QQuantum d-separationandquantum belief propagation
Robert R. [email protected] 18, 2020
Abstract
The goal of this paper is to generalize classical d-separation and classical Belief Prop-agation (BP) to the quantum realm. Classical d-separation is an essential ingredientof most of Judea Pearl’s work. It is crucial to all 3 rungs of what Pearl calls the 3rungs of Causation. So having a quantum version of d-separation and BP probablyimplies that most of Pearl’s Bayesian networks work, including his theory of causality,can be translated in a straightforward manner to the quantum realm.1 a r X i v : . [ qu a n t - ph ] D ec Introduction
Classical Bayesian networks (bnets), d-separation and belief propagation (BP), themain topics of this paper, were first proposed by Judea Pearl and collaborators.Pearl’s results and their history have been amply discussed by him and some coau-thors in the following 4 books: Refs.[2, 3, 4, 5]. Tucci has written an open sourcebook entitled Bayesuvius (see Ref.[6]) about classical bnets. Bayesuvius will be citedfrequently in this paper as a primary source of background information explained inthe same style of notation as this paper.Quantum Bayesian Networks (qbnets) were first proposed by Tucci in Ref.[12].Ref.[12] deals with pure quantum states only. Tucci later generalized qbnets to mixedstates in Ref.[11].The goal of this paper is to generalize classical d-separation and classicalBP to the quantum realm. Tucci has assumed or implied in previous work that d-separation and BP are valid for qbnets, but he has never explicitly proven this. Thispaper is intended to be a first step towards filling that gap.Classical d-separation inspired Tucci to propose the definition of squashedentanglement (SE). d-separation and SE are very closely linked. SE was first proposedby Tucci in a series of 6 papers Refs.[13, 16, 9, 15, 7, 8] spanning the years 1999-2002. Starting around 2004 with Ref.[1] by Christiandl and Winters, other researchersrecognized the importance of SE and began writing papers about it. A more completehistory of SE is given in Ref.[17]. Subsequently, Tucci has written an open sourcesoftware program called Entanglish (see Ref.[10]) for calculating SE.The open-source computer program Quantum Fog (Ref.[14]) written by Tucciassumes that both classical and quantum BP are valid, because it uses the junctiontree (JT) algorithm to do both classical and quantum inference with bnets. The JTalgorithm is a generalization of BP so as to include bnets with loops. Classical d-separation is an essential ingredient of most of Judea Pearl’s work.It is crucial to all 3 rungs of what Pearl calls the 3 rungs of Causation. So having aquantum version of d-separation and BP probably implies that most of Pearl’s bnetwork, including his theory of causality, can be translated in a straightforward mannerto the quantum realm. This is perhaps not too surprising because most of Pearl’sbnet results depend to a large extent on the topology (i.e., graph structure) of theunderlying DAG of a bnet, and that DAG should be essentially the same whether weare considering a quantum phenomenon or its classical limit. Discussed in chapter of Bayesuvius entitled “D-Separation”. Discussed in chapter of Bayesuvius entitled “Message Passing (Belief Propagation)”. Discussed in chapter of Bayesuvius entitled “Juntion Tree Algorithm”. The quantum version of JT used by the current version of Quantum Fog is not quite correctbecause it does not use the vector amplitudes defined in this paper. Notational Conventions and Preliminaries
This paper will employ the same notational conventions as Bayeuvius (Ref.[6]). Hence,if the reader encounters any notation that is not defined in this paper, he/she shouldconsult Bayesuvius (especially its chapter entitled “Notational Conventions and Pre-liminaries”), where it is very likely to be defined.Henceforth, whenever we write [ E ][ h.c. ], where E is some quantum operator,we will mean E E † , where E † is the Hermitian conjugate of E . Also, whenever we write N (! x ), we will mean a normalization constant that is independent of x .Consider a DAG with nx nodes x nx = ( x i ) i =0 , ,...,nx − . A qbnet consists ofsuch a DAG with a TPM (transition probability matrix) attached to each node, butunlike the TPMs for a classical bnet, these ones are complex valued and normal-ized differently. If we represent the TPM of node x j by a