Phase Co-variant Cloning

This example protocol achieves the functionality of Quantum Cloning Machine (QCM). Phase-covariant cloning describes is a special state-dependent cloning which is able to make copies of a specific class of input states which satisfy a certain condition. For qubits, a phase-covariant cloner is a machine which is able to clone equatorial states ( states whose vector lies in the equator $(x-y)$ plane of the Bloch sphere). There are different phase-covariant cloning protocols for qubits. It can be done with or without an extra ancilla state and also asymmetric or symmetric. Generally, the state-dependent cloning protocol is asymmetric, meaning that copies have different fidelity (different qualities compared to the original state) and the symmetric protocol is only a special case of the asymmetric protocol.

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

Assumptions

• We assume that this state-dependent QCM can only copy equatorial states effectively.
• We assume that the transformation is a unitary transformation acting on the Hilbert space of 2 (or 3 for the with ancilla case) qubits.

Outline

Phase Variant cloner can act effectively only on states lying on the equator of Bloch sphere. These states can be described by an angle factor such as $\phi$ and aim of this protocol is to produce optimal copies of these states such that the fidelity of the copies in independent of $\phi$. This task can be done with two types of protocol: without any ancilla (test states) and with ancilla. Both methods produce copies with same optimal fidelity.

Phase-covariant cloning without ancilla

In this case a unitary transformation acts on two input states (the original state and a blank state) and produces a two-qubit state where the subsystem of each of the copies can be extracted from it. The unitary transformation depends on the shrinking factor but not on the phase. This unitary in general acts asymmetrically but it becomes a symmetric case when shrinking factor is equal to $\pi /4$.

Phase-covariant cloning with ancilla

In this case, the transformation acts on three qubits (the original state and a 2-quit ancilla). The protocol is done in three steps:

1. Prepare a Bell state $|\Phi ^{+}\rangle$
2. Perform a 3-qubit unitary
3. Discard the extra state

The fidelity of this protocol is the same as the previous protocol. However, the output states (reduced density matrices of copies) are different.

Notations

• $|\psi (\phi )\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +e^{i\phi }|1\rangle ):$ Input equatorial state
• $T:$ The general map for all the phase-covariant QCMs
• $\eta :$ The shrinking factor, showing how the density matrix of the copies has been changed after the cloning process
• $U_{pc}:$ The unitary transformation of the phase-covariant QCM without ancilla
• $U_{pca}:$ The unitary transformation of the phase-covariant QCM with ancilla
• $\rho _{A},\rho _{B}:$ The reduced density matrix describing the state of subsystem $A(B)$ after the cloning process
• $|\Phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle ):$ The Bell state. One of the maximally entangled states for 2 qubits
• $\mathbb {I} :$ The identity operation (matrix)
• $\sigma _{x},\sigma _{y},\sigma _{z}:$ Pauli Operators X,Y,Z
• $F_{A},F_{B}:$ The fidelity of the subsystem $A(B)$ showing how the first(second) copy is close to the original state. In the symmetric case, fidelity of A and B are equal and the copies are identical.

Properties

• Fidelity Claims

1. Fidelity of the asymmetric case without ancilla case:
   

$F_{A}={\frac {1}{2}}(1+cos\eta )\quad F_{B}={\frac {1}{2}}(1+sin\eta )$

1. Fidelity of the symmetric case without ancilla case: The special case of the asymmetric phase-covariant cloning with $\eta =\pi /4$. This fidelity is larger than the fidelity of the Universal QCM:
   

$F_{A,B}={\frac {1}{2}}(1+{\frac {1}{\sqrt {2}}})\approx 0.8535>{\frac {5}{6}}$

1. Fidelity of the general case with ancilla case: The same as the without ancilla case:
   

$F_{A}={\frac {1}{2}}(1+cos\eta )\quad F_{B}={\frac {1}{2}}(1+sin\eta )$

• Case 1 (without ancilla): The state of each copy which is a subsystem of this two-qubit system is described with a density matrix which can be obtained as below
  

