Perfect NOT transformation and conjugate transformation

The perfect NOT transformation, probabilistic perfect NOT transformation and conjugate transformation are studied. Perfect NOT transformation criteria on a quantum state set $S$ of a qubit are obtained. Two necessary and sufficient conditions for realizing a perfect NOT transformation on $S$ are derived. When these conditions are not satisfied we discuss a probabilistic perfect NOT transformation (gate). We construct a probabilistic perfect NOT machine (gate) by a general unitary-reduction operation. With a postselection of the measurement outcomes, the probabilistic NOT gate yields perfectly complements of the input states. We prove that one can realize probabilistically the NOT gate of the input states secretly chosen from a certain set $S=\{|\Psi_1>, |\Psi_2>,..., |\Psi_n>\}$ if and only if $|\Psi_1>, |\Psi_2>,...,$ and $|\Psi_n>$ are linearly independent. We also generalize the probabilistic NOT transformation to the conjugate transformation in the multi-level quantum system. The lower bound of the best possible efficiencies attained by a probabilistic perfect conjugate transformation are obtained.


INTRODUCTION
The basic building block of any classical information processor is the single bit, which is prepared in one of two possible states, denoted 0 or 1. However, quantum information consists of qubits, each of which has the luxury of being in a superposition of the 0 and 1 states. Since there are an infinite number of superposition states, quantum systems have a much richer and more interesting existence than their classical counterparts. The superposition of states also makes the properties of quantum information quite different from that of its classical counterpart. Whereas the copying of classical information presents no difficulties, owing to the linearity of quantum mechanics, there is a quantum no-cloning theorem [1,2] which asserts that it is impossible to construct a device that will perfectly copy an arbitrary (unknown) state of a two-level particle. However, the quantum nocloning theorem does not rule out the possibility of either imperfect cloning [3,4] or probabilistic cloning [5]. Some applications of cloning have been presented [6][7][8]. With the progress of a quantum information theory, quantum cloning has become a quite interesting field.
There is another difference between classical and quantum information systems. It is very easy to complement a classical bit, i.e., to change the value of a bit, a 0 to a 1 and vice versa. Usually this operation can be accomplished by a NOT transformation (gate). However, in quantum information systems, changing an unknown state |Ψ = α|0 + β|1 of a qubit to its orthogonal complement |Ψ ⊥ = α * |1 − β * |0 that is orthogonal to |Ψ (i.e. inverting the state of a two-level quantum system) is impossible [9,10]. The result is that one can not design a device that will take an arbitrary qubit and transform it into its orthogonal qubit. This is because complex con-jugation of the coefficients in the NOT transformation of a qubit must be accomplished by an antiunitary transformation and cannot be performed by a unitary one. In other words, it is impossible to achieve the perfect NOT gate in quantum information systems.
However, the NOT transformation can be achieved on some states while leaving other states unchanged. Alternatively, there can be a transformation operation that approximates, at best, the NOT gate on all states, called the universal NOT gate [9,10]. In fact, the output of a quantum cloning machine, the ancilla, carries the optimal anticlone of the input state so the universal NOT gate can be accomplished as a by-product of cloning [4].
A combination of unitary evolution together with measurements is an important method in quantum information processing and often achieves very interesting results. It has been used in quantum programming [11], the purification of entanglement [12], quantum teleportation [13] and the preparation of quantum states [14]. Recently, by using this method, Duan and Guo designed a probabilistic quantum cloning machine [5]. With a postselection of the measurement results, the machine outputs perfect copies of the input states.
In this paper, the perfect NOT, probabilistic perfect NOT and conjugate transformations are investigated. We present the criteria for a perfect NOT transformation on a quantum state set S of a qubit. Two necessary and sufficient conditions for realizing a perfect NOT transformation on S are derived and this paper discusses how to build a device to achieve probabilistic perfect NOT transformations when there is no perfect NOT transformation on the state set S. With certain nonzero probabilities of success, this device transforms an arbitrary unknown input state into its orthogonal complement. We also generalize the probabilistic NOT transformation to the conjugate transformation in the multi-level quantum system. Furthermore, the lower bound of the best possible efficiencies attained by a probabilistic perfect conjugate transformation is obtained.

