Quantum cloning of identical mixed qubits

2007 ◽  
Vol 7 (5&6) ◽  
pp. 551-558
Author(s):  
H. Fan ◽  
B.-Y. Liu ◽  
K.-J. Shi
Keyword(s):  
Upper Bound ◽  
Quantum Cloning ◽  
Single Qubit ◽  
Antisymmetric State ◽  
Pure States ◽  
Symmetric States

Quantum cloning of two identical mixed qubits $\rho \otimes \rho$ is studied. We propose the quantum cloning transformations not only for the triplet (symmetric) states but also for the singlet (antisymmetric) state. We can copy these two identical mixed qubits to $M$ ($M\ge 2$) copies. This quantum cloning machine is optimal in the sense that the shrinking factor between the input and the output single qubit achieves the upper bound. The result shows that we can copy two identical mixed qubits with the same quality as that of two identical pure states.

scholarly journals Quantum universality by state distillation

2009 ◽  
Vol 9 (11&12) ◽  
pp. 1030-1052
Author(s):  
B.W. Reichardt
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Fault Tolerant ◽  
Upper Bounds ◽  
Mixed States ◽  
Single Qubit ◽  
Noise Rate ◽  
Pure States ◽  
Rate Threshold ◽  
Stabilizer States

Quantum universality can be achieved using classically controlled stabilizer operations and repeated preparation of certain ancilla states. Which ancilla states suffice for universality? This ``magic states distillation" question is closely related to quantum fault tolerance. Lower bounds on the noise tolerable on the ancilla help give lower bounds on the tolerable noise rate threshold for fault-tolerant computation. Upper bounds show the limits of threshold upper-bound arguments based on the Gottesman-Knill theorem. We extend the range of single-qubit mixed states that are known to give universality, by using a simple parity-checking operation. For applications to proving threshold lower bounds, certain practical stability characteristics are often required, and we also show a stable distillation procedure.}{No distillation upper bounds are known beyond those given by the Gottesman-Knill theorem. One might ask whether distillation upper bounds reduce to upper bounds for single-qubit ancilla states. For multi-qubit pure states and previously considered two-qubit ancilla states, the answer is yes. However, we exhibit two-qubit mixed states that are not mixtures of stabilizer states, but for which every postselected stabilizer reduction from two qubits to one outputs a mixture of stabilizer states. Distilling such states would require true multi-qubit state distillation methods.

An upper bound on the threshold quantum decoherence rate

2004 ◽  
Vol 4 (3) ◽  
pp. 222-228
Author(s):  
A.A. Razborov
Keyword(s):  
Upper Bound ◽  
Random Noise ◽  
Quantum Circuit ◽  
Quantum Decoherence ◽  
Single Qubit ◽  
Constant Rate ◽  
Decoherence Rate

Let $\eta_0$ be the supremum of those $\eta$ for which every poly-size quantum circuit can be simulated by another poly-size quantum circuit with gates of fan-in $\leq 2$ that tolerates random noise independently occurring on all wires at the constant rate $\eta$. Recent fundamental results showing the principal fact $\eta_0>0$ give estimates like $\eta_0\geq 10^{-6}\mbox{--}10^{-4}$, whereas the only upper bound known before is $\eta_0\leq 0.74$.}{In this note we improve the latter bound to $\eta_0\leq 1/2$, under the assumption ${\bf QP}\not\subseteq {\bf QNC^1}$. More generally, we show that if the decoherence rate $\eta$ is greater than 1/2, then we can not even store a single qubit for more than logarithmic time. Our bound also generalizes to the simulating circuits allowing gates of any (constant) fan-in $k$, in which case we have $\eta_0\leq 1-\frac 1k$.

Optimization of coherent attacks in generalizations of the BB84 quantum bit commitment protocol

2002 ◽  
Vol 2 (1) ◽  
pp. 66-96
Author(s):  
R.W. Spekkens ◽  
T. Rudolph
Keyword(s):  
State Estimation ◽  
Quantum State ◽  
Upper Bound ◽  
Upper Bounds ◽  
Entangled State ◽  
Quantum State Estimation ◽  
Quantum Bit ◽  
Bit Commitment ◽  
Single Qubit ◽  
Qubit States

