QQUANTUM INFORMATION GEOMETRY IN THESPACE OF MEASUREMENTS
WARNER A. MILLER
Abstract.
We introduce a new approach to evaluating entangled quan-tum networks using information geometry. Quantum computing is pow-erful because of the enhanced correlations from quantum entanglement.For example, larger entangled networks can enhance quantum key dis-tribution (QKD). Each network we examine is an n -photon quantumstate with a degree of entanglement. We analyze such a state within thespace of measured data from repeated experiments made by n observersover a set of identically-prepared quantum states – a quantum state in-terrogation in the space of measurements. Each observer records a 1 iftheir detector triggers, otherwise they record a 0. This generates a stringof 1’s and 0’s at each detector, and each observer can define a binaryrandom variable from this sequence. We use a well-known informationgeometry-based measure of distance that applies to these binary stringsof measurement outcomes [9, 7, 14], and we introduce a generalization ofthis length to area, volume and higher-dimensional volumes [2]. Thesegeometric equations are defined using the familiar Shannon expressionfor joint and mutual entropy [11]. We apply our approach to three dis-tinct tripartite quantum states: the | GHZ (cid:105) state, the | W (cid:105) state, and aseparable state | P (cid:105) . We generalize a well-known information geometryanalysis of a bipartite state to a tripartite state. This approach providesa novel way to characterize quantum states, and it may have favorablescaling with increased number of photons. INTRODUCTION “No elementary quantum phenomenon is a phenomenon until it is broughtto close by an irreversible act of amplification.” This Niels Bohr-inspiredquantum adage of John Archibald Wheeler, together with the Principle ofComplementarity, is at the very heart of Wheeler’s It-from-Bit framework[13]. In this manuscript, we explore entanglement networks within thisinformation-centric approach. The quantum network we consider in thismanuscript is a quantum state with n photons with varying degrees of en-tanglement. Observers examine the space of measured data from repeatedexperiments on a set of identically-prepared quantum states. Each observerrecords a 1 if their detector triggers, otherwise a ”0” is recorded. This gen-erates a string of 1’s and 0’s at each detector as illustrated in Fig. 1. Thestring of numbers can be represented by a binary random variable. Theobservers may have more than one detector, and therefore each observermay acquire more than one binary random variable. Once these random a r X i v : . [ qu a n t - ph ] D ec WARNER A. MILLER variables are formed, we can apply an information geometry measure of dis-tance, area, volume and n-volumes to the network of observers [9, 7, 2, 14].These measures are defined using the familiar Shannon expression for mutualand conditional entropy [11]. This will be discussed in Sec. 2. In Sec. 4, weapply our approach to three distinct tripartite quantum states: the | GHZ (cid:105) state, the | W (cid:105) state, and a separable state | P (cid:105) . This novel approach pro-vides us with a natural generalization an information geometry model ofBell’s inequality for a bipartite singlet state to a similar analysis tripartitestates [10]. We provide a brief review of this 2-photon geometry in Sec. 3and its generalization in Sec. 5.The power of quantum computing stems from a network of quantumentanglement, and larger networks can enhance quantum key distribution(QKD)[5, 6, 8]. Consequently, a primary focus of this field is to find ascalable method to characterize a quantum state, e.g. a measure of its en-tanglement, and entanglement quality. Such measures have been elusive[8, 4]. Quantum state tomography is impractical as it involves analyzing anexponentially large matrix. This difficulty is not so surprising as the verypower of quantum computing lies with this exponential scaling. We seeka scalable information geometry entanglement measure that is groundedsolely upon the space of measured data from repeated experiments — aquantum state interrogation in the space of measurements. This It-from-Bitapproach is based on a projection of the quantum world by measurement,(i.e. an irreversible act of amplification) onto a classical world of bit stringsof data (bits). After all, any quantum information processing system must“meet” the classical world of information to communicate its informationcontent. It is this network of detectors that we analyze within the “It-from-Bit” framework. The usual conceptual ambiguities that may accompany thequantum measurement process are minimized within this approach; never-theless, the non-classical and non-intuitive features of the quantum remain.The uniqueness of quantum phenomenon are now encoded in the correlationsof our observers bits. The two questions we ask are: “ Can information ge-ometry provide us with a better understanding of the entanglement structureand function in quantum networks? ”; and “
Can this information geometryapproach scale favorably with larger multipartite systems? ”.Scalability is the single salient sign signaling a superior quantum strat-egy. Quantum information processing is driven by exponential scaling, andthis must be a concern for any approach to measure or characterize entan-glement. For this reason, our approach will try to emulate the exponentialscaling suggested by Vedral [12]. The scalability of this approach throughthe Quantum Sanov’s Theorem [12] shows that the fidelity of distinguishingtwo quantum density matrices ρ from ρ improves exponentially with thenumber of measurements, N ,(1) (cid:18) F idelity of ρ → ρ with N measurements (cid:19) = 1 − e − NS ( ρ || ρ ) . UANTUM GEOMETRY OF MEASUREMENTS 3
Figure 1.
An illustration of a simplicial geometry represen-tation of a quantum tripartite state consisting of three pho-tons.