CONDITIONS REQUIRED FOR PERFECT NOT TRANSFORMATIONS
In order to aid the analysis, we first state a Lemma of Duan and Guo [5]: Lemma. If two sets of states |φ 1 , |φ 2 , · · · , |φ n and |φ 1 , |φ 2 , · · · , |φ n satisfy the condition then there exists a unitary operator U such that U |φ i = |φ i , (i = 1, 2, · · · , n). Let S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n } be a set of states of a qubit. When the quantum states of S satisfy based on Lemma we can find a unitary transformation U such that |Ψ ⊥ i = U |Ψ i for i = 1, 2, · · · , n. It is easy to see that Ψ which shows that the condition Eq.(2) is equivalent to Ψ i |Ψ j = Ψ i |Ψ j * for i, j = 1, 2, · · · , n. This, in turn, implies that all innerproducts of the quantum states in the set S are real. Hence we arrive at the following conclusion: Theorem 1. Suppose that S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n } is the set of quantum states. Then a perfect NOT transformation (gate) U on the set S can be realized by a unitary transformation (i.e., there is a unitary transformation U such that U |Ψ i = |Ψ ⊥ i ) if and only if Ψ i |Ψ j = Ψ i |Ψ j * , i, j = 1, 2, · · · , n.
It turns out that if S contains all points of a Bloch sphere [15] of a qubit, one can not realize the perfect NOT transformation on the set S.
Obviously, in Theorem 1 we only consider the case without an ancilla (probe). Now, let us introduce a probe P with the initial state |P (0) . By Lemma, if for arbitrary i, j = 1, 2, · · · , n, then there exists a unitary transformation U , such that It means we can realize the perfect NOT transformation on the quantum state set S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n } with the assistance of the ancilla (probe).
Based on the above argument we obtain following conclusion: Theorem 2. Suppose that the quantum state set S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n } satisfies Ψ i |Ψ j = 0. Then, a perfect NOT transformation (gate) on the state set S can be realized by a unitary transformation acting on the system and a probe if and only if Eq.(13) hold.
Note that, when the quantum state set S contains only two quantum states |Ψ 1 , |Ψ 2 , Eq.(13) can always hold. Therefore, the perfect NOT transformation (gate) on the state set S of two arbitrary quantum states |Ψ 1 , |Ψ 2 can always be realized.
Clearly, if there are no quantum states |P (1) , |P (2) , · · · , |P (n) satisfying Eq.(4) for the quantum state set S, then one can not design a perfect NOT gate for this state set S. In this case one can only consider the universal-NOT or the probabilistic perfect NOT gate. As the universal-NOT gate has been well studied [9,10], in the next section, we will only discuss the probabilistic perfect NOT gate in detail.

PROBABILISTIC PERFECT NOT TRANSFORMATION
The definition of a probabilistic perfect NOT gate is that for a quantum state set S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n }, there is a unitary transformation together with a measurement, which when combined with a postselection of measurement results, makes an arbitrarily unknown input quantum state |Ψ i transform into its orthogonal complement |Ψ ⊥ i with certain nonzero probability of success. That is, for a quantum state set S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n }, if there exists a unitary operation U and a measurement M , which together yield the following evolution: then a probabilistic NOT gate is said to have been built. The combination of a unitary evolution operation and a measurement is very general and can be used to describe any operation in quantum mechanics [16]. Obviously, we can not build a probabilistic NOT gate for any arbitrary quantum state set S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n }, so it is very important to find the conditions that the quantum state set S should be satisfied in order to construct a probabilistic perfect NOT gate.
The unitary evolution of the qubit A and probe P can be described by the following equation where |P 0 and |P (i) are normalized states of the probe P (not generally orthogonal) and |Φ (1) AP , · · · , and |Φ (n) AP are n normalized states of the composite system AP (not generally orthogonal). We assume that in Eq.(15) the coefficients before the states |Ψ ⊥ i |P (i) , and |Φ (i) AP are positive real numbers. Let S 0 be the subspace spanned by the states |P (1) , |P (2) , · · · , |P (n) . In order to realize the probabilistic perfect NOT transformation, we must require that after the unitary evolution a measurement of the probe with a postselection of the measurement results should project its state into the subspace S 0 . After this projection, the state of the system A should be |Ψ ⊥ i . Therefore, all of the states |Φ (i) AP , lie in a space orthogonal to S 0 and can be represented by the following equation With above restriction, inter-inner-products of Eq.(15) yield the following matrix equation where are n×n matrices and E n is the n× n identity matrix. The diagonal efficiency matrix Γ is defined by Γ = diag(γ 1 , γ 2 , ..., γ n ); therefore, . According to result of Duan and Guo [5], Y is a positive-semidefinite matrix. Thus, AP such that Eq.(17) holds. By Lemma the states |Ψ 1 , |Ψ 2 , ..., and |Ψ n are able to be probabilistically transformed to their respective orthogonal complement states. Thus we have the following theorem: Theorem 3. The states |Ψ 1 , |Ψ 2 , ..., and |Ψ n can be probabilistically perfectly transformed to their respective orthogonal complement states if and only if there exist a diagonal positive-definite matrix Γ and |P (i) (i = 1, 2, ..., n) such that the matrix are n × n matrices, and |P (i) (i = 1, 2, · · · , n) are quantum states of a probe.
Theorem 3 is very general, and for the linearly independent quantum state set S we have the conclusion: Theorem 4. The states secretly chosen from the set S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n } can be probabilistically transformed into their respective orthogonal complements by a general unitary-reduction operation, if |Ψ 1 , |Ψ 2 , · · · , and |Ψ n are linearly independent.
Proof : Suppose the Hilbert space of the probe P is an n p -dimensional space, where n p ≥ n + 1. We use |P 0 , |P 1 , ..., and |P n to denote n+ 1 orthonormal states of a probe P . If there exists a unitary operator U that satisfies where |Φ (j) (j = 1, 2, · · · , n) stand for n normalized states of the system (not generally orthogonal) and ϕ i are real numbers, then after the evolution a measurement of the probe P is followed. Eq.(18) is a special case of Eq. (15). The NOT transformation is successful, and the output state of the system is |Ψ ⊥ i , if and only if the measurement outcome of the probe is |P 0 . Evidently, the probability of success ( obtaining |P 0 ) is γ i . For any input state |Ψ i , the probabilistic NOT device should succeed with a nonzero probability. This, in turn, implies that all of the γ i must be positive real numbers. Hence, the evolution (14) can be realized if Eq.(18) holds with positive efficiencies γ i . The n × n inter-inner-products of Eq.(18) yield the equation where the n × n matrices C = [c ij ], . By considering Lemma we know that if there exists a diagonal positive-definite matrix Γ satisfied Eq.(19), then one can realize the unitary evolution (14).
Suppose that the minimum eigenvalue of X (1) is c and the maximum eginvalue of X (⊥) is d. Then there must exist a positive number ε such that c − εd > 0. (20) Let B = (b 1 , b 2 , ..., b n ) T be an arbitrary nonzero n dimensional vector. Then It also follows that where E is the n × n identity matrix. Presume that X (1) and X (⊥) are diagonalized by the unitary matricies U and V , respectively. Eq.(22) can then be rewritten as We use c 1 , c 2 , ..., c n and d 1 , d 2 , ..., d n to denote the eginvalues of matrixes X (1) and X (⊥) , respectively. It is easy to deduce Obviously, there must be a diagonal matrix Γ = diag(γ 1 , γ 2 , ..., γ n ) with γ i > 0 that satisfies Therefore, there is a diagonal matrix √ Γ such that is positive definite. Suppose that the unitary matrix W diagonalizes the Hermitian matrix where all of the eigenvalues m 1 , m 2 , ..., m n are positive real numbers. We can then choose the matrix C in Eq.(19) to be Thus, there exists a diagonal positive definite efficiency matrix Γ such that Eq.(19) holds and the proof of Theorem 4 is complete.

