Distinct correlation properties of entanglement pay the way for novel models of informa-tion and computainforma-tion and reveal the fundamental features of quantum phenomenon. We
have seen in the thesis that the correlation criteria proposed provide a way to analyze the correlation structure of multipartite entanglement. In our preliminary result, the cri-teria can be used to construct Bell inequalities for three four-level systems, which shows that there may exist a family of Bell inequalities for many qudits where each member is comprised of correlators. Furthermore, how to detect genuine multipartite entanglement that are inherent in many-qudit singlet and valence-bound solid states is still open. These entangled states are crucial for quantum information and computation. Then one of our future works is to design a method based on correlators to investigate on the structures of these states. As for entanglement purification, improvement of purification yield will be the next topic. Since the standard protocol for purifying entangled qubit pairs relies on successful controlled-NOT operations and on certain results of measurements, desig-nations of new models that can be achieved deterministically or in a more deterministic way will be helpful for raising the yield of purification. The number of controlled-NOT gates also affects the complexity of an encoder-decoder circuit to perform the five-qubit, single-error correction protocol. The simplest circuit consists of six controlled-NOT gates and three single-qubit operations presented in this thesis and proposed by Braunstein and Smolin might not be improved further. A more convincible proof will be given in the future work. As for generation of entangled pairs or multipartite entangled qubits, preparation of remote entangled states by controlling a media system with a high gen-eration of yield is crucial. How to control the third quantum system in an experiment reliably as the case discussed in the thesis will be investigated elsewhere. For generating entangled photons with SPDC sources, investigations on the quality of entangled photons and coincidence detections are also the future topics for entanglement generations.
Quantum correlations provide novel ways of communications with high security, e.g., the Ekert protocol for key distribution. However, it is worth discussing whether entangle-ment is necessary for quantum communication. In Ref. [193], it has shown that the Ekert protocol [29] based on maximally entangled qubits is equivalent to the scheme of Bennett and Brassard [34] using on nonorthogonal states. It implies that in some quantum
commu-nication tasks utilization of entanglement is not the only method for reliable achievement [194]. Since entanglement can be replaced by separable quantum systems, the processes required for entangled states, e.g., entanglement purification, can be replaced by simple single-qubit operations. We propose an example for quantum secret sharing to discuss it further. With a simple protocol proposed, we show that secret sharing tasks can be performed without assistance of entanglement.
A sender, called Sophie, wants to share a confidential message with her friends Alice and Bob. Instead of giving the whole message to Alice and Bob, Sophie splits the message into two pieces and prepares to send each individual one to Alice and Bob respectively.
The hope of Sophie is that her confidential message can be determined faithfully only when Alice and Bob combine their individual pieces. For protecting the security of the message from selfish actions of any eavesdroppers or dishonest party, Sophie realizes that she cannot send the individual messages to Alice and Bob directly without invoking any secret-sharing protocols.
Quantum mechanics specifies that quantum states can exist in multiple eigenstates simultaneously, i.e., superposition, and measurement of a variable will yield one of the eigenvalues corresponding to the observable with a specific probability and makes a col-lapse of the state vector. Furthermore, quantum states can not be cloned perfectly, and through unitary operations, they can be transformed coherently. An utilization of these quantum mechanical features of physical states and associated operations is sufficient to realize our protocol. In the scenario of quantum secret sharing, Sophie wants Alice to possess a state |sai and Bob to possess another one |sbi. Each party has no information about the state of the other party, and Alice and Bob can share the state |sai ⊗ |sbi only when they combine their own states |sai and |sbi. Let us assume that sa(b) ∈ {0, 1}
and {|0i , |1i} are eigenstates of Pauli matrix σz. A quantum state can be changed from
|0ia⊗ |0ib to |sai ⊗ |sbi by applying unitary transformations to |0ia⊗ |0ib :
|sai ⊗ |sbi = UDUC|0ia⊗ |0ib, (11.1)
where UC and UD are unitary operators. It is worth noting two points involved in the state evolution:
(1) If a specific UC is chosen by Sophie for the state transformation, the operator UD should be consequently fixed for |sai ⊗ |sbi.
(2) We assume that Alice and Bob know that |sai ⊗ |sbi evolves from |0ia ⊗ |0ib. However, since they have no information about both |sai and |sbi before secret sharing, giving them only the operator UC or UD cannot help them to figure |sai ⊗ |sbi out with certainty unless one provides them both UC and UD.
With these two facts, a simple protocol is designed to satisfy the needs of Sophie.
