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A quantum version of Wielandt's inequality

M. Sanz, D. Perez-Garcia, M. M. Wolf, J. I. Cirac

arXiv:0909.5347v2quant-phmath-phmath.MP

TL;DR

The paper asks how many applications of a quantum channel are needed to make every output full rank. It develops a quantum analogue of Wielandt’s inequality and applies the resulting bounds to zero-error capacity and MPS-related questions. The work also draws implications for interaction-range bounds in Hamiltonians with unique MPS ground states.

  • Problem

    The paper seeks a dimension-dependent bound on how many times a quantum channel must be applied to obtain a full-rank output.

    Source: §VI · p. 5

  • Method

    The authors relate quantum primitivity to Kraus-operator spans and derive bounds on the Kraus-span index, which bounds the quantum index of primitivity.

    Source: §I · p. 1, §III · p. 3

  • Results

    The paper derives quantum Wielandt inequalities, including improved bounds under specified conditions on the Kraus-operator span, and applies them to zero-error-capacity and MPS-related results.

    Source: §III · p. 3, §VI · p. 5

  • Takeaways & Limitations

    The inequalities yield dichotomy theorems for zero-error capacity and MPS dimension, and implications for interaction-range bounds of Parent Hamiltonians.

    Source: §IV · p. 4, §V · p. 5, §VI · p. 5

    The bounds on both q and i(A) are not known to be optimal, and q can differ from i(A).

    Source: §VI · p. 5

Abstract

from arXiv · show

In this paper, Wielandt's inequality for classical channels is extended to quantum channels. That is, an upper bound to the number of times a channel must be applied, so that it maps any density operator to one with full rank, is found. Using this bound, dichotomy theorems for the zero--error capacity of quantum channels and for the Matrix Product State (MPS) dimension of ground states of frustration-free Hamiltonians are derived. The obtained inequalities also imply new bounds on the required interaction-range of Hamiltonians with unique MPS ground state.

I. INTRODUCTION

The paper extends Wielandt’s inequality from classical channels to quantum channels by defining quantum primitivity through eventual full-rank outputs. It uses this framework to establish bounds and connect them to zero-error capacity and frustration-free Hamiltonians with MPS ground states.

I. INTRODUCTION

  • Classical primitivity requires some power of a stochastic matrix to have all positive entries; its minimum such exponent bounds how many channel applications guarantee every event has nonzero probability.The paper identifies Wielandt’s inequality as an optimal bound independent of the matrix elements.Source: §I · p. 1
  • The work defines quantum primitivity by requiring that some iterate map every positive semidefinite operator to a positive definite one; the least such iterate is the quantum index of primitivity.The paper considers trace-preserving completely positive linear maps.Source: §I · p. 1
  • The authors relate primitivity to a unique full-rank fixed point and to Kraus operators spanning the full matrix space, then derive a quantum Wielandt inequality.The bound depends on Hilbert-space dimension D and the number d of linearly independent Kraus operators; under certain generic Kraus-operator conditions, tighter inequalities are possible.Source: §I · p. 1
  • The inequalities are applied to zero-error capacity, ground states of frustration-free spin Hamiltonians, and interaction-range bounds for Hamiltonians with unique MPS ground states.The paper reports a zero-error-capacity dichotomy and a proof of a conjecture about frustration-free Hamiltonian ground states.Source: §I · p. 1

II. BASIC NOTIONS

The section defines quantum primitivity and equivalent ways to characterize it, then connects these notions to the classical case. It also establishes how the quantum and Kraus-rank indices relate, including a case where they differ.

II. BASIC NOTIONS

  • The space S_n(A) spans products of exactly n Kraus operators, while H_n(A,ϕ) spans the vectors those products produce from |ϕ⟩.A channel is primitive when H_n(A,ϕ) is the full Hilbert space for every input vector at some n.Source: §II · p. 1
  • For every quantum channel EA, q(EA) ≤ i(A); in the stated example involving the three Pauli matrices, q(EA) = 1 < i(A) = 2.Source: §II · p. 2
  • For a primitive stochastic map and its corresponding quantum channel, the classical primitivity index, quantum index, and Kraus-span index are equal: q(A) = p(A) = i(A).The associated quantum channel reduces to the classical map on diagonal density operators.Source: §II · p. 2
  • A quantum channel is primitive, has eventually full Kraus rank, and is strongly irreducible under equivalent conditions.Strong irreducibility requires a unique eigenvalue of modulus one with a positive definite corresponding eigenoperator.Source: §II · p. 2
  • A quantum channel is primitive exactly when it has eventually full Kraus rank or is strongly irreducible; a non-full-rank fixed operator rules out primitivity.If peripheral eigenvalues are roots of unity, an iterate has multiple eigenoperators at eigenvalue one, contradicting strong irreducibility.Source: §II · p. 2
  • Primitivity can be decided from spectral properties, paralleling the classical criterion of a unique eigenvalue of maximum modulus and a positive definite fixed point.The section’s equivalence result identifies strong irreducibility with primitivity.Source: §II · pp. 2–3

III. QUANTUM WIELANDT’S INEQUALITIES

The section establishes bounds for the quantum index of primitivity and the matrix-span index of primitive quantum channels. It also gives conditions yielding tighter bounds, discusses sharpness, and notes a low-dimensional property of primitive channels.