CONJUGATE TRANSFORMATION OF A MULTI-LEVEL QUANTUM SYSTEM
In this section we discuss conjugate transformation of a multi-level quantum system (qudit). Suppose that the dimension of a Hilbert space for the quantum system is d. An arbitrary quantum state of the system can be written as where α i are complex numbers and {|i } is an orthonormal basis. Let us define a conjugate transformation T as Obviously, a perfect NOT transformation equals U T for a qubit, where U = 0 −1 1 0 is a unitary transformation. We call |Ψ T the conjugate state of quantum state |Ψ . Evidently, one can not design a machine that will take an arbitrary quantum state |Ψ and transform it into its conjugate state |Ψ T because of the need for complex conjugation of the coefficients in the transformation, which must be accomplished by an antiunitary transformation and cannot be performed by a unitary one. By Lemma, we can also assert that this kind transformation is impossible on a general quantum state set S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n } of a qudit, since Ψ i |Ψ j = Ψ T i |Ψ T j = Ψ i |Ψ j * for two arbitrary quantum states in the set S.
However, by the argument similar to the qubit case, we do have the following conclusions: Theorem 1'. A perfect conjugate transformation on the state set S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n } of a qudit can be realized by a unitary transformation if and only if all inner-products of the quantum states in the set S are real.
Theorem 2'. Suppose that the quantum state set S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n } of a qudit satisfies Ψ i |Ψ j = 0. Then a conjugate transformation on the state set S can be realized by a unitary transformation acting on the system and a probe if and only if Eq.(13) hold.
Next we investigate the best possible efficiencies γ i attained by a probabilistic conjugate transformation.

SUMMARY
In conclusion, we have investigated a perfect NOT transformation on a quantum state set S of a qubit and derived two necessary and sufficient conditions for realizing a perfect NOT transformation on S. A probabilistic perfect NOT transformation (gate) was constructed by a general unitary-reduction operation. With a postselection of the measurement outcomes, the probabilistic NOT gate was shown to yield perfect respective orthogonal complements of the input states. We also show that one can construct a probabilistic perfect NOT gate of the input states secretly chosen from a certain set S = {|Ψ 1 , |Ψ 2 , · · · , |Ψ n } if |Ψ 1 , |Ψ 2 , · · · , and |Ψ n are linearly independent. Furthermore, we generalize the probabilistic NOT transformation to the conjugate transformation in a multi-level quantum system. The lower bound of the best possible efficiencies attained by a probabilistic perfect conjugate transformation was obtained.