Firstly, Sophie can randomly choose a UC from a set of operators and apply it to |0ia⊗
|0ib, and then she send each individual qubit to Alice and Bob. It is clear that UC is unknown to both Alice and Bob. When both of Alice and Bob have received the qubits, according the operator UC chosen, Sophie announce which UD should be used by Alice and Bob. In an ideal situation where any eavesdropper and cheat are excluded from considerations, Alice and Bob can reconstruct the state |sai and |sbi with certainty if they follow Sophie’s instruction for UD. When considering eavesdropping, if any selfish actions of eavesdroppers or dishonest party change Sophie’s preparation UC|0ia ⊗ |0ib
in transit, the subsequent operation UD shall not transfer the qubits to |sai ⊗ |sbi and then the message can not be reconstructed with certainty. This effect on the secret states can be utilized to expose eavesdroppers. For instance, in our protocol |0ia ⊗ |1ib and
|1ia⊗ |0ib represent the logical bits 0 and 1 respectively, whereas the states |0ia⊗ |0ib and
|1ia⊗ |1ib are used to detect eavesdroppers, which means that Sophie shall be aware of eavesdroppers when she find that the result of combination of Alice and Bob is |0ia⊗ |0ib
or |1ia⊗ |1ib and is not consistent with her designation of |sai ⊗ |sbi.
With the idea introduced above, the quantum secret-sharing protocol is specified by six steps:
S1. Sophie randomly choose two local unitary operators Ca and Cb, where Ca, Cb ∈
{X+, X−, Y+, Y−} and
and applies Ca and Cb to the states |0ia and |0ib respectively, i.e., she prepares a product state |Ψi = |Ψai ⊗ |Ψbi, where
Ψa(b)
= Ca(b)|0ia(b).
S2. Sophie sends the qubits |Ψai and |Ψbi to Alice and Bob respectively.
S3. Through classical communication, Sophie confirms that both parties have received the qubits.
S4. Sophie announce which kinds of operators, say Da and Db, should be used by Alice and Bob to reconstruct a secret state |si = |sai⊗ |sbi . The set of operators (Ca, Cb) chosen in S1 restricts the choices of local operators in this step. If Ca(b) ∈ {U+, U−} where U = X or Y , Da(b), the operator applied by Alice (Bob), should also be in the set of operator {U+, U−}. The type + or − for U+ and U− depends on the choice of Sophie.
When Sophie has made her decision, she broadcast the choices of Da and Db in public.
The designation of (sa, sb) depends on Sophie’s message for sharing. To share a logical bit 0 or 1 with Alice and Bob, her designation is (sa = 0, sb = 1) or (sa = 1, sb = 0) respectively. Furthermore, Sophie can design (sa = 0, sb = 0) or (sa = 0, sb = 1) to examine whether Alice and Bob inform her of their results faithfully (refer to S6).
S5. First, Alice and Bob have to transform their qubits by Da and Db respectively.
When the transformations are performed, they measure their qubits in the respective orthonormal bases {|0ia, |1ia} and {|0ib, |1ib} and eventually get the output states |s′ai and |s′bi. In an ideal situation without eavesdropping, they have s′a= sa and s′b = sb with certainty. Then, they combine their results of measurement and have s′as′b.
S6. Alice and Bob identify whether their combined message is even: s′as′b ∈ {00, 11}
or odd: s′as′b ∈ {01, 10}. If the message is even, both of them should inform Sophie of this
result via classical communication. If s′as′b = 01(10), they share a bit 0 (1) with Sophie and keep a secret.
With the above protocol, quantum secret sharing can be achieved without entangle-ment. A detailed discussion of the security of the proposed protocol will be given elsewhere [195]. In addition to quantum communication, it has been shown that sophisticated quan-tum search can be performed without entanglement and the quanquan-tum interference alone suffices to reduce the complexity of query requirement [196]. The experimental demon-strations by optical implementation is reported in Ref. [197]. It would be interesting and important to find other quantum mechanical procedures that require no entanglement source or generate no multipartite correlations of quantum states at any time step.
Tightness of Bell inequalities
Every tight Bell inequality fulfills the following conditions [115]:
Condition 1. All the generators of the convex polytope must belong either to the half-space or to the hyperplane.
Condition 2. There must be 4d(d − 1) linear independent generators among the ones that belong to the hyperplane.
On the other hand, non-tight Bell inequalities satisfy only the first condition. Then, we will examine the proposed Bell inequality by these conditions for tightness. Since we have proven that the proposed Bell inequality fulfills the first condition in the third section. Then we proceed to consider the second condition for the Bell inequality. All the generators of the convex polytope are written as
G=
which can also be represented as the following form by the defined variables shown in Eq.
(2.57):