probability amplitude A ( x j | pa ( x j )), then A ( x j | pa ( x j )) must satisfy the normalization condition (cid:88) x j (cid:12)(cid:12) A ( x j | pa ( x j )) (cid:12)(cid:12) = 1 . (1)The amplitude for the full DAG is defined as in the classical case, by multiplying theTPMs of all the nodes: A ( x nx ) = (cid:89) j A ( x j | pa ( x j )) . (2)Note that (cid:88) x nx | A ( x nx ) | = 1 . (3)Suppose a., b. are disjoint multinodes in x nx . Denote the complement multin-ode of a. by a .c = x nx − a. Then we define the vector amplitude | A (cid:105) ( a. ) and theprobability P ( a. ) by | A (cid:105) ( a. ) = | A ( a. ) (cid:105) = | a. (cid:105) (cid:88) a c. A ( a., a. c ) | a. c (cid:105) , (4)with boundary cases | A (cid:105) ( x nx ) = A ( x nx ), | A (cid:105) = (cid:80) x nx A ( x nx ) | x nx (cid:105) . and P ( a. ) = (cid:88) a. c | A ( a., a. c ) | . (5)Note that (cid:80) a. P ( a. ) = 1 so P ( a. ) is a bona fide probability distribution as the notationimplies. Note also that (cid:104) A ( a. ) | A ( a. ) (cid:105) = P ( a. ) , (6)so you can think of | A (cid:105) ( a. ) as being a generalized square root of P ( a. ).3e define the conditional vector amplitude | A (cid:105) ( b. | a. ) by | A (cid:105) ( b. | a. ) = | A ( b. | a. ) (cid:105) = | A (cid:105) ( b., a. ) | A (cid:105) ( a. ) . (7)This is analogous to the definition of conditional probability, P ( b. | a. ) = P ( b.,a. ) P ( a. ) . Notethat | A (cid:105) ( b. | a. ) isn’t really a ket; it’s a tuple of two kets, but it is convenient torepresent it as a ket. We can also define the dual (cid:104) A | ( b. | a. ) of | A (cid:105) ( b. | a. ) so that thefollowing equations are satisfied: (cid:104) A | ( b. | a. ) = (cid:104) A ( b. | a. ) | = (cid:104) A | ( b., a. ) (cid:104) A | ( a. ) , (8) (cid:104) A ( b. | a. ) | A ( b. | a. ) (cid:105) = (cid:104) A ( b., a. ) | A ( b., a. ) (cid:105)(cid:104) A ( a. ) | A ( a. ) (cid:105) = P ( b., a. ) P ( a. ) = P ( b. | a. ) . (9)Hence, you can think of | A (cid:105) ( b. | a. ) as being a generalized square root of P ( b. | a. ).Suppose a., b., c. are disjoint multinodes such that a. ∪ b. ∪ c. = x nx Then (cid:88) b. | A (cid:105) ( a., b. ) = (cid:88) b. (cid:88) c. | A (cid:105) ( a., b., c. ) (10)= | A (cid:105) ( a. ) (11)This is analogous to the probability marginalization rule (cid:80) b. P ( a., b. ) = P ( a. ).Note that (cid:88) b. | A (cid:105) ( a. | b. ) | A (cid:105) ( b. ) = (cid:88) b. | A (cid:105) ( a., b. ) (12)= | A (cid:105) ( a. ) . (13)This is analogous to the probability splitting rule P ( a. ) = (cid:80) b. P ( a. | b. ) P ( b. ).Suppose a. and e. are disjoint multinodes, with e. representing evidence. Notethat | A (cid:105) ( a. | e. ) = | A (cid:105) ( a., e. ) | A (cid:105) ( e. ) (14)= | A (cid:105) ( e. | a. ) | A (cid:105) ( a. ) | A (cid:105) ( e. ) . (15)This is analogous to the classical Bayes rule P ( a. | e. ) = N (! a. ) P ( e. | a. ) P ( a. ).Let P λ be the set of all probability distributions P λ ( λ ) with λ ∈ S λ .Let H x represent a Hilbert space spanned by an orthonormal basis | x (cid:105) , where x ∈ S x . Let H x,y = H x ⊗ H y . 4et D x be the set of all density matrices acting on H x . Likewise, let D x,y bethe set of all density matrices acting on H x,y . If ρ x,y ∈ D x,y , then ρ x will denote thepartial trace tr y ρ x,y .Let D x,y,λ d be the set of all density matrices in D x,y,λ which are diagonal in λ .In other words, ρ x,y,λ d ∈ D x,y,λ d if it is of the form ρ x,y,λ d = (cid:88) λ P λ ( λ ) | λ (cid:105)(cid:104) λ | ρ λx,y , (16)where ρ λx,y ∈ D x,y for all λ and P λ ∈ P λ .classical quantumEntropy H ( x ) = − (cid:80) x P ( x ) ln P ( x ) S ρ ( x ) = S ( ρ ) = − tr x ( ρ ln ρ )for ρ ∈ D x ConditionalEntropy H ( x | y ) = − (cid:80) x,y P ( x, y ) ln P ( x | y ) S ρ ( x | y ) = − S ρ ( y ) + S ρ ( x, y )for ρ ∈ D x,y MutualInformation H ( x : y ) = (cid:80) x,y P ( x, y ) ln P ( x,y ) P ( x ) P ( y ) S ρ ( x : y ) = S ρ ( x ) + S ρ ( y ) − S ρ ( x, y )for ρ ∈ D x,y ConditionalMutualInformation H ( x : y | λ ) = (cid:80) x,y,λ P ( x, y, λ ) ln P ( x,y | λ ) P ( x | λ ) P ( y | λ ) S ρ ( x : y | λ ) = S ρ ( x | λ ) + S ρ ( y | λ ) − S ρ ( x, y | λ )for ρ ∈ D x,y,λ Table 1: Definitions of various Entropies and Informations.Table 1 gives the definitions of various entropies and informations used inclassical Shannon Information Theory (SIT), and their counterparts in quantum SIT.