III. QUANTUM WIELANDT’S INEQUALITIES

  • Some S_n(A) contains an operator with nonzero trace for n≤D^2−d+1; dimension growth under primitivity underpins this lemma.Source: §III · p. 3
  • For primitive channels, q(EA)≤i(A) and generally i(A)≤(D^2−d+1)D^2.Source: §III · p. 3
  • If the Kraus span contains an invertible element, i(A)≤D^2−d+1; with a non-invertible element having a nonzero eigenvalue, i(A)≤D^2.Source: §III · p. 3
  • The bounds’ sharpness is unknown; an example has i(A)=D^2−D, exceeding the classical Wielandt bound when D>2.Source: §III · p. 3
  • For D=2,3, S1(A) always contains an element with a nonzero eigenvalue, so the general bound is not needed without one of the other bounds.Source: §III · p. 4

IV. ZERO-ERROR CAPACITY

The section establishes a zero-error-capacity dichotomy for quantum channels with a full-rank fixed point: capacity persists at every power, or vanishes by the quantum primitivity threshold. The threshold depends only on the Hilbert-space dimension and Kraus-operator count.

IV. ZERO-ERROR CAPACITY

  • Zero-error capacity is the largest asymptotic code rate achievable with exactly zero decoding error, rather than error merely approaching zero.The section notes its relevance when errors cannot be tolerated or when only finitely many channel uses are available.Source: §IV · p. 4
  • C0(E^q(E)) is zero or C0(E^n) is at least 1 for every n, for a channel with a full-rank fixed point.The threshold q(E) is bounded by (D^2−d+1)D^2 and depends only on D and d, not on the channel itself.Source: §IV · p. 4
  • If the channel has multiple fixed points, or a unit-modulus eigenvalue besides its unique full-rank fixed point, a classical bit or qubit can be encoded at every power.In the latter case, the eigenvalue is a root of unity, so some power of the channel has multiple fixed points.Source: §IV · p. 4
  • If the channel has only one fixed point and no other unit-modulus eigenvalues, it is primitive, and C0(E^n) is zero for n ≥ q(E).The power E^q(E) maps outputs to full-rank states.Source: §IV · p. 4

V. FRUSTRATION-FREE HAMILTONIANS AND MATRIX PRODUCT STATES

The section applies quantum Wielandt bounds to parent Hamiltonians and to ground states of frustration-free spin chains. It gives an interaction-length condition for a unique, gapped MPS ground state and a dichotomy concerning MPS dimension for unique frustration-free ground states.

V. FRUSTRATION-FREE HAMILTONIANS AND MATRIX PRODUCT STATES

  • A frustration-free Hamiltonian has a ground state that minimizes the energy locally, and parent-Hamiltonian terms are constructed as projectors onto the orthogonal complement of an MPS image.The section situates MPS as a variational family used to approximate ground states of local one-dimensional Hamiltonians.Source: §V · pp. 4–5
  • When N − L ≥ i(A), the local term's kernel contains the stated MPS-generated states, implying that |ψA⟩ remains a ground state for every N′ > N and is unique there.The argument uses injectivity for lengths at least i(A).Source: §V · p. 5
  • For a parent Hamiltonian, interaction range L > i(A) ensures that |ψA⟩ is its unique ground state and that a spectral gap exists above it.The bound supplies an interaction-length condition for obtaining a parent Hamiltonian with these properties.Source: §V · p. 5
  • Theorem 5 concerns frustration-free Hamiltonians with a unique ground state for every system size and gives a dichotomy involving MPS representations with matrix size D × D.The supplied theorem passage also states that, for prime N, there are many lengths for which the ground state has no MPS representation with small matrices.Source: §V · p. 5
  • The quantum Wielandt inequality bounds the block injectivity length by O(D^4), while the number of blocks is at most D.These bounds enter the theorem's argument relating system length and MPS dimension.Source: §V · p. 5

VI. CONCLUSIONS AND OPEN PROBLEMS

The paper gives dimension-dependent bounds for making a quantum channel’s Choi matrix full rank and applies the inequalities to zero-error capacity and Matrix Product State questions. It also identifies open problems involving direct bounds on the quantum index of primitivity and links to quantum graph theory.

VI. CONCLUSIONS AND OPEN PROBLEMS

  • Dimension-dependent bounds on channel applications to obtain a full-rank Choi matrix support results on zero-error capacity and Matrix Product States.These include a dichotomy theorem for zero-error capacity, a demonstrated conjecture about frustration-free Hamiltonian ground states, and implications for Parent Hamiltonian interaction range.Source: §VI · p. 5
  • Bounds on the quantum index of primitivity q and the index i(A) remain distinct targets because q need not equal i(A).The paper notes that some applications, including those in the MPS context, require bounds on i(A), while others such as Theorem 2 use q; optimal bounds for both remain open.Source: §VI · p. 5
  • A quantum-graph analogue remains open, despite the close relationship between classical stochastic matrices and graph theory.The paper notes attempts to relate quantum channels and quantum graphs but says no well-defined analogue exists in the quantum context.Source: §VI · p. 5
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