1 Introduction
Nies and Scholz formalized the concept of an infinite sequence of qubits and referred to it as a state [Reference Nies and Stephan11]. We formalize a notion of measurement of a state in a computable basis. We define a state to be ‘quantum measurement random’ if it yields a Martin-Löf random bitstring with probability one when measured in any computable basis. Our main theorem is that there is a quantum measurement random state which is not quantum Martin-Löf random. Its proof goes through two general results. The first extends a result of V. Vovk on Martin-Löf randomness for computable measures while the second is a combinatorial lemma on Kronecker products.
1.1 Classical and quantum algorithmic randomness
Cantor space (
$2^\omega $
) is the set of infinite sequences of bits equipped with a compact topology and the uniform (Lebesgue) measure. Some elements of
$2^\omega $
possess regularities or patterns while others are ‘random’: The infinite bitstring
$101010\ldots $
has a regularity, namely that the ones and zeroes alternate. By contrast, a bitstring obtained by tossing a fair coin infinitely often almost surely lacks any regularities. Algorithmic randomness quantifies the sense in which the latter bitstring is more ‘random’ than the former [Reference Calude5, Reference Downey and Hirschfeldt7, Reference Nies8, Reference Shen, Uspensky and Vereshchagin13]. An element of
$2^\omega $
is said to be random if it possesses no effective regularities.Footnote
1
A key idea in algorithmic randomness is that an element of
$2^\omega $
possesses an effective regularity if and only if it is contained in an effective null set.Footnote
2
Thus an element of
$2^\omega $
is random iff it is in no effective null set. Different levels of effectivity give different levels of randomness.
Martin-Löf randomness is arguably the most important notion of algorithmic randomness. Martin-Löf random bitstrings are incompressible and resist concise descriptions in that they have high initial segment prefix-free Kolmogorov complexity (K) [Reference Calude5, Reference Downey and Hirschfeldt7, Reference Nies8, Reference Shen, Uspensky and Vereshchagin13]. Schnorr randomness is a notion of randomness strictly weaker than Martin-Löf randomness. It has characterizations in terms of a version of K defined using computable measure machines. Quantum algorithmic randomness is a generalization of algorithmic randomness to the quantum setting [Reference Nies and Scholz10]. While algorithmic randomness studies the randomness of infinite sequences of bits, quantum algorithmic randomness studies the randomness of infinite sequences of qubits. Nies and Scholz [Reference Nies and Scholz10] formalized a notion of an infinite qubitstring (an infinite sequence of qubits) and referred to it as a state. They also defined quantum Martin-Löf randomness and quantum Solovay randomness for states. Quantum Schnorr randomness was defined in [Reference Bhojraj3].
The theory of quantum algorithmic randomness has nice parallels with that of classical algorithmic randomness: Any
$X \in 2^\omega $
gives a corresponding state
$\rho _X$
. Quantum Martin-Löf randomness and quantum Schnorr randomness generalize Martin-Löf randomness and Schnorr randomness, respectively, in the sense that X is Martin-Löf random iff
$\rho _X$
is quantum Martin-Löf random and X is Schnorr random iff
$\rho _X$
is quantum Schnorr random. Quantum Martin-Löf randomness is equivalent to quantum Solovay randomness and is strictly stronger than quantum Schnorr randomness [Reference Bhojraj3]. A quantum version of the law of large numbers is satisfied by quantum Schnorr random states [Reference Bhojraj3]. Weak quantum Solovay and quantum Schnorr states can be characterized using QK, a quantum analogue of K. These characterizations mirror the characterizations of Solovay random and Schnorr random bit sequences in terms of K [Reference Bhojraj2]. The quantum algorithmic randomness of states is also related to the von-Neumann entropy of its initial segments [Reference Bhojraj4]. This article introduces quantum measurement randomness, a new notion of quantum algorithmic randomness based on quantum measurement.
1.2 Motivation
An m dimensional quantum system is represented by a density matrix on
$\mathbb {C}^m$
which is an
$m \times m$
Hermitian, positive semidefinite matrix with trace one (see [Reference Chuang and Nielsen6, Section 2.4]). A ‘qubit’ is a two dimensional quantum system and is the quantum analogue of the classical ‘bit’. A system of n qubits is a
$2^n$
dimensional quantum system and thus is given by a density matrix on
$\mathbb {C}^{2^n}$
. A state
$\rho =(\rho _n)_{n=1}^{\infty }$
is a coherent sequence such that
$\rho _n$
is a density matrix on
$\mathbb {C}^{2^n}$
and represents the first n qubits of
$\rho $
(see Definition 2.3 for the details).
