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 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 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:

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 |a\rangle = cos\theta |0\rangle + sin\theta |1\rangle,}
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 |b\rangle = sin\theta |0\rangle + cos\theta |1\rangle,}
where 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 \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 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 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:
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 \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

Input: $|a\rangle ^{\otimes N}\otimes |0\rangle ^{\otimes M-N}$ or $|b\rangle ^{\otimes N}\otimes |0\rangle ^{\otimes M-N}$
Output: $|\alpha _{NM}\rangle =(A+B)|a^{M}\rangle +(A-B)|b^{M}\rangle$ or $|\beta _{NM}\rangle =(A-B)|a^{M}\rangle +(A+B)|b^{M}\rangle$

1. Perform a unitary on N input states of the form $|x\rangle ^{\otimes N}$, with $x=a,b$ and also on $(M-N)$ blank states, which its action is discribed by:
   

$|\alpha _{NM}\rangle =U_{NM}(|a\rangle ^{\otimes N}\otimes |0\rangle ^{\otimes M-N})=(A+B)|a^{M}\rangle +(A-B)|b^{M}\rangle$,
$|\beta _{NM}\rangle =U_{NM}(|b\rangle ^{\otimes N}\otimes |0\rangle ^{\otimes M-N})=(A-B)|a^{M}\rangle +(A+B)|b^{M}\rangle$,

where,
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 A = \frac{1}{2}\sqrt{\frac{1 + S^N}{1 + S^M}}, \quad B = \frac{1}{2}\sqrt{\frac{1 - S^N}{1 - S^M}}}

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.