State Dependent N-M Cloning

This example protocol achieves the functionality of Quantum Cloning. A state-dependent cloner is a Quantum Cloning Machine (QCM) which is dependent on the input states. The main purpose of a state-dependent cloner is to design a QCM which produces better copies (compared to the universal QCMs) by having some partial information about the input states. We investigate the general $N \rightarrow M$ approximate state-dependent cloner which transforms two set of non-orthogonal states (consisting of N states) to M identical clones with a special constraint on the inner product of the final states. For the special case of $1 \rightarrow 2$ cloner, it is shown that, in general, this machine acts much better than the universal cloning protocol for two special set of states.

Tags: Non-Universal Cloning, State-Dependent Cloning, Building Blocks, Quantum Cloning, Universal Cloning, asymmetric cloning, copying quantum states, Quantum Functionality, Specific Task,symmetric or Optimal or Symmetric Cloning, Probabilistic Cloning

Assumptions

• We assume that no ancillary states (test qubits) are not needed for this state-dependent protocol.
• We assume that this state-dependent QCM is symmetric.
• We assume that the transformation is unitary, acting on the Hilbert space of $M$ qubits and thus, the following relation holds between the inner product of input and output states:
  

$\langle \alpha _{NM}|\beta _{NM}\rangle =(\langle a|b\rangle )^{N}=S^{N}$

• We assume that protocol is optimal on the input states meaning that it will give the maximum possible fidelity value on average for a given set of states.

Outline

This state-dependent QCM can only act effectively on two sets of states. These states are non-orthogonal (orthogonal case is trivial). For a state-dependent cloner, the only thing that we need is to perform a transformation which takes N identical input states and $M-N$ blank states, to M copies such that it yields the best copies for the given special input states. It is important to note that this transformation is useful when we have a prior information about the input states.

Notations Used

• $|a\rangle ,|b\rangle :$ Two nonorthogonal input states
• $S:$ The inner (scalar) product of two input states $|a\rangle$ and $|b\rangle$
• $|\alpha _{NM}\rangle ,|\beta _{NM}\rangle ,$ Output states produced by the state-dependent QCM acting on input states
• $F_{g}^{opt}(N,M):$ The global optimal fidelity for $N\rightarrow M$ state-dependent cloning protocol
• $F_{sd}(N,M):$ The fidelity of the subsystem for $N\rightarrow M$ state-dependent cloning protocol

Properties

• State-dependent cloning machine only transforms a set of two non-orthogonal input states, parametrized as follows:

$|a\rangle =cos\theta |0\rangle +sin\theta |1\rangle ,$
$|b\rangle =sin\theta |0\rangle +cos\theta |1\rangle ,$
where $\theta \in [0,\pi /4]$ and their scalar product (or inner product) is specified as $S=\langle a|b\rangle =sin2\theta$.

• The condition that transformation should be unitary adds following constraint on the scalar product of final states:
  

$\langle \alpha _{NM}|\beta _{NM}\rangle =(\langle a|b\rangle )^{N}=S^{N}$

• Fidelity Claims
  • Optimal global fidelity: The average global fidelity of the machine for both sets of input states in terms of the inner(scalar) product of the two sets will be:
    

$F_{g}^{opt}(N,M)={\frac {1}{2}}(1+S^{N+M}+{\sqrt {1-S^{2N}}}{\sqrt {1-S^{2M}}})$

• • Subsystem fidelity: The fidelity of the density matrix of the subsystem with the original state $a$ in terms of the inner product (similar fidelity for the subsystem b):
    

$F_{sd}(N,M)=\langle a|\rho _{\alpha }|a\rangle =A^{2}(1+S^{2}+2S^{M})+B^{2}(1+S^{2}-2S^{M})+2AB(1-S^{2})$
where the reduced density matrix of the subsystem $a$ (The quantum state of a subsystem in density matrix representation) is described as:
$\rho _{\alpha }=(A+B)^{2}|a\rangle \langle a|+(A-B)^{2}|b\rangle \langle b|+(A^{2}-B^{2})S^{M-1}(|a\rangle \langle b|+|b\rangle \langle a|)$
A and B are presented in the Pseudo Code section.

Pseudo Code

General Case For more generality, we use the density matrix representation of the states which includes mixed states as well as pure states. For a simple pure state $|\psi \rangle$ the density matrix representation will be $\rho _{\psi }=|\psi \rangle \langle \psi |$. Let us assume the initial qubit to be in an unknown state $\rho _{\psi }$. Our task is to clone this qubit universally, i.e. input-state independently, in such a way, that we can control the scaling of the original and the clone at the output. In other words, we look for output which can be represented as below:
$\rho _{a}^{out}=s_{0}\rho _{\psi }+{\frac {1-s_{0}}{2}}{\hat {I}}$
$\rho _{b}^{out}=s_{1}\rho _{\psi }+{\frac {1-s_{1}}{2}}{\hat {I}}$
Here we assume that the original qubit after the cloning is “scaled” by the factor $s_0$, while the copy is scaled by the factor $s_{1}$. These two scaling parameters are not independent and they are related by a specific inequality.

