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arXiv:quant-ph/0603237 · Physical Review A

State estimation from a pair of conjugate qudits

Xiang-Fa Zhou, Yong-Sheng Zhang, Guang‐Can Guo

Published Oct 23, 2006• 4 citations
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Abstract

We show that, for $N$ parallel input states, an antilinear map with respect to a specific basis is essentially a classical operation. We also consider the information contained in a phase-conjugate pair $\ensuremath{\mid}\ensuremath{\phi}⟩\ensuremath{\mid}{\ensuremath{\phi}}^{*}⟩$, and prove that there is more information about a quantum state encoded in a phase-conjugate pair than in a parallel pair.

DOI
10.1103/physreva.74.042324

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We show that, for NNN parallel input states, an antilinear map with respect to a specific basis is essentially a classical operation. We also consider the information contained in a phase-conjugate pair \ensuremath∣\ensuremathϕ⟩\ensuremath∣\ensuremathϕ∗⟩\ensuremath{\mid}\ensuremath{\phi}⟩\ensuremath{\mid}{\ensuremath{\phi}}^{*}⟩\ensuremath∣\ensuremathϕ⟩\ensuremath∣\ensuremathϕ∗⟩, and prove that there is more information about a quantum state encoded in a phase-conjugate pair than in a parallel pair.
We show that, for NNN parallel input states, an anti-linear map with respect to a specific basis is essentially a classical operator. We also consider the information contained in phase-conjugate pairs ∣phi>∣phi∗>|\\phi > |\\phi^*>∣phi>∣phi∗>, and prove that there is more information about a quantum state encoded in phase-conjugate pairs than in parallel pairs.
In the context of multiparameter quantum estimation theory, we investigate the construction of linear schemes in order to infer two classical parameters that are encoded in the quadratures of two quantum coherent states. The optimality of the scheme built on two phase-conjugate coherent states is proven with the saturation of the quantum Cramér–Rao bound under some global energy constraint. In a more general setting, we consider and analyze a variety of n-mode schemes that can be used to encode n classical parameters into n quantum coherent states and then estimate all parameters optimally and simultaneously.
We extend the concept of probabilistic unambiguous discrimination of quantum states to quantum state estimation. We consider a scenario where the measurement device can output either an estimate of the unknown input state or an inconclusive result. We present a general method how to evaluate the maximum fidelity achievable by the probabilistic estimation strategy. We illustrate our method on two explicit examples: estimation of a qudit from a pair of conjugate qudits and phase covariant estimation of a qubit from N copies. We show that by allowing for inconclusive results it is possible to reach estimation fidelity higher than that achievable by the best deterministic estimation strategy.
Quantum channels can be characterized by their action on an orthogonal operator basis, where these operators are related to observable properties of the quantum system. For qudit and multimode bosonic systems, this is encoded respectively in the Heisenberg--Weyl transfer matrix estimated from the Choi-state, and in the characteristic-function transfer map estimated from a two-mode squeezed vacuum based Choi-state. We derive sample-complexity bounds for estimating entries of the transfer matrix/map to additive accuracy εεε with success probability ≥1−δ\ge1-δ≥1−δ, under different resources: access to the complex-conjugate channel E∗\mathcal{E}^*E∗ and/or parallel access to ccc copies. In all settings, the learner uses parallel channel calls with adaptively chosen, ancilla-assisted input states and measurements. Absolute values of transfer-matrix entries can be learned efficiently with simultaneous access to E\mathcal{E}E and E∗\mathcal{E}^*E∗, with tight scaling ε−4ε^{-4}ε−4. Without conjugate access, any c<dc<dc<d copies are insufficient for efficient learning, requiring sample complexity exponential in the number of (ddd-level) qudits nnn (for prime ddd). Efficiency is recovered at c=dc=dc=d, with tight scaling ε−2dε^{-2d}ε−2d. For bosonic systems, exponential sample complexity holds in terms of an effective dimension induced by an energy constraint for all c=O(1/ε)c=O(1/ε)c=O(1/ε). Although the task is learning a particular state, these bounds carry stronger implications than standard state-learning bounds since the learner controls the inputs and has ancillary assistance. This establishes a hierarchy of channel-learning resources: self-complex-conjugate channels require two-copy ancilla-assisted access for efficient learning, while for every square-free ddd, some channels require ddd-copy access. As a corollary, we derive tighter lower bounds for state learning with limited multi-copy access.
We address the problem of broadcasting NNN copies of a generic qubit state to M>NM>NM>N copies by estimating its direction and preparing a suitable output state according to the outcome of the estimate. This semiclassical broadcasting protocol is more restrictive than a general one, since it requires an intermediate step where classical information is extracted and processed. However, we prove that a suboptimal superbroadcasting, namely broadcasting with simultaneous purification of the local output states with respect to the input ones, is possible. We show that in the asymptotic limit of $M\ensuremath{\rightarrow}\ensuremath{\infty}$ the purification rate converges to the optimal one, proving the conjecture that optimal broadcasting and state estimation are asymptotically equivalent. We also show that it is possible to achieve superbroadcasting with simultaneous inversion of the Bloch vector direction (universal NOT). We prove that in this case the semiclassical procedure of state estimation and preparation turns out to be optimal. We finally analyze semiclassical superbroadcasting in the phase-covariant case.
Robust quantum computation with ddd-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high-fidelity two-qudit gates. We first describe how to implement parallel single-qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth (time) of the parallel gate sequence is equal to the height of an associated tree. The implementation of a given unitary can then optimize the tradeoff between gate time and resources used. These ideas are illustrated for qudits encoded in the ground hyperfine states of the alkali-metal atoms 87Rb^{87}\mathrm{Rb}87Rb and 133Cs^{133}\mathrm{Cs}133Cs. Second, we provide a protocol for implementing parallelized nonlocal two-qudit gates using the assistance of entangled qubit pairs. Using known protocols for qubit entanglement purification, this offers the possibility of high-fidelity two-qudit gates.