Projective measurement (henceforth simply referred to as ‘measurement’) of a density matrix is one of the most important and fundamental operations in quantum mechanics (see [Reference Chuang and Nielsen6, Section 2.2.5] and [Reference Wilde15, Sections 3.4 and 3.5.5]). We formulate an analogous notion of measurement for a state which is a sequence of density matrices. This motivates our formalization of a notion of measurement of a state in Section 3.
Measuring a qubit yields
$0$
or
$1$
each with a certain probability. An obvious way to measure
$\rho $
is to measure its first qubit, followed by measuring its second qubit, followed by measuring its third qubit, and so on. That is, the
$n^{th}$
qubit of
$\rho $
is measured after the first
$n-1$
qubits have been measured in order. As each measurement yields
$0$
or
$1$
, measuring the first n qubits of
$\rho $
yields an element of
$\{0,1\}^n$
. Letting n tend to infinity thus yields an element of
$2^\omega $
.
As each measurement is probabilistic, measuring the first n qubits of
$\rho $
yields a probability distribution on
$\{0,1\}^n$
, where the probability of
$\sigma \in \{0,1\}^n$
is the probability of obtaining
$\sigma $
when the first n qubits of
$\rho $
are measured. Letting n tend to infinity yields a distribution on
$2^\omega $
.
We elaborate on this idea: Let
$B=(B_n)_n$
be an infinite sequence of orthonormal bases of
$\mathbb {C}^2$
. Measuring the first qubit of
$\rho $
using
$B_1$
yields
$0$
or
$1$
thus inducing a distribution
$\mu _1$
on
$\{0,1\}$
. Measuring the second qubit of
$\rho $
using
$B_2$
after the first qubit has been measured using
$B_1$
similarly induces a distribution
$\mu _2$
on
$\{0,1\}^2$
. Continuing this, sequentially measuring the first n qubits of
$\rho $
induces a distribution
$\mu _n$
on
$\{0,1\}^n$
. Moreover, the coherence of
$\rho _{n+1}$
and
$\rho _n$
(see Definition 2.3) implies that
$\mu _{n+1}$
extends
$\mu _n$
in the natural way. The state
$\rho $
thus induces an infinite sequence of distributions
$(\mu _n)_n$
such that
$\mu _{n+1}$
extends
$\mu _n$
. This sequence induces a distribution on Cantor space.
The above intuitive idea motivates our formalization of the process of qubit-wise measuring
$\rho $
in Section 3.
If measuring
$\rho $
yields a non-random element of
$2^\omega $
with high probability, then the behavior of
$\rho $
on measurement exhibits a regularity. For example if measuring
$\rho $
yields
$1010\ldots $
with probability one, then the behavior of
$\rho $
on measurement is highly regular. By contrast, if measuring
$\rho $
yields a Martin-Löf random element of
$2^\omega $
with probability one, then the behavior of
$\rho $
on measurement is random and irregular. It is plausible to consider states of the former kind to be random while those of the latter variety to be non-random. This motivates our definition of quantum measurement randomness in Section 4. A state is defined to be quantum measurement random if measuring it yields a Martin-Löf random with probability one.Footnote
3
1.3 Outline and main results
Section 2 introduces the relevant background from algorithmic randomness and quantum theory. Section 3 formalizes how measurement of a state in a computable basis induces a distribution on
$2^{\omega }$
. Section 4 introduces quantum measurement randomness. Section 4.1 gives an overview of the relationships between quantum measurement randomness and other quantum randomness notions. Section 4.3 shows that quantum measurement random states satisfy a quantum version of the law of large numbers. This is necessary for legitimizing quantum measurement randomness as a plausible randomness notion, as satisfying the law of large numbers is a weak form of randomness which any reasonable notion of randomness should imply.
Section 5 isolates two important results central to our main result in Section 6. These are general results which may be intrinsically mathematically interesting independently of their specific applications in Section 6. The first (Theorem 5.2) concerns Martin-Löf randomness for arbitrary computable measures. The second (Lemma 5.3) concerns Kronecker products of vectors in
$\mathbb {C}^2$
.
Section 6 contains the main result: There is a quantum measurement random state which is not quantum Martin-Löf random.