Stage 1 Cloner State Preparation

1. Prepare the original qubit and two additional blank qubits $m$ and $n$ in pure states: $|0\rangle _{m_{0}}\otimes |0\rangle _{n_{0}}\equiv |00\rangle$
2. Prepare $|\psi \rangle _{m_{1},n_{1}}=c_{1}|00\rangle +c_{2}|01\rangle +c_{3}|10\rangle +c_{4}|11\rangle$, where the complex $c_{i}$ coefficients will be specified so that the flow of information between the clones will be as desired.
   At this stage the original qubit is not involved, but this preparation stage will affect the fidelity of the clones at the end of the process.
3. To prepare the $|\psi \rangle _{m_{1},n_{1}}$ state, a Unitary gate must be performed so that:

$U|00\rangle =|\psi \rangle _{m_{1},n_{1}}$

• Use following relations to specify $c_j$ in terms of $a$ and $b$:
  

$c_{1}={\sqrt {\frac {s_{0}+s_{1}}{2}}}$, $c_{2}={\sqrt {\frac {1-s_{0}}{2}}}$, $c_{3}=0$, $c_{4}={\sqrt {\frac {1-s_{1}}{2}}}$
these $c_{j}$ satisfy the scaling equations and also the normalization condition of the state $|\psi \rangle _{m_{1},n_{1}}$. They are being used to control the flow of information between the clones

Stage 2 Cloning Circuit

• The cloning circuit consists of four CNOT gates acting on original and pre-prepared qubits from stage 2. We call the original qubit $|\psi \rangle _{in}$, ``first qubit", the first ancillary qubit of $|\psi\rangle_{m_1,n_1}$, ``second qubit" and the second one, ``third qubit". The CNOT gates will act as follows:

1. First CNOT acts on first and second qubit while the first qubit is control and the second qubit is the target.
2. Second CNOT acts on first and third qubit while the first qubit is control and the third qubit is the target.
3. Third CNOT acts on first and second qubit while the second qubit is control and the first qubit is the target.
4. Forth CNOT acts on first and third qubit while the third qubit is control and the first qubit is the target.

• Mathematically the cloning part of the protocol can be shown as:
  

$|\psi \rangle _{out}=P_{3,1}P_{2,1}P_{1,3}P_{1,2}|\psi \rangle _{in}|\psi \rangle _{m_{1},n_{1}}$

Stage 3 Discarding ancillary state

• Discard one of the extra states. The output states will be the first and second (or third) output.
  
  

Special case with bell state:
Stage 1 Cloner state preparation

1. Prepare the original qubit and two additional blank qubits $m$ and $n$ in pure states: $|0\rangle _{m_{0}}\otimes |0\rangle _{n_{0}}\equiv |00\rangle$
2. Prepare Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi\rangle_{m_1,n_1} = a|\Phi^{+}\rangle_{m_1,n_1} + b|0\rangle_{m_1} |+\rangle_{n_1}} , where $|\Phi ^{+}\rangle$ is a Bell state and $|+\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )$. In this case, the density matrix representation of the output states will be:
   

$\rho _{a}^{out}=(1-b^{2})|\psi \rangle \langle \psi |+{\frac {b^{2}}{2}}{\hat {I}}$
$\rho _{b}^{out}=(1-a^{2})|\psi \rangle \langle \psi |+{\frac {a^{2}}{2}}{\hat {I}}$
Stage 2 Cloning Circuit

• The cloning circuit is exactly the same as the general case. after the cloning circuit, the output state will be:
  

$a|\psi \rangle _{in}|\Phi ^{+}\rangle _{m,n}+b|\psi \rangle _{m}|\Phi ^{+}\rangle _{in,n}$ The reduced density matrix of two clones A and B can be written in terms of their fidelities:
$\rho _{a}^{out}=F_{a}|\psi \rangle \langle \psi |+(1-F_{a})|\psi ^{\perp }\rangle \langle \psi ^{\perp }|$
$\rho _{b}^{out}=F_{b}|\psi \rangle \langle \psi |+(1-F_{b})|\psi ^{\perp }\rangle \langle \psi ^{\perp }|$
Stage 3 Discarding ancillary state

• The same as the general case.

Further Information

One of the most important applications of quantum cloning is to analyze the security of Quantum Key Distribution (QKD) protocols. Usually, an eavesdropper is supposed in QKD protocols os assumed to be able to perform any attacks, including copying of states used by parties involved in the protocol. Special states which are being used in different QKD protocols will lead to different state-dependent cloners for analyzing cloning attacks.