1.5.5. Quantum copying: Beyond the no-cloning theorem[16]

We analyze the possibility of copying (that is, cloning) arbitrary states of a quantum-mechanical spin-1/2 system.

1.5.5.1. INTRODUCTION

The ideal copying process is described by the transformation

$$|s\rangle_a|Q\rangle_x\rightarrow|s\rangle_a|s\rangle_b|\tilde{Q}\rangle_x\tag{1.1}$$

where $|s\rangle_a$ is the in state of the original mode and $|Q\rangle_x$ is the in state of the copying device.

It have been shown that if a particle in an arbitrary mixed state is sent into a device and two particles emerge, it is impossible for the two reduced density matrices of the two-particle state to be identical to the input density matrix. Nevertheless, it is still an open question how well one can copy quantum states, i.e., when ideal copies are not available how close the copy state (out state in the mode b) can be to the original state (i.e., $|s \rangle_a$ ). The other question to answer is what happens to the original state $|s \rangle_a$ after the copying.

With the advent of quantum communication, e.g., quantum cryptography, and quantum computing, understanding the limits of the manipu- lations we can perform on quantum information becomes important.

1.5.5.2. WOOTTERS-ZUREK QUANTUM-COPYING MACHINE AND NONCLONING THEOREM

In their paper Wootters and Zurek analyzed the copying process defined by the transformation relation on basis vectors $|0\rangle_a$ and $|1\rangle_a$ : $$|0\rangle_a|Q\rangle_x\rightarrow|0\rangle_a|0\rangle_b|Q_0\rangle_x$$

$$|1\rangle_a|Q\rangle_x\rightarrow|1\rangle_a|1\rangle_b|Q_1\rangle_x$$

we can assume that $$\sideset{_x}{_x} {\langle Q|Q\rangle}=\sideset{_x}{_x} {\langle Q_0|Q_0\rangle}=\sideset{_x}{_x} {\langle Q_1|Q_1\rangle}=1$$ The Wootters-Zurek (WZ) quantum-copying machine (QCM) is defined in such a way that the basis vectors $|0\rangle_a$ and $|1\rangle_a$ are copied (that is, cloned) ideally, that is, for these states the relation (1.1) is fulfilled.

The Hilbert-Schmidt norm of an operator  is given by $$\|\hat{A}\|_2=[\text{Tr}(\hat{A}^\dagger\hat{A})]^{1/2}$$ Our distance between the density matrices $\hat{\rho}_1$ and $\hat{\rho}_2$ is then $$D=(\|\hat{\rho}_1-\hat{\rho}_2\|_2)^2$$ Other measures of the similarity of two density matrices have been used. Schumacher has advocated the use of fidelity which is defined as $$F=\text{Tr}(\hat{\rho}_1^{1/2}\hat{\rho}_2\hat{\rho}_1^{1/2})^{1/2}\tag{2.1}$$

1.5.5.3. INPUT-STATE-INDEPENDENT QUANTUM-COPYING MACHINE

When using this 'universal' quantum-copying machine (UQCM) superposition states $$|s\rangle_a=\alpha|0\rangle_a+\beta|1\rangle_a$$ are copied equally well for any value of a in the sense 2 that the distances $D_a=\text{Tr}[\hat{\rho}_a^{(out)}-\hat{\rho}_a^{(id)}]^2$ and $D_{ab}=\text{Tr}[\hat{\rho}_{ab}^{(out)}-\hat{\rho}_{ab}^{(id)}]^2$ do not depend on the parameter $\alpha$.