The geometry of the triangle emerges from a space ofmeasurements that are parameterized by the six parametersof the detectors of our three observers. The parameters arethe two angles on Bloch sphere characterizing each detector.In this manuscript, and for clarity of presentation only, werestrict our three observers, Alice ( A ), Bob ( B ) and Charlie( C ), to measurements made within the subspace linearly po-larized light (the equator of the Bloch sphere). This space ofmeasurement can be spanned by three angular parameters,one for each of the three detectors, α , β and γ . With thechoice of the detectors made, the three observers record a 1if their detector gets triggered otherwise they record a 0. Weanalyze the entropy of each of these binary strings as wellas calculate the mutual and conditional entropy of pairs andtriplets of the detectors. The geometric quantities are func-tions of these entropies. Therefore, the information geometryof this triangle ABC is determined by; (1) the quantum state | Ψ (cid:105) , and (2) the three angles, α , β and γ defining orientationof Alice, Bob and Charlie’s detectors; respectively. We cancalculate the length of each edge, and the area of the triangle.This will be discussed in Sec. 2. WARNER A. MILLER
Here, S ( ρ || ρ ) = T r ( ρ log ρ − ρ log ρ ) is a relative entropy. We areanimated by this “information thermodynamic” structure, although we arefar from making progress on this front. We outline here an approach to theanswers to our two questions that is based on quantum state interrogationin a space of measurements. We introduce potential measures that utilizequantum information geometry, and look forward in the future to verifyif our “large N ” thermodynamic-like collection of bits from measurementsshare the structure of the exponential scaling suggested by Vedral [12].2. Outline of Information Geometry: From Distances to Areaand Volumes
Information geometry is defined through the entropy of a network of ran-dom variables. For example if we examine the binary outcomes of Alice’sdetector in Fig. 1, we can obtain the probability distribution for Alice’s( A ) detector labeled α . In particular, we separately summing the numberof times the detector fires and gives a 1, and the number non-detections 0that she measured. We then divide each of these by the total number ofmeasurements. In this way we can assign a binary random variable, A , toAlice’s 19 measurements (Fig. 1).(2) A = (cid:26) with probability / with probability / . Whereas probability measures uncertainty about the occurrence of a singleevent, entropy provides a measure the uncertainty of a collection of events.If X i is a s -state random variable, then(3) (cid:18) Entropyof X i (cid:19) = H X i := − s (cid:88) χ i =1 p ( x i ) log p ( x i ) . Here p ( x i ) = p ( X = x i ) is the probability that the random variable has thevalue x i . In this manuscript, we use only binary random variables ( s = 2).The entropy is the largest when our uncertainty of the value of the randomvariable is complete (e.g. uniform distribution of probabilities), and theentropy is zero if the random variable always takes on the same value,(4) 0 ≤ H X i ≤ log ( s ) . In this sense, entropy is a measure of our ignorance. We will make use ofthe mutual entropy and conditional entropy over an ensemble of randomvariables. The mutual entropy is defined over the joint probability distribu-tions,(5) H ABC = − (cid:88) i,j,k p ( a i , b j , c k ) log p ( a i , b j , c k ) , UANTUM GEOMETRY OF MEASUREMENTS 5 and the conditional entropy is defined over conditional probability distribu-tions, H A | B = − (cid:88) i,j p ( b j ) log p ( a i | b j ) , (6) H A | BC = − (cid:88) i,j,k p ( b j , c k ) log p ( a i | b j , c k ) . (7)Here the probability that A = a i , B = b j and C = c k is the joint probability p ( a i , b j , c k ), and the probability that A = a i given that we know a priorithat B = b j and C = c k is the conditional probability p ( a i | b j , c k ), and theseare related by(8) p ( a i | b j c k ) = p ( a i , b j , c k ) p ( b j , c k ) . We use use an extension of the Shannon-based information distance definedby Rokhlin[9] and Rajski[7],(9) (cid:18)
Length ofEdge AB (cid:19) = D AB := H A | B + H B | A = 2 H AB − H A − H B , to construct a geometric triangle from these measurement outcomes. Here inEq. 9, A and B are binary random variables derived from a joint probabilitydistribution p ( A = a i , B = b i ) = p ( a i , b i ) with i ∈ { i, s } . This informationdistance has desirable properties.(1) It is constructed to be symmetric, D AB = D AB .(2) It obeys the triangle inequality, D AB ≤ D Ac + D CB .(3) It is nonnegative, D AB ≤
0, and equal to 0 when A “=” B .Furthermore, if A and B are uncorrelated to each other then,(10) D AB = 2 ( H A + H B ) − H A − H B = H A + H B , and D AB is bounded,(11) 0 ≤ D AB ≤ H A + H B ≤ s. In addition to the three edge lengths of the triangle in Fig. 1, we canassign an information area that we developed earlier with Caves, Kheyfets,Lloyd, Miller, Schumacher and Wotters [2], A ABC := H A | BC H B | CA + H B | CA H C | AB + H C | AB H A | BC = 3 H ABC − H AB + H BC + H AC ) H ABC + ( H AC H BC + H AB H AC + H AB H BC ) . (12)This can be generalized to higher-dimensional simplexes, e.g. the informa-tion volume for a tetrahedron can be defined as,(13) V ABCD := H A | BCD H B | CDA H C | DAB + H B | CDA H C | DAB H D | ABC + H C | DAB H D | ABC H B | CDA + H D | ABC H A | BCD H B | CDA . WARNER A. MILLER
For classical probability distributions these formulas are well defined andhave all of the requisite symmetries, positivity, bounds and structure usuallyrequired for such formulae. In particular,(14) 0 ≤ D AB ≤ H A + H B ≤ s (cid:124) (cid:123)(cid:122) (cid:125) s -state r.v’s , and(15) 0 ≤ A ABC ≤ s ) , with their minimum values taken when the random variables are completelycorrelated, and their maximum values obtained when the random variablesare completely uncorrelated. We have shown that for classical probabilitydistributions these formulas are well defined and have all requisite sym-metries, positivity, bounds and structure required for such formulae [2].However, for probability distributions based on measurements of quantumsystems these assumptions must be weakened, and triangle inequalities maybe violated. We will outline such an example in Sec. 3.We are confident that these novel formulae, Eqns. 12-13, can provide anew characterization of quantum states and their degree of entanglement.We may need far fewer measurements than one would think to distinguishthese states. In particular, the Quantum Sanov’s Theorem [12] shows thatthe fidelity of distinguishing two quantum states ρ from ρ improves ex-ponentially with the number of measurements, N as seen in Eq. 1. This“information thermodynamic” feature may lay credence our scalability as-sumption; however this requires further investigation.3. Quantum State Interrogation of the Singlet State: AReview
For a bipartite quantum system we can explore the information geometrythrough the distance formula given in Eq. 9. One can look for a relation-ship between the entanglement and its geometry. Based on this approachSchumacher examined the relationship between the violation of the Bellinequality for a singlet state and the triangle inequality in information ge-ometry [10]. This is illustrated in Fig. 2. Here we review his results in detailas this is the simplest non-trivial application of this formalism. We providemany identical copies of a singlet state,(16) | S (cid:105) = 1 √ | (cid:108)(cid:108)(cid:105) + | ↔↔(cid:105) ) , and two observers Alice and Bob as shown in Fig. 2. Alice receives thephoton propagating to the left, and Bob receives the photon traveling tothe right. Alice choses randomly one of two detectors. Alice’s first detector, (cid:104) α | , is a linear polarizer rotated clockwise from the vertical state | (cid:108)(cid:105) byan angle α , and her second detector is rotated by an angle α . Similarly,Bob’s first and second detectors are rotated by β and β ; respectively. We UANTUM GEOMETRY OF MEASUREMENTS 7
Figure 2.
