Quantum inferring acausal structures and the Monty Hall problem

This paper presents a quantum version of the Monty Hall problem based upon the quantum inferring acausal structures, which can be identified with generalization of Bayesian networks. Considered structures are expressed in formalism of quantum information theory, where density operators are identified with quantum generalization of probability distributions. Conditional relations between quantum counterpart of random variables are described by quantum conditional operators. Presented quantum inferring structures are used to construct a model inspired by scenario of well-known Monty Hall game, where we show the differences between classical and quantum Bayesian reasoning.


Introduction
Probability theory is very useful mathematical theory, which has a lot applications in many areas of of science and engineering.Probabilistic modelling has strong impact in development of artificial intelligence, providing tools for knowledge representation, knowledge management and reasoning [1].Many problems related with computer vision, speech recognition, extraction of information or diagnosis of diseases can be modelled by probabilistic graphical models which structures describe conditional dependencies between random variables [2].The next part of this paper presents the generalization of classical conditional distributions and its applications in graphical models in the field of quantum information theory.
Cerf and Adami in [3,4] introduced a quantum conditional amplitude operator as a extension of conditional probability distributions.This approach allowed to complement of relationship between Shannon conditional entropy and von Neumann conditional entropy.Next, Warmuth and Kuzmin in [5] used the conditional operators for generalization of probability theory into quantum information theory.
Leifer in [6] used conditional states in the context of quantum channels.He connected classical conditional probabilities and bipartite density operators with variant of well-known Jamio lkowski isomorphism.Later with Spekkens [7,8] presented formalism of quantum inferring structures.They described two cases of the structures: causal (one system at two times) and acausal (two systems at a single time).
In [9] quantum conditional states was used to problem of causal inference for quantum variables.The results of research show that an advantages of inferring causal structures arise from entanglement and coherence.
2 Formalism of quantum information

Dirac notation
Throughout this paper complex Euclidean space with Dirac notation is used.Symbol |ψ ∈ C n called as ket, denotes a complex n-dimensional column vector.The conjugate transpose of a ket (Hermitian transpose) is denoted by bra ψ| = (|ψ ) † .A space of linear operators from C n to C n is denoted by L(C n ).
In any n-dimensional complex Euclidean space C n exists standard orthogonal basis, which is expressed as set of normed, pairwise orthogonal vectors: Hence, every vector |ψ ∈ C n can be expressed as the linear combination Consider two vectors |ψ The inner product (scalar product) between those two vectors is denoted by: The outer product between |ψ , |φ is expressed as where |i j| ∈ L(C n ) is a matrix, that every its element is zero except element at position (i + 1, j + 1), which is one.
The tensor product between |ψ , |φ is defined as the n 2 dimensional vector where {|i ⊗ |j } forms orthonormal base on C n 2 .Throughout this paper, abbreviation |ij := |i ⊗ |j is used .

Density operators
Density matrix is a complex matrix ρ ∈ L(C n ) which is Hermitian semidefinite positive and trace one [10]. where A mixed state which is not separable is called entangled.

Measurement
Suppose we have k measurement operators A i , which satisfy the completeness relation The quantum measurement performed on the initial state ρ produces i-th outcome with probability p i = Tr(A i ρA † i ) and transforms ρ into Suppose that ρ is a operator from L(C nA ⊗ C nB ) and the measurement is performed on the second subsystem system.The resulting density matrix can be expressed as where Tr(( is the probability of obtaining of ρi .

Generalization of probability theory
As was proposed in [7] we can replace a random variable A with a quantum system C nA and probability distribution P (A) on that variable with density matrix ρ A from L(C nA ).Suppose that Then corresponding density matrix has the form Next, generalized random variables will be called as quantum variables.
If ρ A ∈ L(C nA ) and ρ B ∈ L(C nB ) represent some distributions, then joint probability distribution can be defined as If matrix ρ AB is given, then we can marginalize it with such formula According to [7], we introduce following product . Quantum conditional state can be defined as Moreover, the formula below holds Quantum conditional state can be considered as generalization of conditional probability distributions.Consider quantum joint distribution ρ AB .If the variable B obtains value b, then the state ρ AB is transformed to The comma in index of ρ A,B=b is used only for readability.The marginal distribution of the variable A is given as Suppose that ρ ∈ L(C nA ⊗ C nB ), then we can consider tree cases of relationships between variables A and B.