2 Background
We state some preliminaries relevant to the article.
2.1 Notation
Let
$2^{\omega }$
denote the collection of infinite sequences of bits, let
$2^n$
denote the set of bit strings of length n, let
$2^{<\omega } := \bigcup _{n} 2^n$
and let
$2^{\leq \omega }:= 2^{<\omega } \cup 2^{ \omega }$
. If
$C \subseteq 2^{<\omega }$
, let 
be the set of all
$X \in 2^{\omega }$
such that an initial segment of X is in C. For all
$n \in \omega $
, let
$X(n)$
denote the
$n^{th}$
bit of a bitstring
$X \in 2^{\leq \omega }$
. We index the positions in a string starting from
$1$
instead of from
$0$
. We use ‘distribution’ and ‘measure’ interchangeably to mean a probability distribution. The uniform measure on
$2^{\omega }$
denoted by
$\lambda $
is the measure induced by letting the measure of 
to be
$2^{-|\tau |}$
for each
$\tau \in 2^{<\omega }$
. For any measure m on
$2^\omega $
, we will abuse notation by writing
$m(\tau )$
instead of the more cumbersome 
for
$\tau \in 2^{<\omega }$
. By ‘projection’ we will always mean a Hermitian projection.
2.2 Quantum theory
As per the bra–ket notation, a vector
$v \in \mathbb {C}^d$
is denoted as
$|v\rangle $
and its dual is denoted by
$\langle v|$
. In a fixed basis of
$\mathbb {C}^d$
,
$|v\rangle $
is a column vector while
$\langle v|$
is a row vector which is equal to
$((|v\rangle )^*)^T$
(see [Reference Chuang and Nielsen6, Sections 2.1.1 and 2.1.4]). Let I denote the identity matrix on
$\mathbb {C}^2$
.
A density matrix is a complex, Hermitian, positive semidefinite matrix with trace one. A two-dimensional quantum system is called a qubit and is represented by a density matrix on
$\mathbb {C}^2$
. An n-qubit quantum system is a
$2^n$
-dimensional quantum system lives in the tensor product
$\bigotimes _{i=1}^n \mathbb {C}^2 = \mathbb {C}^{2^n}$
. It is represented by a density matrix on
$\mathbb {C}^{2^n}$
.
Definition 2.1. The set
$\{|0\rangle , |1\rangle \}$
is a fixed orthonormal basis for
$\mathbb {C}^2$
(see [Reference Chuang and Nielsen6, Section 2.2.1]). For any
$n>0$
, the standard basis (or, the computational basis) of
$\mathbb {C}^{2^{n}}$
is the orthonormal basis,
$$\begin{align*}\bigg\{\bigotimes_{i=1}^n |\sigma(i)\rangle: \sigma \in 2^n\bigg\}.\end{align*}$$
For convenience, we denote
$\bigotimes _{i=1}^n |\sigma (i)\rangle $
simply by
$|\sigma \rangle $
.
All matrices, henceforth, are with respect to the standard basis.
Definition 2.2. Let
$m>0$
. A projective measurement on
$\mathbb {C}^m$
is a finite set
$\{\Pi _j\}_j$
, where each
$\Pi _j$
is a projection on
$\mathbb {C}^m$
and
$\sum _j \Pi _j $
is the identity operator. By ‘measurement’ we will always mean a projective measurement.
Suppose that d, a density matrix on
$\mathbb {C}^m$
, is measured using
$\{\Pi _j\}_j$
. The measurement obeys the following measurement axioms (see [Reference Wilde15, p. 126]):
-
(1) The outcome j occurs with probability
$\text {Tr}(\Pi _j d)$
. -
(2) If the outcome j occurs, then the state after the measurement (the ‘post-measurement’ state) is described by the density matrix
$\dfrac {\Pi _j d \Pi _j}{\text {Tr}(\Pi _j d)}$
.
We sometimes say that ‘
$\Pi _i$
is the outcome of the measurement’ when we really mean that the ‘outcome of the measurement is the integer i which labels the operator
$\Pi _i$
in
$\{\Pi _j\}_j$
’.
2.3 Classical and quantum algorithmic randomness
The classical algorithmic randomness notions are from [Reference Downey and Hirschfeldt7, Reference Nies8, Reference Shen, Uspensky and Vereshchagin13] and the quantum algorithmic randomness notions are from [Reference Bhojraj3, Reference Nies and Scholz10].