We illustrate here the information geometry ofa singlet state analyzed by Schumacher [10]. There are twoobservers, Alice and Bob, that are detecting the 2-photonsfrom the singlet state | S (cid:105) . Alice has two detectors, one linearpolarizer rotated an angle α away from vertical, the otherdetector is rotated an angle α , and similarly for Bob. Anensemble of singlet states are prepared and Alice and Bobrandomly choose one or the other detector. This leads aftermany measurements to four binary random variables, A , A , B and B . in the bottom of the figure, we show thequadrilateral formed by these for random variables. We canuse Eq. 9 to calculate the four distances D ’s shown on theedges. We cannot connect the diagonals as they are mutuallyexclusive; therefore, we can not define an information area.perform this calculation for arbitrary angles and for a symmetric photonsinglet state. Schumacher considered an anti-symmetric spin 1 / α = 0, α = π/ β = π/ β = 3 π/ A , A , B and B , andthey are all equally likely.(17) p ( A = 0) = p ( A = 1) = p ( A = 0) = p ( A = 1) = p ( B = 0) = p ( B = 1) = p ( B = 0) = p ( B = 1) = 12 . We then calculate two consecutive local measurements on each pair of detec-tors in order to determine the four sets of conditional probabilities ( A - B , WARNER A. MILLER A - B , A - B and A - B ) In particular for the two detectors A and B ,(18) p ( A = 0 | B = 0) = cos ( β − α ) , p ( A = 1 | B = 0) = sin ( β − α ) ,p ( A = 0 | B = 1) = sin ( β − α ) , p ( A = 1 | B = 1) = cos ( β − α ) ,p ( B = 0 | A = 0) = cos ( β − α ) , p ( B = 1 | A = 0) = sin ( β − α ) ,p ( B = 0 | A = 1) = sin ( β − α ) , p ( B = 1 | A = 1) = cos ( β − α ) . The conditional probability expressions for A and B are the same as thosein Eq. 18 except we must substitute β → β . For A and B we modifyEq. 18 with α → α
2, and for A and B we substitute both angles, i.e. α → α β → β . The joint probabilities can be recovered fromEqs. 17-18 using Eq. 37. In this example, each joint probability is just halfits conditional probability.We are now in a position to use Eq. 3 and Eq. 6 to calculate the entropy,the conditional entropy as well as the information distance in Eq. 9. We findthat the entropies are maximal and consistent with the complete uncertaintyin the outcome of each measurement by A or B . This is reflected in Eq. 17)where(19) H A = H B = −
12 log 12 −
12 log 12 = 1 . The joint entropies are more interesting, and can be obtained from Eq. 18and Eq. 37, H A B = 1 − sin ( β − α ) log (cid:0) sin ( β − α ) (cid:1) − cos ( β − α ) log (cid:0) cos ( β − α ) (cid:1) , (20) H A B = 1 − sin ( β − α ) log (cid:0) sin ( β − α ) (cid:1) − cos ( β − α ) log (cid:0) cos ( β − α ) (cid:1) , (21) H A B = 1 − sin ( β − α ) log (cid:0) sin ( β − α ) (cid:1) − cos ( β − α ) log (cid:0) cos ( β − α ) (cid:1) , (22) H A B = 1 − sin ( β − α ) log (cid:0) sin ( β − α ) (cid:1) − cos ( β − α ) log (cid:0) cos ( β − α ) (cid:1) . (23)We find the four lengths of the quadrilateral in the lower part of Fig. 2 usingEqs. 10, 20 and 19, D A B = − ( β − α ) log (cid:0) sin ( β − α ) (cid:1) − ( β − α ) log (cid:0) cos ( β − α ) (cid:1) , D A B = − ( β − α ) log (cid:0) sin ( β − α ) (cid:1) − ( β − α ) log (cid:0) cos ( β − α ) (cid:1) , D A B = − ( β − α ) log (cid:0) sin ( β − α ) (cid:1) − ( β − α ) log (cid:0) cos ( β − α ) (cid:1) , D A B = − ( β − α ) log (cid:0) sin ( β − α ) (cid:1) − ( β − α ) log (cid:0) cos ( β − α ) (cid:1) . If the quadrilateral formed by the four detectors as illustrated in Fig. 2was embedded in a Euclidean surface, then the direct route A → B shouldalways be greater than or equal to the indirect route A → B → A → B ,(24) D A B ≤ D A B + D A B + D A B . UANTUM GEOMETRY OF MEASUREMENTS 9
However, Schumacher showed that this triangle inequality is violated forcertain angles [10]. For our symmetric 2-photon singlet state we obtainthe same violation. In particular, we find a maximal violation within asymmetric sub-space where three of the pairwise detectors have the samedifference in their relative angular settings, whilst the relative angular settingbetween the direct connection between A and B is three times larger,(25) β − α = β = α = β − α ≈ . , and therefore,(26) β − α = 3( β − α ) . This yields a violation in the triangle inequality which Schumacher suggestsis an information geometry realization of the violation of the Bell Inequalityfor the maximally entangled singlet state | S (cid:105) . Here,(27) D A B (cid:124) (cid:123)(cid:122) (cid:125) . (cid:54)≤ D A B + D A B + D A B (cid:124) (cid:123)(cid:122) (cid:125) . . While we can further explore this bipartite example, it will be reported in ourfuture work. Nevertheless, this bipartite example of information geometrymotivates us to begin to explore tripartite states. We will outline an analysisof two entangled tripartite states and a seperable state in the next section.4.