The next box comes from the chapter of Bayesuvius entitled “D-Separation”.
Claim 1 (Classical d-separation Theorem)Suppose
A., B., Z. are disjoint multinodes of a DAG G . A. ⊥ G B. | Z. iff H ( A. : B. | Z. ) = 0 for all P compatible with G . The proof of this theorem will not be presented here. It isn’t presented in thecurrent version of Bayesuvius either. To see it, you will have to look in Ref.[2], andin the original papers cited therein.The definition of classical d-separation (i.e., of A. ⊥ G B. | Z. ) only dependson the topology of the DAG. We will define quantum d-separation exactly as it is5efined classically. Whether the bnet has probabilities or amplitudes for its TPMsdoes not make a difference at the level of the definition of d-separation. It onlybecomes important when trying to find a quantum analogue of the whole classicald-separation theorem which is stated in the box above. The remainder of this sectionwill be dedicated to finding a quantum analogue to the whole box above.We start by using the definitions introduced in the previous section to concludethat: S ρ ( x : y | λ d ) = (cid:88) λ P ( λ ) (cid:104) S ( ρ λx ) + S ( ρ λy ) − S ( ρ λx,y ) (cid:105) (17)for ρ ∈ D x,y,λ d . Claim 2 S ρ ( x : y | λ d ) = 0 iff ρ λx,y = ρ λx ρ λy for all λ . proof: S ( ρ λx ) + S ( ρ λy ) − S ( ρ λx,y ) = 0 iff ρ λx,y = ρ λx ρ λy QED
Next, we express each density matrix ρ λx,y as ρ λx,y = U DU † , (18)where U is a unitary matrix and D is a diagonal matrix with non-negative diagonalentries. Now let A ( x, y | x , y , λ ) = (cid:104) x |(cid:104) y | U | x (cid:105)| y (cid:105) (19)and (cid:104) x |(cid:104) y | D | x (cid:105)| y (cid:105) = P ( x , y | λ ) . (20)Hence, ρ λx,y = (cid:88) x ,y (cid:104) (cid:88) x,y | x (cid:105)| y (cid:105) A ( x, y | x , y , λ ) (cid:105) P ( x , y | λ ) (cid:104) h.c. (cid:105) (21)To make our future expressions more concise, define the two abbreviations R = ( x , y ), R = ( x, y ). Then ρ x,y,λ d = (cid:88) R ,λ (cid:104) | λ (cid:105) (cid:88) R | R (cid:105) A ( R | R , λ ) (cid:112) P ( R | λ ) (cid:112) P ( λ ) (cid:124) (cid:123)(cid:122) (cid:125) A ( R,R ,λ ) (cid:105)(cid:104) h.c. (cid:105) . (22)From the definitions of conditional probabilities and conditional amplitudes, we get A ( R | R , λ ) = A ( y | x, R , λ ) A ( x | R , λ ) (23)6 (cid:15) (cid:15) (cid:22) (cid:22) x (cid:15) (cid:15) (cid:111) (cid:111) (cid:8) (cid:8) λ (cid:0) (cid:0) (cid:31) (cid:31) (cid:94) (cid:94) (cid:64) (cid:64) y x (cid:111) (cid:111) y (cid:15) (cid:15) x (cid:15) (cid:15) λ (cid:0) (cid:0) (cid:31) (cid:31) (cid:94) (cid:94) (cid:64) (cid:64) y x ( a ) ( b )Figure 1: S ( x : y | λ d ) is nonzero for the qbnet of panel ( a ) but zero for the qbnet ofpanel ( b ).and P ( R | λ ) = P ( y | x , λ ) P ( x | λ ) . (24)Therefore, A ( R, R , λ ) = A ( y | x, R , λ ) A ( x | R , λ ) (cid:112) P ( y | x , λ ) P ( x | λ ) P ( λ ) . (25)Eq.