The most general quantum-copying transformation rules for pure states on a two-dimensional space can be written as $$|0 \rangle_a|Q \rangle_x\rightarrow\sum^1_{k,l=0}|k \rangle_a|l \rangle_b|Q_{kl} \rangle_x,$$ $$|1 \rangle_a|Q \rangle_x\rightarrow\sum^1_{m,n=0}|m \rangle_a|n \rangle_b|Q_{mn} \rangle_x,$$ where the states $|Q_{mn} \rangle_x$ are not necessarily orthonormal for all possible values of $m$ and $n$. The general copying transformation is very complex and it involves many free parameters $\sideset{x}{x}{ \langle Q_{kl}|Q_{mn}\rangle }$ which characterize the copying machine. In what follows we will concentrate our attention on one particular copying transformation which fulfills our demands as described above. We propose the transformation $$|0 \rangle_a|Q \rangle_x\rightarrow|0 \rangle_a|0 \rangle_b|Q_0 \rangle_x+[|0 \rangle_a|1 \rangle_b+|1 \rangle_a|0 \rangle_b]|Y_0 \rangle_x$$ $$|1 \rangle_a|Q \rangle_x\rightarrow|1 \rangle_a|1 \rangle_b|Q_1 \rangle_x+[|0 \rangle_a|1 \rangle_b+|1 \rangle_a|0 \rangle_b]|Y_1 \rangle_x$$ which is an obvious generalization of the WZ QCM. Due to the unitarity of the transformation (3.2) the following relations hold: $$\sideset{x}{x}{ \langle Q_i|Q_i \rangle}+2\sideset{x}{x}{ \langle Y_i|Y_i \rangle}=1,\quad i=0,1$$ $$\sideset{x}{x}{ \langle Y_0|Y_1 \rangle}=\sideset{x}{x}{ \langle Y_1|Y_0 \rangle}=0$$ There are still many free parameters to specify, therefore we will further assume that the copying-machine state vectors $|Y_i \rangle_x$ and $|Q_i \rangle_x$ are mutually orthogonal: $$\sideset{x}{x}{ \langle Q_i|Y_i \rangle}=0, \quad i=0,1$$ and that $$\sideset{x}{x}{ \langle Q_0|Q_1 \rangle}=0$$ The Hilbert-Schmidt norm $$D_a=2 \xi^2(4 \alpha^4-4 \alpha^2+1)+2 \alpha^2(1- \alpha^2)(\eta-1)^2$$ where we have introduced the notation $$\sideset{x}{x}{ \langle Y_0|Y_0 \rangle}=\sideset{x}{x}{ \langle Y_1|Y_1 \rangle}=\xi$$ $$\sideset{x}{x}{ \langle Y_0|Q_1 \rangle}=\sideset{x}{x}{ \langle Q_0|Y_1 \rangle}=\sideset{x}{x}{ \langle Q_1|Y_0 \rangle}=\sideset{x}{x}{ \langle Y_1|Q_0 \rangle}=\eta/2$$ We find that if the parameters j and h are related as $$\eta=1-2 \xi$$ then the norm D a is input-state independent and it takes the value $$D_a=2 \xi^2$$ $$\frac{\partial}{\partial \alpha^2}D_{ab}^{(2)}=0$$ we find $\xi =1/6$. For this value of$\xi$ the norm $D_{ab}$is $\alpha$independent and its value is equal to 2/9.

1.5.5.3.1. Some properties of the UQCM

1. Any measurement performed on mode $b$ affects the state of mode $a$.
2. Once we have found the basis in which both density operators $\hat{\rho}^{(id)}$ and $\hat{\rho}^{(out)}$ are diagonal we can easily find the value of the fidelity parameter $F$ a as introduced by Schumacher. The fidelity parameter which we are interested in is given by Eq.(2.1) with $\hat{\rho}_1=\hat{\rho}_a^{(id)}$ and $\hat{\rho}_2=\hat{\rho}_a^{(out)}$ In our case the fidelity is equal to a constant value $\sqrt{5/6}$ for all input states. We can conclude that the UQCM has that universal property to be input-state independent, that is, all pure states are copied equally well.

1.5.5.4. MEASUREMENT OF THE ORIGINAL AND THE COPY STATE AT THE OUTPUT OF QCM

In summary the output from the UQCM has the following property. If any projection is measured in the b mode the unconditioned a-mode ensemble which results is close to the ideal output state, i.e., the input state, and can be used to find the expectation value of any a-mode operator in the ideal output state. In addition, the b-mode measurement provides us with information about the input state.

1.5.5.5. COPYING STATES IN THE NEIGHBORHOOD OF GIVEN STATE

What we can do with the output of this copying machine is to use it to calcu- late the expectation values of any operator which annihilates this state.

1.5.5.6. CONCLUSIONS

A problem with all of these machines is that the copy and original which appear at the output are entangled. This means that a measurement of one affects the other. We found, however, that a nonselective measurement of one of the output modes will provide information about the input state and not disturb the reduced density matrix of the other mode. Therefore the output of these copying machines is useful.