Extending from Bipartite to Tripartite Quantum Networks
In this section we examine the information geometry of a tripartite statefunction — a generalization of the bipartite work of Schumacher that wasdiscussed in the previous section [10]. We will focus on the informationgeometry for three distinct states, one seperable quantum state and twowell studied entangled states. In particular, we examine the following threestates:(1) | ψ (cid:105) = | GHZ (cid:105) = √ ( | (cid:108)(cid:108)(cid:108)(cid:105) + | ↔↔↔(cid:105) );(2) | ψ (cid:105) = | W (cid:105) = √ ( | (cid:108)(cid:108)↔(cid:105) + | (cid:108)↔(cid:108)(cid:105) + | ↔(cid:108)(cid:108)(cid:105) ); and(3) | ψ (cid:105) = | P (cid:105) = | (cid:108)(cid:108)(cid:108)(cid:105) .In the next three subsections, Sec. 4.1-4.3, we examine the geometry of Fig. 1for each of these states. Once again, we will restrict ourselves to only linearpolarization measurements on the equator of the Bloch sphere. In Sec. 5, wedescribe a octagonal network for our tripartite system that is the analogueof the quadrilateral in Fig 2 for the bipartite system of Schumacher [10].4.1. Quantum State Interrogation: the | GHZ (cid:105)
State.
We analyze theinformation geometry of the triangle shown in Fig: 1 for the Greenberger,Horne & Zeilinger (GHZ) tripartite state,(28) | Ψ (cid:105) = ⇒ | GHZ (cid:105) = 1 √ | (cid:108)(cid:108)(cid:108)(cid:105) + | ↔↔↔(cid:105) ) We will calculate the three edge lengths using the techniques introducedin Sec. 2 and applied in Sec: 3. We will also calculate the information areaEq. 12 for this triangle. This is something we could not do with the bipartitesystem in Sec. 3.We consider three observers Alice ( A ), Bob ( B ) and Charlie ( C ) as shownin Fig. 3. A , B and C measure the GHZ state using their choice of detectors, M A = (cos ( α ) σ z + sin ( α ) σ x ) ⊗ I ⊗ I (29) M B = I ⊗ (cos ( β ) σ z + sin ( β ) σ x ) ⊗ I (30) M C = I ⊗ I ⊗ (cos ( γ ) σ z + sin ( γ ) σ x ) ;(31)respectively. Here the σ ’s are the usual Pauli matrices. The probability of A measuring a photon is(32) p ( A ) = tr (cid:16) M † A M A ρ GHZ (cid:17) , where ρ GHZ = | GHZ (cid:105)(cid:104)
GHZ | is the density matrix for the GHZ state. If theinitial state was | GHZ (cid:105) then after A ’s measurement the state would be leftin(33) | GHZ A (cid:105) = M A | GHZ (cid:105) (cid:113) (cid:104)
GHZ | M † A M A | GHZ (cid:105) . For the remainder of this section we will set α = 0.The eight joint probabilities from the three measurements on this entan-gled state M C M B M A | GHZ (cid:105) are: p ( A = 1 , B = 1 , C = 1) = cos ( β ) cos ( γ ) , p ( A = 0 , B = 1 , C = 1) = sin ( β ) sin ( γ ) ,p ( A = 1 , B = 1 , C = 0) = cos ( β ) sin ( γ ) , p ( A = 0 , B = 1 , C = 0) = sin ( β ) cos ( γ ) ,p ( A = 1 , B = 0 , C = 1) = sin ( β ) cos ( γ ) , p ( A = 0 , B = 0 , C = 1) = cos ( β ) sin ( γ ) ,p ( A = 1 , B = 0 , C = 0) = sin ( β ) sin ( γ ) , p ( A = 0 , B = 0 , C = 0) = cos ( β ) cos ( γ )(34)Tracing these joint probability over each observer yields the pairwise jointprobabilities, p ( A = 1 , B = 1) = cos ( β ) ,p ( A = 1 , B = 0) = sin ( β ) ,p ( A = 1 , C = 1) = cos ( γ ) ,p ( A = 1 , C = 0) = sin ( γ ) ,p ( A = 0 , B = 1) = sin ( β ) ,p ( A = 0 , B = 0) = cos ( β ) ,p ( A = 0 , C = 1) = sin ( γ ) ,p ( A = 0 , C = 0) = cos ( γ ) ,p ( B = 1 , C = 1) = (cid:0) cos ( β ) cos ( γ ) + sin ( β ) sin ( γ ) (cid:1) ,p ( B = 1 , C = 0) = (cid:0) cos ( β ) sin ( γ ) + sin ( β ) cos ( γ ) (cid:1) ,p ( B = 0 , C = 1) = (cid:0) sin ( β ) cos ( γ ) + cos ( β ) sin ( γ ) (cid:1) ,p ( B = 0 , C = 0) = (cid:0) cos ( β ) cos ( γ ) + sin ( β ) sin ( γ ) (cid:1) . (35) UANTUM GEOMETRY OF MEASUREMENTS 11