(25) can be represented graphically by the qbnet Fig.1( a ). The TPMs,printed in blue, for the 2 qbnets ( a ) and ( b ) of Fig.1, are as follows. A ( λ ) = (cid:112) P ( λ ) (26) A ( x | λ ) = (cid:112) P ( x | λ ) (27) A ( y | x , λ ) = (cid:26) (cid:112) P ( y | x , λ ) for Fig.1( a ) (cid:112) P ( y | λ ) for Fig.1( b ) (28) A ( x | R , λ ) = (cid:26) A ( x | R , λ ) for Fig.1( a ) A ( x | x , λ ) for Fig.1( b ) (29) A ( y | x, R , λ ) = (cid:26) A ( y | x, R , λ ) for Fig.1( a ) A ( y | y , λ ) for Fig.1( b ) (30)It’s easy to check that • x ⊥ G y | λ is false and S ( x : y | λ d ) (cid:54) = 0 in Fig.1( a ), whereas7 x ⊥ G y | λ is true and S ( x : y | λ d ) = 0 in Fig.1( b ).So far, we have shown how, given any density matrix ρ ∈ D x,y,λ , one canconstruct a qbnet. This method of constructing a qbnet from a density matrix ρ (orvice versa, constructing a ρ from a qbnet) can be generalized to finding a qbnet for any ρ ∈ D x nx . for arbitrary nx . Now we argue that the proof of the quantum d-separationtheorem should be formally identical to the proof of the classical d-separation theorem.The only difference between the proofs is that whenever a probability occurs in theclassical proof, it must be replaced by a vector amplitude in the quantum proof. Ofcourse, probabilities and vector amplitudes are normalized differently, but that shouldnot change the form of the proofs. Note that we have been careful to show that vectoramplitudes can be conditioned and satisfy a splitting rule, just like probabilities do.Also, we have been careful to define d-separation A. ⊥ G B. | Z. to be identical for theclassical and quantum cases. Hence, we argue that, without looking at the details ofthe proof of the classical d-separation theorem, one can conclude that the followingtheorem must be true: Claim 3 (Quantum d-separation Theorem)Suppose
A., B., Z. are disjoint multinodes of a DAG G . A. ⊥ G B. | Z. iff S ρ ( A. : B. | Z. d ) = 0 for all ρ compatible with G . Define the squashed entanglement of a density matrix ρ x,y by E sq ( ρ x,y ) = 12 min ρ ∈D S ρ ( x : y | λ d ) (31)where D = { ρ ∈ D x,y,λ d | tr λ d ρ x,y,λ d = ρ x,y } . Then the quantum d-separationtheorem immediately(?) implies the following. Claim 4
Suppose
A., B. are disjoint multinodes of a DAG G ,( A. ⊥ G B. | Z. for some Z. such that A., B., Z. are disjoint multinodes of G)iff E sq ( ρ A.,B. ) = 0 . The rules (and their proof) of classical BP can be found in the Bayesuvius chapterentitled “Message Passing (Belief Propagation)”. Just like the proof of the classicald-separation theorem, the proof of the rules for classical BP relies on 3 ingredients: • the topology of a DAG 8 the definition of conditional probabilities as ratios of joint probabilities • the splitting rule for probabilitiesSince these 3 ingredients are also available in the quantum side if we replace prob-abilities by vector amplitudes, we can conclude that the rules for quantum BP areformally the same as those for classical BP, modulus the replacement of probabilitiesby vector amplitudes. The difference in normalization of probabilities and vectoramplitudes does not make the rules for classical and quantum BP different becausethese rules are defined up to a normalization constant.In Bayesuvius, the BP chapter entitled “Message Passing (Belief Propaga-tion)” considers a general case of classical BP (viz., BP