It is well known that no quantum bit commitment protocol is unconditionally secure. Nonetheless, there can be non-trivial upper bounds on both Bob's probability of correctly estimating Alice's commitment and Alice's probability of successfully unveiling whatever bit she desires. In this paper, we seek to determine these bounds for generalizations of the BB84 bit commitment protocol. In such protocols, an honest Alice commits to a bit by randomly choosing a state from a specified set and submitting this to Bob, and later unveils the bit to Bob by announcing the chosen state, at which point Bob measures the projector onto the state. Bob's optimal cheating strategy can be easily deduced from well known results in the theory of quantum state estimation. We show how to understand Alice's most general cheating strategy, (which involves her submitting to Bob one half of an entangled state) in terms of a theorem of Hughston, Jozsa and Wootters. We also show how the problem of optimizing Alice's cheating strategy for a fixed submitted state can be mapped onto a problem of state estimation. Finally, using the Bloch ball representation of qubit states, we identify the optimal coherent attack for a class of protocols that can be implemented with just a single qubit. These results provide a tight upper bound on Alice's probability of successfully unveiling whatever bit she desires in the protocol proposed by Aharonov et al., and lead us to identify a qubit protocol with even greater security.

scholarly journals Improved upper bounds on the stabilizer rank of magic states

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 606
Author(s):  
Hammam Qassim ◽  
Hakop Pashayan ◽  
David Gosset
Keyword(s):  
Relative Error ◽  
Upper Bound ◽  
Linear Codes ◽  
Upper Bounds ◽  
Quantum Circuits ◽  
Symmetric Product ◽  
Simulation Algorithm ◽  
Single Qubit ◽  
Classical Algorithm ◽  
Stabilizer States

In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of m copies of the magic state |T⟩=2−1(|0⟩+eiπ/4|1⟩) in the limit of large m. In particular, we show that |T⟩⊗m can be exactly expressed as a superposition of at most O(2αm) stabilizer states, where α≤0.3963, improving on the best previously known bound α≤0.463. This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an n-qubit Clifford + T circuit U with m uses of the T gate to within a given inverse polynomial relative error using a runtime poly(n,m)2αm. We also provide improved upper bounds on the stabilizer rank of symmetric product states |ψ⟩⊗m more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and m instances of any (fixed) single-qubit Z-rotation gate with runtime poly(n,m)2m/2. We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.

scholarly journals Two-Qubit Bloch Sphere

Physics ◽  
2020 ◽  
Vol 2 (3) ◽  
pp. 383-396
Author(s):  
Chu-Ryang Wie
Keyword(s):  
Density Matrix ◽  
Reduced Density Matrix ◽  
Entanglement Measure ◽  
Bloch Sphere ◽  
Bipartite State ◽  
Single Qubit ◽  
Non Local ◽  
Pure States ◽  
Reduced Density ◽  
Simple Rotation

Three unit spheres were used to represent the two-qubit pure states. The three spheres are named the base sphere, entanglement sphere, and fiber sphere. The base sphere and entanglement sphere represent the reduced density matrix of the base qubit and the non-local entanglement measure, concurrence, while the fiber sphere represents the fiber qubit via a simple rotation under a local single-qubit unitary operation; however, in an entangled bipartite state, the fiber sphere has no information on the reduced density matrix of the fiber qubit. When the bipartite state becomes separable, the base and fiber spheres seamlessly become the single-qubit Bloch spheres of each qubit. Since either qubit can be chosen as the base qubit, two alternative sets of these three spheres are available, where each set fully represents the bipartite pure state, and each set has information of the reduced density matrix of its base qubit. Comparing this model to the two Bloch balls representing the reduced density matrices of the two qubits, each Bloch ball corresponds to two unit spheres in our model, namely, the base and entanglement spheres. The concurrence–coherence complementarity is explicitly shown on the entanglement sphere via a single angle.

Dynamics of coherence under Markovian and non-Markovian environments

2017 ◽  
Vol 31 (35) ◽  
pp. 1750329 ◽  
Author(s):  
Zhong-Xiao Wang ◽  
Teng Ma ◽  
Shu-Hao Wang ◽  
Tie-Jun Wang ◽  
Chuan Wang
Keyword(s):  
Density Matrix ◽  
Upper Bound ◽  
Quantum Coherence ◽  
Quantum Discord ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Initial State ◽  
Single Qubit ◽  
Markovian Processes ◽  
Markovian Dynamics

The behavior of quantum coherence is studied under Markovian and non-Markovian dynamics for open quantum systems. For single qubit systems, we show that the coherence depending on the off-diagonal elements of the density matrix is the upper bound of the coherence depending on the relative entropy under both Markovian and non-Markovian processes. For two-qubit systems, in both Markovian and non-Markovian processes, quantum discord and coherence show less sensitivity to the initial state than quantum entanglement. We also find that the quantum discord has similar behaviors with coherence under both Markovian and non-Markovian dynamics.

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