Finally, tracing the joint probability over all pairs of observers gives us thesix probabilities for the measurement outcomes of A , B and C to be(36) p ( A = 0) = 1 / , p ( A = 1) = 1 / p ( B = 0) = 1 / , p ( B = 1) = 1 / p ( C = 0) = 1 / , p ( C = 1) = 1 / . The pairwise conditional probabilities can be recovered from these pairwisejoint probabilities since(37) p ( A = i | B = j ) = p ( A = i, B = j ) p ( B = j ) . However, since p ( A = i ) = p ( B = i ) = P ( C = i ) = 1 / ∀ i ∈ { , } then the joint probabilities for the | GHZ (cid:105) state are just half the conditionalprobabilities.We are now in a position to use Eqs. 3-6 to calculate the entropy, theconditional entropy as well as the information distance in Eq. 9. The entropyof our observers are maximal, H A = 1 , (38) H B = 1 , (39) H C = 1 , (40)and the joint entropy between pairs of our observers are,(41) H AB = 1 − sin ( β ) log(sin ( β )) − cos ( β ) log(cos ( β )) ,H AC = 1 − sin ( γ ) log(sin ( γ )) − cos ( γ ) log(cos ( γ )) ,H BC = 1 − (cid:0) cos ( β ) cos ( γ ) + sin ( β ) sin ( γ ) (cid:1) log (cid:0) cos ( β ) cos ( γ ) + sin ( β ) sin ( γ ) (cid:1) − (cid:0) sin ( β ) cos ( γ ) + cos ( β ) sin ( γ ) (cid:1) log( (cid:0) sin ( β ) cos ( γ ) + cos ( β ) sin ( γ ) (cid:1) ) . Finally, we use Eq. 37 to find the joint entropy H ABC of A , B and C ,(42) H ABC = 1 − cos ( β ) cos ( γ ) log(cos ( β ) cos ( γ )) − cos ( β ) sin ( γ ) log(cos ( β ) sin ( γ )) − sin ( β ) cos ( γ ) log(sin ( β ) cos ( γ )) − sin ( β ) sin ( γ ) log(sin ( β ) sin ( γ )) . The three lengths of the edges of the triangle for this GHZ state arederived from Eq. 9, and are D AB = − ( β ) log (cid:0) sin ( β ) (cid:1) − ( β ) log (cid:0) cos ( β ) (cid:1) , (43) D AC = − ( γ ) log (cid:0) sin ( γ ) (cid:1) − ( γ ) log (cid:0) cos ( γ ) (cid:1) , (44) D BC = − (cid:0) cos ( β ) cos ( γ ) + sin ( β ) sin ( γ ) (cid:1) log (cid:0) cos ( β ) cos ( γ ) + sin ( β ) sin ( γ ) (cid:1) (45) − (cid:0) sin ( β ) cos ( γ ) + cos ( β ) sin ( γ ) (cid:1) log( (cid:0) sin ( β ) cos ( γ ) + cos ( β ) sin ( γ ) (cid:1) ) . (46)To determine the area of the triangle formed by A , B and C , we use thedefinition in Eq. 12. We have all the entropies and the conditional entropieswe need for this calculation by using the chain rule for multiple random Figure 3.
The information geometry triangle formed byour three observers, A , B , and C . These observers share thephotons from a | GHZ (cid:105) state shown in red. Each of the threeblack edges can be assigned an information length (we show D AB ), and the resulting triangle can be assigned an area, A | GHZ (cid:105)
ABC . variables,(47) H ABC = H A + H B | A (cid:124) (cid:123)(cid:122) (cid:125) H AB + H C | AB to solve for H C | AB . The information triangle area A | GHZ (cid:105)
ABC can be obtainedby Eq, 12 using our expressions for the joint entropies. Since the informationarea is an involved function of the two detector angles β and γ , we will notdisplay this explicitly. However we evaluate it numerically, and show ourresults in Fig. 3. It is interesting to us that this entangled state, there is arelatively large region where the Euclidean area is close to the informationarea. The information area is well behaved with a local maximum at β = γ = π/
4. At this particular maximum, the geometry of the triangle is andisosceles triangle and is embeddable in the Euclidean plane. Its embeddedarea (3) achieves the upper bound for the area formula in Eq. 15 and is
UANTUM GEOMETRY OF MEASUREMENTS 13 different from the Euclidean area ( ∼ . D AB = D AC = D BC = 2 , (48) A | GHZ (cid:105)
ABC = 3 , (49)as illustrated in Fig. 4. Figure 4.
The left plot is the information area, A | GHZ (cid:105)
ABC ( β, γ ), the center plot is the Euclidean area ( A E )based on the three information distances in Eq. 43, and theright plot is the ratio of these two areas. This ratio of areasform a plateau with a concave center. There is no violationof the triangle inequality. The plots range is β, γ ∈ { , π/ } .The Euclidean area plotted in the middle box of Fig. 4 is defined usingHeron’s formulae,(50) A E = 14 (cid:112) ( D AB + D AC − D BC )( D AB − D AC + D BC )( −D AB + D AC + D BC )( D AB + D AC + D BC ) . The missing sections of the domain is where the triangle inequality is vi-olated. Eq. 50 is ideally suited to detect triangle inequality violations asthe radical becomes imaginary. Perhaps, this is not so surprising since thethree-tangle obtains its maximum permitted value of unity for the | GHZ (cid:105) state [3]. The tangle is the square of the concurrence. We will look at the | W (cid:105) state in the next section whose three-tangle is <
1, but whose pairwise-tangle is maximal and greater than the pairwise tangle for the | GHZ (cid:105) state[1].4.2.
