) M ) + S ( ρ M ) , where I is the identity channel acting on the memory.Therefore, in the presence of memory, the expression ofIPC takes the following form. I memC = I mem − I mem = S ( ρ AM ) − S ( ρ AM ) . (27)To compare the IPC with and without memory, we com-pare Eq.(15) with Eq.(27), which gives the following I C − I memC = [ S ( ρ A (cid:48) ) − S ( ρ A (cid:48) M )] − [ S ( ρ A ) − S ( ρ AM )]= I coh ( M (cid:105) A (cid:48) ) − I coh ( M (cid:105) A ) ≤ . (28)Here, I coh ( M (cid:105) A ) = S ( ρ A ) − S ( ρ AM ) is the coherent infor-mation that is non-increasing under the action of quan-tum channels [17, 22, 23]. This analysis tells us thatthe IPC is increasing in the presence of memory, which seems contrary to the intuition that memory reduces theincompatibility.Next, we compute the IPC in the modified gamewith our approach. In our case, after the X measure-ment, the extractable classical information from ρ A isthe mutual information of the quantum-classical ensem-ble ρ CA = (cid:80) x | x (cid:105) C (cid:104) x | ⊗ ρ xA (see sec.II D). However, nowit is conditioned on the memory ρ M . Therefore, in thepresence of memory the extractable information will bethe mutual information between ρ C = (cid:80) x | x (cid:105) C (cid:104) x | and ρ A , conditioned on the memory ρ M via the tripartiteclassical-quantum state ρ CAM = (cid:80) x p x | x (cid:105) C (cid:104) x | ⊗ ρ xAM ,i.e., X mem = S ( A : C | M )= S ( A | M ) + S ( C | M ) − S ( AC | M )= S ( ρ AM ) − S ( M ) + S ( ρ CM ) − S ( ρ CAM ) . Here, we have simply expanded the conditional en-tropies to get the final form. Also, after Eve performsher measurement Y on the subsystem ρ A , the remain-ing mutual information between ρ A (cid:48) and ρ C conditionedon the memory ρ M , via the classical-quantum ensemble (cid:80) x p x | x (cid:105) C (cid:104) x | ⊗ ρ xA (cid:48) M is given by X mem = S ( A (cid:48) : C | M )= S ( A (cid:48) | M ) + S ( C | M ) − S ( A (cid:48) C | M )= S ( ρ A (cid:48) M ) − S ( M ) + S ( ρ CM ) − S ( ρ CA (cid:48) M ) . Therefore the IPC, using our approach in presence ofmemory, takes the following form I mem ( C ) = I mem − I mem = S ( ρ AM ) − S ( ρ A (cid:48) M ) − S ( ρ ACM ) + S ( ρ A (cid:48) CM )= S ( ρ AM ) − S ( ρ A (cid:48) M ) − (cid:88) x S ( ρ xAM ) + (cid:88) x S ( ρ xA (cid:48) M )= S ( ρ AM ) − S ( ρ A (cid:48) M ) + (cid:88) x S ( ρ xA (cid:48) ) . (29)In the above calculations we have used the fact that ρ xA are pure states so that ρ xAM and ρ xA (cid:48) M are bipartiteproduct states. Now, if we compare the IPC without andwith memory in from Eq.(26) and Eq.(29) respectively,we have I ( C ) − I mem ( C )= [ S ( ρ A ) − S ( ρ AM )] − [ S ( ρ A (cid:48) ) − S ( ρ A (cid:48) M )]= I coh ( M (cid:105) A ) − I coh ( M (cid:105) A (cid:48) ) ≥ . (30)Thus, we find that using our approach, the IPC isnon-increasing in the presence of memory. It follows theintuition that the presence of memory should reducethe incompatibility. On comparing Eq.(28) with Eq.30,we find that I C − I memC = − ( I ( C ) − I mem ( C )). Thisrelation strongly indicates that the information leakagecontent I C envisaged in ref.[18], is not capable of fullycapturing the problems we face in a typical quantuminformation processing scenarios. This analysis alsovalidates our approach for quantifying the IPC. Example 1 (Comparison of incompatibilities of a phys-ical context with two different memories) . Suppose, σ in = α | λ (cid:105) (cid:104) λ | + β | λ (cid:105) (cid:104) λ | be a qubit state and S x = {| + (cid:105) (cid:104) + | , |−(cid:105) (cid:104)−|} and S z = {| (cid:105) (cid:104) | , | (cid:105) (cid:104) |} be the sharpspin measurements along x and z directions respectively,where {| λ (cid:105) , | λ (cid:105)} are the eigen basis of σ in . Here ≤ α, β ≤ and α + β = 1 . Now, we take our physical con-text as C = ( σ in , S z , S x ) . We will consider the followingcase where Alice is using memories M keeping input state σ in fixed:Suppose, Alice is using a qubit memory M such that σ in,M = p | ψ in,M (cid:105) (cid:104) ψ in,M | + − p I AM , where ≤ p ≤ , | ψ (cid:105) in,M = √ α (cid:48) | λ (cid:105) | λ (cid:48) (cid:105) + √ β (cid:48) | λ (cid:105) | λ (cid:48) (cid:105) , I AM = I × , ≤ α (cid:48) , β (cid:48) ≤ , α (cid:48) + β (cid:48) = 1 , {| λ (cid:105) (cid:48) , | λ (cid:105) (cid:48) } are the eigenbasis of σ M and σ M = T r ρ i n [ σ in,M ] . Alice chooses α (cid:48) , β (cid:48) and p such that α = pα (cid:48) + 1 − p β = pβ (cid:48) + 1 − p hold. Then, Tr M [ σ in,M ] = σ in . For example, when α = and β = , one possible choice is p = , α (cid:48) = and β (cid:48) = . The state of the memory is σ M = Tr A ( σ in,M ) = α | λ (cid:48) (cid:105) (cid:104) λ (cid:48) | + β | λ (cid:48) (cid:105) (cid:104) λ (cid:48) | . Let, q xy = (cid:104) x | λ y (cid:105) where x ∈ { , , + , −} and y ∈ { , } . The bipartiteensemble, created by the S z measurement of Alice, is { p (cid:48) i , σ iAM } where, p (cid:48) i = Tr [( | i (cid:105) (cid:104) i | ⊗ I ) σ in,M ] = p [ α (cid:48) | q i | + β (cid:48) | q i | ] + − p and σ iAM = ( | i (cid:105)(cid:104) i |⊗ I ) σ in,M ( | i (cid:105)(cid:104) i |⊗ I ) Tr [( | i (cid:105)(cid:104) i |⊗ I ) σ in,M ] and i ∈ { , } . Now it can be easily checked that σ iAM = | i (cid:105) (cid:104) i |⊗ [ p | φ (cid:48) i (cid:105) (cid:104) φ (cid:48) i | + − p p (cid:48) i I ] where | φ i (cid:105) = √ p i ( √ α (cid:48) q i | λ (cid:105) + √ β (cid:48) q i | λ (cid:105) ) . The post-measurement average bipartitestate is σ AM = (cid:80) i p (cid:48) i σ iAM . Clearly, σ A = Tr M σ AM = p (cid:80) i p (cid:48) i | i (cid:105) (cid:104) i | + (1 − p ) I . After, Eve’s S x measurement on A part, the average bipartite state will become σ A (cid:48) M = I ⊗ σ M where σ M = [ p (cid:80) i p i | φ (cid:48) i (cid:105) (cid:104) φ (cid:48) i | + (1 − p ) I ] = σ M and the average state of A part becomes σ A (cid:48) = I . So, thereduction in information leak is given as I ( C ) − I M ( C )= [ S ( σ A ) − S ( σ AM )] − [ S ( σ (cid:48) A ) − S ( σ A (cid:48) M )]= [ S ( σ A ) − S ( σ AM )] − [ S ( σ (cid:48) A ) − S ( σ A (cid:48) ) − S ( σ M )]= S ( σ M ) + S ( σ A ) − S ( σ AM ) = I ( A : M ) σ AM . (33) Now, consider a special case where | λ (cid:105) , | λ (cid:105) are theeigen basis of σ y , α = and β = . In this case, | q ij | = ∀ i ∈ { , } and ∀ j ∈ { , } . Also, fromequation (31) we get α (cid:48) = p − p . Clearly, α (cid:48) ≥ only for p ≥ . We plot the leakage difference I ( C ) − I M ( C ) with respect to p in Fig. (1) . To quantify the amount ConcurrenceLeakage DifferenceMutual Information p FIG. 1. (Colour Online) Plot of Concurrence and mutual in-formation of σ in,M and the leakage difference vs the parameter p . It can be seen that Concurrence and mutual information of σ in,M and the leakage difference is monotonically increasingwith respect to the parameter p . of memory we use the concurrence measure(ref. [24])and the mutual information of the initial bipartite state σ in,M . From Fig. (1) we get that with increment of p ,concurrence and mutual information of σ in,M and theleakage difference, are monotonically increasing with p .We can also say that the information leakage difference isa monotonically increasing function of both concurrenceand mutual information in the state σ in,M . Equivalently,we can say that the leakage with memory is monotonicallydecreasing with increasing value concurrence and mutualinformation. It can be observed from Fig. (1) , the leak-age difference is non-zero for the region p (cid:47) . wherethe concurrence is vanishing. In this region the non-zeroleakage difference can be attributed to the non-vanishingmutual information. The example (1) suggests us to write down the follow-ing conjecture:
Conjecture 1.1.
With increment of correlation betweenthe memory and the input state, information leakagemonotonically decreases.
Therefore, we conclude based on the validity of theconjecture, that the presence of more memory correlationhelps in reducing the leakage.
V. CONCLUSION
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