$\rho _{A}=Tr_{B}(|\psi (\phi )_{F}\rangle \langle \psi (\phi )_{F}|)$
$\rho _{B}=Tr_{A}(|\psi (\phi )_{F}\rangle \langle \psi (\phi )_{F}|)$

$\circ$ The symmetric case occurs when $\eta =\pi /4$

• Case 2 (with ancilla case): The unitary transformation can also be described in terms of the fidelity of one of the copies and Pauli operators:
  

$U_{pca}=F_{A}\mathbb {I} _{ABM}+(1-F_{A})\sigma _{z}\otimes \sigma _{z}\otimes \mathbb {I} +{\sqrt {F_{A}(1-F_{A})}}(\sigma _{x}\otimes \sigma _{x}+\sigma _{y}\otimes \sigma _{y})\otimes \mathbb {I}$

Pseudo Code

General Information

• The to-be-cloned states of the phase-covariant cloner are equatorial states of the form:
  

$|\psi (\phi )\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +e^{i\phi }|1\rangle )$

• Equatorial states could be written in density matrix representation as:
  

$|\psi (\phi )\rangle \langle \psi (\phi )|={\frac {1}{2}}(\mathbb {I} +cos\phi \sigma _{x}+sin\phi \sigma _{y})$
where $\sigma _{x}$ and $\sigma _{y}$ are Pauli Operators.

• The action of a general phase-covariant QCM is described by a map $T$ acting as follows:
  

$T(|\psi (\phi )\rangle \langle \psi (\phi )|)=\eta |\psi (\phi )\rangle \langle \psi (\phi )|+(1-\eta ){\frac {\mathbb {I} }{2}}$
where $\eta$ is the shrinking factor. This relation holds for all $\phi$, which guarantees that the cloning machine to act equally well on all the equatorial states. Now we investigate different types of the protocol:

Phase-covariant cloning without ancilla

Input: $|\psi (\phi )\rangle |0\rangle$ Output: ${\frac {1}{\sqrt {2}}}(|00\rangle +cos\eta e^{i\phi }|10\rangle +sin\eta e^{i\phi }|01\rangle )$

1. Perform a unitary transformation described as follows:
   

$U_{pc}|00\rangle =|00\rangle$
$U_{pc}|10\rangle =cos\eta |10\rangle +sin\eta |01\rangle$

Phase-covariant cloning with ancilla

Stage 1 Cloner state preparation

1. Prepare a Bell state $|\Phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )$

Stage 2 Cloner transformation
Input: $|\psi (\phi )\rangle _{A}|\Phi ^{+}\rangle _{BM}$
Output: $U_{pca}|\psi (\phi )\rangle _{A}|\Phi ^{+}\rangle _{BM}$

1. Perform the unitary transformation described as follows:
   

$U_{pca}|0\rangle |0\rangle |0\rangle =|0\rangle |0\rangle |0\rangle$
$U_{pca}|1\rangle |0\rangle |0\rangle =(cos\eta |1\rangle |0\rangle +sin\eta |0\rangle |1\rangle )|0\rangle$
$U_{pca}|0\rangle |1\rangle |1\rangle =(cos\eta |0\rangle |1\rangle +sin\eta |1\rangle |0\rangle )|1\rangle$
$U_{pca}|1\rangle |1\rangle |1\rangle =|1\rangle |1\rangle |1\rangle$

Stage 3 Discarding ancillary state

1. Discard the extra state. mathematically, trace out the ancilla

Further Information

The phase-covariant QCM has a remarkable application in quantum cryptography since it is used for some of the eavesdropping strategies on the BB84 QKD. This is due to the fact that states which are being used as BB84 states, are equatorial states and these are the only states that the eavesdropper is interested in. Both protocols mentioned in Pseudo Code can be used for this analysis.