Quantum State Interrogation: the | W (cid:105) State.
Following the lasttwo subsections, we briefly outline the information geometry of the triangleshown in Fig: 1 for the | W (cid:105) state,(51) | Ψ (cid:105) = ⇒ | W (cid:105) = 1 √ | (cid:108)↔↔(cid:105) + | ↔(cid:108)↔(cid:105) + | ↔↔(cid:108)(cid:105) ) . We will calculate the three edge lengths using the techniques introduced inSec. 2 and applied in Sec: 3. We will also calculate the information areaEq. 12 for this triangle. For the rest of this section we set α = 0.Again we consider the same three observers Alice ( A ), Bob ( B ) and Char-lie ( C ). A , B and C measure the | W (cid:105) state with measurement operatorsgiven in Eq. 29. We also set α = 0 for the remainder of the section.The eight joint probabilities from the three measurements on this entan-gled state M C M B M A | W (cid:105) are: p ( A = 1 , B = 1 , C = 1) = sin ( β ) sin ( γ ) , p ( A = 0 , B = 1 , C = 1) = sin ( β + γ ) ,p ( A = 1 , B = 1 , C = 0) = sin ( β ) cos ( γ ) , p ( A = 0 , B = 1 , C = 0) = cos ( β + γ ) ,p ( A = 1 , B = 0 , C = 1) = cos ( β ) sin ( γ ) , p ( A = 0 , B = 0 , C = 1) = cos ( β + γ ) ,p ( A = 1 , B = 0 , C = 0) = cos ( β ) cos ( γ ) , p ( A = 0 , B = 0 , C = 0) = sin ( β + γ ) . (52)Tracing these joint probability over each observer yields the pairwise jointprobabilities, p ( A = 1 , B = 1) = sin ( β ) ,p ( A = 1 , B = 0) = cos ( β ) ,p ( A = 1 , C = 1) = sin ( γ ) ,p ( A = 1 , C = 0) = cos ( γ ) ,p ( A = 0 , B = 1) = ,p ( A = 0 , B = 0) = ,p ( A = 0 , C = 1) = ,p ( A = 0 , C = 0) = ,p ( B = 1 , C = 1) = (cid:0) sin ( β ) sin ( γ ) + sin ( β + γ ) (cid:1) ,p ( B = 1 , C = 0) = (cid:0) sin ( β ) cos ( γ ) + cos ( β + γ ) (cid:1) ,p ( B = 0 , C = 1) = (cid:0) cos ( β ) sin ( γ ) + cos ( β + γ ) (cid:1) ,p ( B = 0 , C = 0) = (cid:0) cos ( β ) cos ( γ ) + sin ( β + γ ) (cid:1) . (53)Finally, tracing the joint probability over all pairs of observers gives us thesix probabilities for the measurement outcomes of A , B and C ,(54) p ( A = 0) = 2 / , p ( A = 1) = 1 / p ( B = 0) = 2 / , p ( B = 1) = 1 / p ( C = 0) = 2 / , p ( C = 1) = 1 / . The pairwise conditional probabilities can be recovered from these pair-wise joint probabilities since using Eq 37.We are now in a position to use Eqs. 3-6 to calculate the entropy, theconditional entropy as well as the information distance in Eq. 9. The entropyof our observers are all equal,(55) H A = H B = H C = log (3) − . UANTUM GEOMETRY OF MEASUREMENTS 15
The joint entropy between pairs of our observers are,(56) H AB = log(3) − sin ( β ) log(sin ( β )) − cos ( β ) log(cos ( β )) ,H AC = log(3) − sin ( γ ) log(sin ( γ )) − cos ( γ ) log(cos ( γ )) ,H BC = log(3) − (cid:0) sin ( β ) sin ( γ ) + sin ( β + γ ) (cid:1) log (cid:0) sin ( β ) sin ( γ ) + sin ( β + γ ) (cid:1) − (cid:0) sin ( β ) cos ( γ ) + cos ( β + γ ) (cid:1) log (cid:0) sin ( β ) cos ( γ ) + cos ( β + γ ) (cid:1) − (cid:0) cos ( β ) sin ( γ ) + cos ( β + γ ) (cid:1) log (cid:0) cos ( β ) sin ( γ ) + cos ( β + γ ) (cid:1) − (cid:0) cos ( β ) cos ( γ ) + sin ( β + γ ) (cid:1) log (cid:0) cos ( β ) cos ( γ ) + sin ( β + γ ) (cid:1) . Finally, we use Eq. 37 to find the joint entropy H ABC of A , B and C (57) H ABC = log(3) − sin ( β ) sin ( γ ) log(sin ( β ) sin ( γ )) − sin ( β ) cos ( γ ) log(sin ( β ) cos ( γ )) − cos ( β ) sin ( γ ) log(cos ( β ) sin ) − cos ( β ) cos ( γ ) log(cos ( β ) cos ( γ )) − sin ( β + γ ) log(sin ( β + γ )) − cos ( β + γ ) log(cos ( β + γ )) − cos ( β + γ ) log(cos ( β + γ )) − sin ( β + γ ) log(sin ( β + γ )) . The three lengths of the edges of the triangle for this | W (cid:105) state are derivedfrom Eq. 9, and are(58) D AB = − sin ( β ) log (cid:0) sin ( β ) (cid:1) − cos ( β ) log (cid:0) cos ( β ) (cid:1) , D AC = − sin ( γ ) log (cid:0) sin ( γ ) (cid:1) − cos ( γ ) log (cid:0) cos ( γ ) (cid:1) , D BC = − sin ( β ) sin ( γ ) log(sin ( β ) sin ) − sin ( β ) cos ( γ ) log(sin ( β ) cos ( γ )) − cos ( β ) sin ( γ ) log(cos ( β ) sin ) − cos ( β ) cos ( γ ) log(cos ( β ) cos ( γ )) − sin ( β + γ ) log(sin ( β + γ )) − cos ( β + γ ) log(cos ( β + γ )) − cos ( β + γ ) log(cos ( β + γ )) − sin ( β + γ ) log(sin ( β + γ )) . To determine the area of the triangle formed by A , B and C we use thedefinition in Eq. 12. We have all the entropies, conditional entropies, andchain rule for multiple random variables in Eq. 47 to solve for H C | AB , etc. .The information triangle area A | W (cid:105) ABC can be obtained by Eq, 12 using ourexpressions for the joint entropies. As it is an involved function of the twodetector angles β and γ , we will not display this explicitly. However weevaluate it numerically and illustrate our results in Fig. 5. It is interestingthat this entangled state is significantly different than the | GHZ (cid:105) state. Ithas a saddle point rather than a local minimum. The information area iswell behaved with a local saddle point at α = β = π/
4. The Euclideanarea based on Eq. 50 is well defined over the entire range indicating thatthe information triangles never violate the triangle inequality. At the saddlepoint in the geometry, the triangle is embeddable in the Euclidean planeand its embedded area (1 . D AB = D AC = 2 , (59) D BC = 1 . , and (60) A | W (cid:105) ABC = 0 . . (61) as illustrated in Fig. 5. It would be interesting to see if this is related Figure 5.
