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 $ε$ with success probability $\ge1-δ$ , under different resources: access to the complex-conjugate channel $\mathcal{E}^*$ and/or parallel access to $c$ 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 $\mathcal{E}$ and $\mathcal{E}^*$ , with tight scaling $ε^{-4}$ . Without conjugate access, any $c<d$ copies are insufficient for efficient learning, requiring sample complexity exponential in the number of ( $d$ -level) qudits $n$ (for prime $d$ ). Efficiency is recovered at $c=d$ , with tight scaling $ε^{-2d}$ . For bosonic systems, exponential sample complexity holds in terms of an effective dimension induced by an energy constraint for all $c=O(1/ε)$ . 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 $d$ , some channels require $d$ -copy access. As a corollary, we derive tighter lower bounds for state learning with limited multi-copy access.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 $ε$ with success probability $\ge1-δ$, under different resources: access to the complex-conjugate channel $\mathcal{E}^*$ and/or parallel access to $c$ 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 $\mathcal{E}$ and $\mathcal{E}^*$, with tight scaling $ε^{-4}$. Without conjugate access, any $c<d$ copies are insufficient for efficient learning, requiring sample complexity exponential in the number of ($d$-level) qudits $n$ (for prime $d$). Efficiency is recovered at $c=d$, with tight scaling $ε^{-2d}$. For bosonic systems, exponential sample complexity holds in terms of an effective dimension induced by an energy constraint for all $c=O(1/ε)$. 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 $d$, some channels require $d$-copy access. As a corollary, we derive tighter lower bounds for state learning with limited multi-copy access.

quant-pharXiv:2608.05307Aug 7, 20260 citations

Optimal full estimation of qubit mixed states (18 pages)

Manuel A. Ballester, E. Bagán

Physical Review AJan 1, 20060 citations

State estimation from a pair of conjugate qudits

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

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.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.

arXiv:quant-ph/0603237Physical Review AOct 23, 20064 citations

ε

with success probability

\ge1-δ

, under different resources: access to the complex-conjugate channel

\mathcal{E}^*

and/or parallel access to

c

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

\mathcal{E}

and

\mathcal{E}^*

, with tight scaling

ε^{-4}

. Without conjugate access, any

c<d

copies are insufficient for efficient learning, requiring sample complexity exponential in the number of (

d

-level) qudits

n

(for prime

d

). Efficiency is recovered at

c=d

, with tight scaling

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)

. 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

d

, some channels require

d

-copy access. As a corollary, we derive tighter lower bounds for state learning with limited multi-copy access.

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.