Independent random variables. There exist ρ
such that ρ = ρ A ⊗ ρ B -then variables are independent in classical way.

Inferring acausal structures
Suppose that ρ B|A and ρ A are given.The state ρ B|A characterizes conditional relationship between variables A and B, it can be interpreted as belief about variables.In this case, the state for system B can be expressed as this formula in classical probability theory is called as the law of total probability.Fig. 1 illustrate graphical interpretation of reasoning over given ρ B|A and ρ A and it can be considered as simply Bayesian network.
The conditional independence relationship associated with graphical structure of Bayesian network allow to rewrite the generalized joint distribution of A and B as follows Consider structure (Fig. 2) described by state ρ A and conditional states ρ B|A and ρ C|A .To obtain joint probabilities of A and B or A and C we use formulas The joint probability distribution of variables A, B and C can be obtained in following way In similar way, more complicated models can be prepared.

Acausal structure in Monty Hall game
Suppose the player is on a game show and he has to choose one of three doors: behind one of them there is a car, behind others, goats.After his choice, the host (who knows, where the prize is) opened another, which has a goat.Player is again able to choose a door.Is better for him to change the door or to keep his decision?Let ρ A be probability distribution of prize being behind any door (more precisely: where player think it is) and ρ B probability distribution of choosing the door for the first time by player.Let ρ C|AB be host's selection of door probability distribution -it is conditional probability, because host knows where the prize is and which door was chosen by player and he uses this knowledge.
Using eq. ( 14) it is easy to compute joint probability of all states Let as assume that player chooses b-th door (doors are numbered 0, 1 and 2).It means that the measurement |b b| was performed on system B. That fact can be used for updating probability distribution ρ ABC in following way Moreover, there is also measurement |c c| = |b b| made on system C, because host does not open the door which player has chosen.Then we have new conditional state Probability distribution of variable A after those measurements is given by

Non-entangled case
Let ρ A and ρ B be uniform distributions and ρ C|AB be distribution defined in Table 1 ρ There is no entanglement between A and B, and the inferring structure can be represented as in Fig. 3.
Suppose, that player choose door 0 and subsequently the host opens door 1.Then, using eq.( 21)-(25) we get Solution above means, that it is twice better to change the door.The result is the same as in the classical case.Dependencies with an entanglement between prize position and player decision of choosing door.

Entangled case
Let ρAB , ρAB ∈ L(C 3 ⊗ C 3 ) = L(C 9 ) be defined as After marginalizing those matrices we get hence they match distributions from the previous section.That inferring structure can be described by Fig.
Suppose player can choose one of two cases below:  3. host opens the door 2 (we make measurement |0 0| on subsystem C).
Suppose first or second case takes place.Graphs intersects only in points (0, 1) and (1, 0).It means that using trivial mixtures (λ = 0 or λ = 1), it is not necessary to wait for host response and moreover player can point the right door directly after choosing for the first time.However, in other cases (when λ ∈ (0, 1)), player can decrease the probability of finding the prize behind door 0, if he wait for a host move.At the same time he increases the probability of finding prize behind door 2.
If hosts opens door 2, we can compute that player again knows exactly where the prize is hidden.
We supposed, that player chooses at the first time door 0, but similar computations leads to the same conclusions in other cases.

Conclusions
Probability theory can be adapted and generalized into quantum systems.Classical random variables can be expressed as diagonal density matrices of separable states.Proposed formalism of probability theory is more general than classical approach.We show that quantum entanglement can be used for construction of reasoning structures based on the concept of probabilistic graphical models.
In Monty Hall example, we show that there exist differences between classical and quantum approaches for probabilistic reasoning.

Figure 1 :
Figure 1: Graphical interpretation of relationship between variables represented by quantum states.The distribution of B is expressed as ρ B = Tr A (ρ B|A ⋆ ρ A ).

Figure 2 :
Figure 2: Inferring structure with three random variables.
Figure 4:Dependencies with an entanglement between prize position and player decision of choosing door.

4 .|2 = 1 .
Using the same matrix ρ C|AB as in section 5.1 and assuming again that |b = |0 and |c = |1 : It can be interpreted as follows: the choice of player has impact of the position of prize.Now define a mixture of the two operators above:

Figure 5 :
Figure 5: Probability of finding prize behind door 0. Dashed line describe the case with opening the door 1 by host.It means that measurement was performed on subsystem C. Solid line shows the case without opening the door.

Table 1 :
Probability of choosing door by host C B A