The left plot is the information area A | W (cid:105) ABC ( β, γ ),the center plot is the Euclidean area ( A E ) based on the threeinformation distances in Eq. 58, and the right plot is theratio, where we see a diagonal ridge line. The plots range is β, γ ∈ { , π/ } .to the large two-way tangle for this state. We can conclude two thingsthus far, (1) triangle inequality violations do not appear here, and (2) theinformation area of the W (cid:105) state is qualitatively different from the | GHZ (cid:105) state. Obviously, more needs to be done to measure entanglement.4.3.
Quantum State Interrogation: a Separable State | P (cid:105) . Followingthe last two subsections, we briefly outline the information geometry of thetriangle shown in Fig: 1 for the separable state,(62) | Ψ (cid:105) = ⇒ | P (cid:105) = | (cid:108)(cid:108)(cid:108)(cid:105) . We calculate the three edge lengths using the techniques introduced in Sec. 2and applied in Sec: 3. We will also calculate the information area for thistriangle that is given by Eq. 12.Again we consider three observers Alice ( A ), Bob ( B ) and Charlie ( C ). A , B and C measure the separable state using their choice of detectors definedin Eq. 29. For the rest of this section we also set α = 0.The only four non-vanishing joint probabilities from the three measure-ments on this separable state M C M B M A | P (cid:105) are, p ( A = 1 , B = 1 , C = 1) = cos ( β ) cos ( γ ) , (63) p ( A = 1 , B = 1 , C = 0) = cos ( β ) sin ( γ ) , (64) p ( A = 1 , B = 0 , C = 1) = sin ( β ) cos ( γ ) , (65) p ( A = 1 , B = 0 , C = 0) = sin ( β ) sin ( γ ) . (66) UANTUM GEOMETRY OF MEASUREMENTS 17
Tracing the joint probability successively over the three observers gives usthe eight non-vanishing pairwise joint probabilities, p ( A = 1 , B = 1) = cos ( β ) , (67) p ( A = 1 , B = 0) = sin ( β ) , (68) p ( A = 1 , C = 1) = cos ( γ ) , (69) p ( A = 1 , C = 0) = sin ( γ ) , (70) p ( B = 1 , C = 1) = cos ( β ) cos ( γ ) , (71) p ( B = 1 , C = 0) = cos ( β ) sin ( γ ) , (72) p ( B = 0 , C = 1) = sin ( β ) cos ( γ ) , (73) p ( B = 0 , C = 0) = sin ( β ) sin ( γ ) . (74)Finally, tracing the joint probability over all pairs of observers gives us thesix probabilities for the measurement outcomes of A , B and C ,(75) p ( A = 0) = 0 , p ( A = 1) = 1 ,p ( B = 0) = sin ( β ) , p ( B = 1) = cos ( β ) ,p ( C = 0) = sin ( γ ) , p ( C = 1) = cos ( γ ) . The conditional probabilities can be recovered from these probabilities usingEq.37 and we find the following three sets of conditional probabilities ( A - B , A - C and B - C ):(76) p ( A = 0 | B = 0) = 0 p ( A = 1 | B = 0) = 1 p ( A = 0 | B = 1) = 0 p ( A = 1 | B = 1) = 1 p ( B = 0 | A = 0) = N A p ( B = 1 | A = 0) = N Ap ( B = 0 | A = 1) = sin ( β ) p ( B = 1 | A = 1) = cos ( β ) ;(77) p ( A = 0 | C = 0) = 0 p ( A = 1 | C = 0) = 1 p ( A = 0 | C = 1) = 0 p ( A = 1 | C = 1) = 1 p ( C = 0 | A = 0) = N A p ( C = 1 | A = 0) = N Ap ( C = 0 | A = 1) = sin ( γ ) p ( C = 1 | A = 1) = cos ( γ ) ;(78) p ( B = 0 | C = 0) = sin ( β ) p ( B = 1 | C = 0) = cos ( β ) p ( B = 0 | C = 1) = sin ( β ) p ( B = 1 | C = 0) = cos ( β ) p ( C = 0 | B = 0) = sin ( γ ) p ( C = 1 | B = 0) = cos ( γ ) p ( C = 0 | B = 1) = sin ( γ ) p ( C = 1 | B = 1) = cos ( γ ) . We are now in a position to use Eqs. 3-6 to calculate the entropy, theconditional entropy as well as the information distance in Eq. 9. The entropy of our observers are H A = 0 , (79) H B = − sin ( β ) log(sin ( β )) − cos ( β ) log(cos ( β )) , (80) H C = − sin ( γ ) log(sin ( γ )) − cos ( γ ) log(cos ( γ )) , (81)(82)and the joint entropy between pairs of our observers are,(83) H AB = − sin ( β ) log(sin ( β )) − cos ( β ) log(cos ( β )) ,H AC = − sin ( γ ) log(sin ( γ )) − cos ( γ ) log(cos ( γ )) ,H BC = − sin ( γ ) sin ( β ) log(sin ( γ ) sin ( β )) − cos ( γ ) sin ( β ) log(cos ( γ ) sin ( β )) − sin ( γ ) cos ( β ) log(sin ( γ ) cos ( β )) − cos ( γ ) cos ( β ) log(cos ( γ ) cos ( β )) . Finally, we use Eq. 37 to find the joint entropy H ABC of A , B and C , itsimplifies to H ABC = H BC , (84)where H BC is given explicitly in Eq. 83. We find the three lengths of theedges of the triangle for this separable state from Eq. 9,(85) D AB = − sin ( β ) log(sin ( β )) − cos ( β ) log(cos ( β )) , D AC = − sin ( γ ) log(sin ( γ )) − cos ( γ ) log(cos ( γ )) , D BC = − ( γ ) sin ( β ) log(sin ( γ ) sin ( β )) − ( γ ) sin ( β ) log(cos ( γ ) sin ( β )) − ( γ ) cos ( β ) log(sin ( γ ) cos ( β )) − ( γ ) cos ( β ) log(cos ( γ ) cos ( β )) − sin ( β ) log(sin ( β )) − cos ( β ) log(cos ( β )) − sin ( γ ) log(sin ( γ )) − cos ( γ ) log(cos ( γ )) . The area of the triangle for this separable state using Eq. 12 and Eq. 84 is,(86) A | P (cid:105) ABC ( β, γ ) = H BC − H AB H BC − H AC H BC + H AB H AC . It is interesting that for this separable state, the Euclidean area is essen-tially zero for all values of β and γ . The information area is well behavedwith a maximum at β = γ = π/
