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A magic criterion (almost) as nice as PPT, with applications in distillation and detection

Zhenhuan Liu, Tobias Haug, Qi Ye, Zi-Wen Liu, Ingo Roth

Abstract

We introduce a mixed-state magic criterion, the Triangle Criterion, which plays a role for magic analogous to the Positive Partial Transposition (PPT) criterion for entanglement: it combines strong detection capability, a clear geometric interpretation, and an operational link to magic distillation. Using this criterion, we uncover several new features of multi-qubit magic distillation and detection. We prove that genuinely multi-qubit magic distillation protocols are strictly more powerful than all single-qubit schemes by showing that the Triangle Criterion is not stable under tensor products, in sharp contrast to the PPT criterion. Moreover, we show that, with overwhelming probability, multi-qubit magic states with relatively low rank cannot be distilled by any single-qubit distillation protocol. We derive an upper bound on the minimal purity of magic states, which is conjectured to be tight with both numerical and constructive evidences. Using this minimal-purity result, we predict the existence of unfaithful magic states, namely states that cannot be detected by any fidelity-based magic witness, and reveal fundamental limitations of mixed-state magic detection in any single-copy scheme.

A magic criterion (almost) as nice as PPT, with applications in distillation and detection

Abstract

We introduce a mixed-state magic criterion, the Triangle Criterion, which plays a role for magic analogous to the Positive Partial Transposition (PPT) criterion for entanglement: it combines strong detection capability, a clear geometric interpretation, and an operational link to magic distillation. Using this criterion, we uncover several new features of multi-qubit magic distillation and detection. We prove that genuinely multi-qubit magic distillation protocols are strictly more powerful than all single-qubit schemes by showing that the Triangle Criterion is not stable under tensor products, in sharp contrast to the PPT criterion. Moreover, we show that, with overwhelming probability, multi-qubit magic states with relatively low rank cannot be distilled by any single-qubit distillation protocol. We derive an upper bound on the minimal purity of magic states, which is conjectured to be tight with both numerical and constructive evidences. Using this minimal-purity result, we predict the existence of unfaithful magic states, namely states that cannot be detected by any fidelity-based magic witness, and reveal fundamental limitations of mixed-state magic detection in any single-copy scheme.

Paper Structure

This paper contains 7 sections, 16 theorems, 44 equations, 2 figures.

Table of Contents

  1. The number of neighbouring stabilizer states
  2. Equivalence between two formulations
  3. Multi-qubit magic distillation
  4. The minimal purity of magic state
  5. Absolute non-resourceful ball
  6. Unfaithful magic state
  7. Detectability of Triangle Criterion and multi-qubit magic

Key Result

Theorem 1

Given any quantum state $\rho$, if there exist three stabilizer pure states $\{\psi_1,\psi_2,\psi_3\}$ with $\Tr(\psi_i\psi_j)=1/2$ for all $i\neq j$ such that then $\rho$ is not a mixture of stabilizer states. This criterion detects all single-qubit mixed magic states and all multi-qubit pure magic states.

Figures (2)

  • Figure S1: Circuit to distil $\rho^{\otimes 2}$ to a single qubit state $\rho^\prime$, which can be further distilled using magic state distillation protocols. This circuit uses two CNOT gates, a Hadamard gate, and a Pauli-$Y$ gate.
  • Figure S2: Probability of detecting magic using a single triangle witness operator $W_{ijk}$, i.e. observing $\text{tr}(\rho W_{ijk})<0)$. Here, we sample states $\rho\sim \pi_{d,k}$ and plot against traced-out dimension $k$ for different qubit number $n$, where state dimension $d=2^n$.

Theorems & Definitions (24)

  • Definition 1
  • Theorem 1: Triangle Criterion
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Conjecture 1
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 14 more