for polytrees (BP-Gen)) anda special case of classical BP (viz., BP for bipartite bnets (BP-BB)). We end thissection with 2 subsections dedicated to the quantum analogues of the rules for BP-Gen and BP-BB. Those 2 subsections are simply exact quotes from the BP chapterin Bayesuvius, except that all P ’s have been crossed out and replaced by | A (cid:105) ’s. Let a na = ( a i ) i =0 , ,...,na − denote the parents of x and b nb = ( b i ) i =0 , ,...,nb − its children.Define π x ( x ) = (cid:88) a na (cid:0)(cid:0)(cid:18) | A (cid:105) P ( x | a na ) (cid:89) i π x ⇐ a i ( a i ) (32)= E a na [ (cid:0)(cid:0)(cid:18) | A (cid:105) P ( x | a na )] (33)(boundary case: if x is a root node, use π x ( x ) = (cid:0)(cid:0)(cid:18) | A (cid:105) P ( x ).) and λ x ( x ) = (cid:89) i λ b i ⇒ x ( x ) . (34)(boundary case: if x is a leaf node, use λ x ( x ) = 1.) • RULE 1: (red parent) λ x ⇒ a i ( a i ) (cid:124) (cid:123)(cid:122) (cid:125) OUT = N (! a i ) (cid:88) x λ x ( x ) (cid:124) (cid:123)(cid:122) (cid:125) IN (cid:88) ( a k ) k (cid:54) = i (cid:0)(cid:0)(cid:18) | A (cid:105) P ( x | a na ) (cid:89) k (cid:54) = i π x ⇐ a k ( a k ) (cid:124) (cid:123)(cid:122) (cid:125) IN (35)= N (! a i ) (cid:88) x λ x ( x ) E ( a k ) k (cid:54) = i [ (cid:0)(cid:0)(cid:18) | A (cid:105) P ( x | a na )] (36)= N (! a i ) E ( a k ) k (cid:54) = i E x | a na λ x ( x ) (37)9boundary case: if x is a root node, use λ x ⇒ a i ( a i ) = N (! a i ).) • RULE 2: (red child) π b i ⇐ x ( x ) (cid:124) (cid:123)(cid:122) (cid:125) OUT = N (! x ) π x ( x ) (cid:124) (cid:123)(cid:122) (cid:125) IN (cid:89) k (cid:54) = i λ b k ⇒ x ( x ) (cid:124) (cid:123)(cid:122) (cid:125) IN (38)(boundary case: if x is a leaf node, use π b i ⇐ x ( x ) = N (! x ) π x ( x ) .)In the above equations, if the range set of a product is empty, then define the productas 1; i.e., (cid:81) k ∈∅ F ( k ) = 1. Claim:
Define
BEL ( t ) ( x ) = N (! x ) λ ( t ) x ( x ) π ( t ) x ( x ) . (39)Then lim t →∞ BEL ( t ) ( x ) = (cid:0)(cid:0)(cid:18) | A (cid:105) P ( x | (cid:15) ) . (40)This says that the belief in x = x converges to (cid:0)(cid:0)(cid:18) | A (cid:105) P ( x | (cid:15) ) and it equals the product ofmessages received from all parents and children of x = x . Traversing an x (i.e., root) node. For i = 0 , , . . . , nx −
1, if α ∈ nb ( i ), then, m ( t ) α ⇐ i ( x i ) = (cid:89) β ∈ nb ( i ) − α m ( t − β ⇒ i ( x i ) , (41)whereas if α / ∈ nb ( i ) m ( t ) α ⇐ i ( x i ) = m ( t − α ⇐ i ( x i ) . (42)2. Traversing an f (i.e., leaf ) node. For α = 0 , , . . . , nf −
1, if i ∈ nb ( α ), then m ( t ) α ⇒ i ( x i ) = (cid:88) ( x k ) k ∈ nb ( α ) − i f α ( x nb ( α ) ) (cid:89) k ∈ nb ( α ) − i m ( t − α ⇐ k ( x k ) (43)= E ( t − x k ) k ∈ nb ( α ) − i [ f α ( x nb ( α ) )] , (44)10hereas if i / ∈ nb ( α ) m ( t ) α ⇒ i ( x i ) = m ( t − α ⇒ i ( x i ) . (45)In the above equations, if the range set of a product is empty, then define theproduct as 1; i.e., (cid:81) k ∈∅ F ( k ) = 1. Claim: (cid:0)(cid:0)(cid:18) | A (cid:105) P ( x i | (cid:15) ) = lim t →∞ N (! x i ) (cid:89) α ∈ nb ( i ) m ( t ) α ⇒ i ( x i ) (46)and (cid:0)(cid:0)(cid:18) | A (cid:105) P ( x nb ( α ) | (cid:15) ) = lim t →∞ N (! x nb ( α ) ) f α ( x nb ( α ) ) (cid:89) k ∈ nb ( α ) m ( t ) α ⇐ k ( x k ) . (47)In the classical case, f α ( x nb ( α ) ) stands for a real valued function (e.g., P ( f α =1 | x nb ( α ) )), whereas in the quantum case, it stands for a vector amplitude (e.g., | A (cid:105) ( f α =1 | x nb ( α ) )). A Appendix: Reduced qbnet λ (cid:0) (cid:0) (cid:30) (cid:30) Y X (cid:111) (cid:111) λ (cid:0) (cid:0) (cid:30) (cid:30) Y X ( a ) ( b )Figure 2: The 2 qbnets in Fig.1 can sometimes be reduced to these 2 qbnets.Let X = ( x, x ) and Y = ( y, y ). Note that if, as indicated in Eq.(49), A ( x | R , λ ) is independent of y , then the 2 qbnets in Fig.1 can be reduced to the2 qbnets in Fig.2, The node TPMs, printed in blue, of the qbnets in Fig.2, are asfollows: A ( λ ) = (cid:112) P ( λ ) (48) A ( x, x | (cid:26)(cid:26) y , λ ) = (cid:40) A ( x | (cid:26)(cid:26)(cid:62) x R , λ ) (cid:112) P ( x | λ ) for Fig.2( a ) A ( x | x , λ ) (cid:112) P ( x | λ ) for Fig.2( b ) (49)11 ( y, y | x, x , λ ) = (cid:26) A ( y | x, R , λ ) (cid:112) P ( y | x , λ ) for Fig.2( a ) A ( y | y , λ ) (cid:112) P ( y | λ ) for Fig.2( b ) (50) References [1] Matthias Christandl and Andreas Winter. “squashed entanglement”: An ad-ditive entanglement measure.
Journal of Mathematical Physics , 45(3):829–840,Mar 2004.[2] Judea Pearl.
Probabilistic Inference in Intelligent Systems . Morgan Kaufmann,1988.[3] Judea Pearl.
Causality: Models, Reasoning, and Inference, Second Edition . Cam-bridge University Press, 2013.[4] Judea Pearl, Madelyn Glymour, and Nicholas P Jewell.
Causal inference instatistics: A primer . John Wiley & Sons, 2016.[5] Judea Pearl and Dana Mackenzie.
The book of why: the new science of causeand effect . Basic Books, 2018.[6] Robert R. Tucci. Bayesuvius (book). https://github.com/rrtucci/Bayesuvius/raw/master/main.pdf .[7] Robert R. Tucci. Entanglement of bell mixtures of two qubits. https://arxiv.org/abs/quant-ph/0103040 .[8] Robert R. Tucci. Entanglement of distillation and conditional mutual informa-tion. https://arxiv.org/abs/quant-ph/0202144 .[9] Robert R. Tucci. Entanglement of formation and conditional information trans-mission. https://arxiv.org/abs/quant-ph/0010041 .[10] Robert R. Tucci. Entanglish (software). https://github.com/rrtucci/Entanglish .[11] Robert R. Tucci. An introduction to quantum Bayesian networks for mixedstates. https://arxiv.org/abs/1204.1550 .[12] Robert R. Tucci. Quantum Bayesian nets. https://arxiv.org/abs/quant-ph/9706039 . 1213] Robert R. Tucci. Quantum entanglement and conditional information transmis-sion. https://arxiv.org/abs/quant-ph/9909041 .[14] Robert R. Tucci. Quantum Fog (software). https://github.com/artiste-qb-net/quantum-fog .[15] Robert R. Tucci. Relaxation method for calculating quantum entanglement. https://arxiv.org/abs/quant-ph/0101123 .[16] Robert R. Tucci. Separability of density matrices and conditional informationtransmission. https://arxiv.org/abs/quant-ph/0005119 .[17] Wikipedia. Squashed entanglement. https://en.wikipedia.org/wiki/Squashed_entanglementhttps://en.wikipedia.org/wiki/Squashed_entanglement