4. In this particular case the geometry of thetriangle collapses in the Euclidean plane to a line. At this global maximumin information area we find, D AB = D AC = 1 , (87) D BC = D AB + D AC = 2 , and (88) A | P (cid:105) ABC = 1 . (89)This is illustrated in Fig. 6.5. From Bipartite Quadrilateral to Tripartite Octagon:Future Prospects
What we outlined in this manuscript is the beginning of our explorationof our approach to information geometry. We do not have a definitive entan-glement measure or actual scalability results. However, the analysis of the
UANTUM GEOMETRY OF MEASUREMENTS 19
Figure 6.
The left plot is the information area ( A | P (cid:105) ABC ( β, γ )), the center plot is the Euclidean area ( A E )based on the three information distances in Eq. 85, and theright plot is the ratio, where we see A E ∼ e A ABC . The plotsrange is β, γ ∈ { , π/ } .triangles for the | GHZ (cid:105) , | W (cid:105) and separable state | P (cid:105) yielded qualitativelyunique features. The relative differences between the perimeters of each ofthe triangles and the corresponding information area suggests that a cur-vature measure might be useful for differentiating quantum states, and inparticular the separable state | (cid:108)(cid:108)(cid:108)(cid:105) showed an interesting feature of nearlyzero area from Heron’s formulae. We also observed that the triangle inequal-ity was not violated for these entangled states in the measurement space weconsidered. In particular, there were no violations for the | GHZ (cid:105) and, weobserved that there were never triangle inequality violations for the | W (cid:105) state. It is clear that an exhaustive exploration is needed, and this seemsfeasible. We have the six Bloch sphere detector angles to explore, as well asthe need to explore a sampling of the space of symmetric tripartite states.It is equally clear that new measures and ideas are needed. We discuss apromising avenue in the remainder of this section.We have generalized the Schumacher approach from bipartite states totripartite states. If each of our observers, A , B and C can choose randomlybetween two separate detectors then the triangle becomes an octahedron asillustrated in Fig. 7. For the bipartite system each of the two observers hadtwo detectors so the line connecting the observer became the quadrilateral il-lustrated in the lower half of Fig. 2. In our generalization to tripartite states,each of our three observers has two detectors, and the triangle connectingthem becomes an octahedron as shown in the right box of Fig. 7. In thisconfiguration we can calculate the 12 lengths of the edges of the octahedron,we can also calculate the 8 triangle areas. We can then ask questions as tothe embeddability of the octahedron in Euclidean and Minkowski space, its Figure 7.
This figure illustrates a generalization of Schu-macher’s quantum information geometry of a bipartite stateto our information geometry of a tripartite state. Each of thethree observers have two detectors (left box). The quadrilat-eral of Schumacher (Fig 2) generalizes naturally to an oc-tagon (right box). We can use our formalism to study itsinformation geometry. We can use this arrangement to ex-plore some curvature and embeddability properties of theoctahedron as indicators of the properties of the quantumstate, | Ψ (cid:105) .curvature, and other measures. This work is in progress and will be reportedat a later date. ACKNOWLEDGMENTS
This research benefited form discussions with P. M. Alsing and his group,and from discussions with M. Corne and S. Mostafanazhad Aslmarand. Wethank AFRL/RITA and the Griffiss Institute for providing a stimulating re-search environment and support under the Summer Faculty Fellowship Pro-gram. This research was supported under AFOSF/AOARD grant
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