Definition 2.3. A state,
$\rho =(\rho _n)_{n=1}^\infty $
is an infinite sequence of density matrices such that
$\rho _{n} \in \mathbb {C}^{2^{n} \times 2^{n}}$
and
$\forall n>1$
,
$PT_{\mathbb {C}^{2}}(\rho _n)=\rho _{n-1}$
.
Here
$PT_{\mathbb {C}^{2}}$
denotes the partial trace which ‘traces out’ the last qubit from
$\mathbb {C}^{2^n}$
(see [Reference Chuang and Nielsen6, Section 2.4.3]). The state
$\rho $
represents an infinite sequence of qubits whose first n qubits are described by
$\rho _n$
. The definition requires
$\rho $
to be coherent in the sense that for all n,
$\rho _n$
, when ‘restricted’ via the partial trace to its first
$n-1$
qubits, has the same measurement statistics as the state on
$n-1$
qubits given by
$\rho _{n-1}$
. For any
$\sigma \in 2^{<\omega }$
let
$|\sigma \rangle $
denote
$\bigotimes _{i} |\sigma (i)\rangle $
. Any element of
$2^\omega $
can be thought of as a state: If
$X\in 2^\omega $
, the state
$\rho _{X}=(\rho _{n})_n$
given by
$\rho _{n}=|X\upharpoonright n \big>\big <X\upharpoonright n |$
is the quantum analog of X.
Definition 2.4. Let
$\tau =(\tau _n)_{n=1}^\infty $
be the state given by
$\tau _n = 2^{-n}\bigotimes _{i=1}^n I$
.
Recall that a complex algebraic number is the root of a polynomial with rational coefficients.
Definition 2.5. A special projection is a Hermitian projection matrix with complex algebraic entries.
By a result of Rabin, there is a 1–1 function from the complex algebraic numbers to
$\omega $
such that the field operations in the image are computable. This function may be used to identify a special projection with a natural number (see [Reference Nies and Scholz10, p. 8]). We define a sequence of special projections to be computable when the corresponding sequence of natural numbers is computable.Footnote
4
A state
$\rho =(\rho _{n})_n$
is computable if each
$\rho _n$
has complex algebraic entries and the sequence
$(\rho _{n})_n$
is computable.
Definition 2.6. A quantum
$\Sigma _{1}^0$
class (q-
$\Sigma _{1}^0$
) G is a computable sequence
$G=(p_{n})_{n=1}^{\infty }$
such that
$p_n$
is a special projection on
$\mathbb {C}^{2^n}$
and range
$(p_n \otimes I) \subseteq $
range
$(p_{n+1})$
for all n.
This is the quantum analogue of a
$\Sigma _{1}^0$
class in the classical setting.
Definition 2.7. A
$\Sigma _{1}^0$
class
$S \subseteq 2^{\omega }$
is any set of the form, 
, where C is a computably enumerable set of strings.Footnote
5
Definition 2.8. If
$G=(p_{n})_n$
is a q-
$\Sigma _{1}^0$
class and
$\rho =(\rho _n)_n$
is a state define
$\rho (G):=\lim _{n}(\text {Tr}(\rho _n p_n))$
.
Definition 2.9. For a state
$\rho =(\rho _{n})_{n}$
and q-
$\Sigma _{1}^0$
class
$G=(p_{n})_{n}$
,
$\rho (G):=\lim _{n} \text {Tr} (\rho _{n}p_n)$
.
Recall the definition of a Martin-Löf test (ML-test).
Definition 2.10. An ML-test is a computable sequence,
$(S_{m})_{m=1}^\infty $
of
$\Sigma _{1}^{0}$
classes such that
$\lambda (S_m) \leq 2^{-m}$
for all m. A
$X\in 2^\omega $
fails
$(S_m)_m$
if
$X \in S_m$
for all m and is Martin-Löf random if it does not fail any ML-test. Let
$\textbf {ML}$
denote the set of Martin-Löf random elements of
$2^\omega $
.
Its quantum generalization is as follows.
Definition 2.11. A quantum Martin-Löf test (q-ML test) is a computable sequence
$(G_{m})_{m=1}^\infty $
of q-
$\Sigma _{1}^{0}$
classes such that
$\tau (G_m) \leq 2^{-m}$
for all m. A state
$\rho $
is said to fail the q-ML test
$(G_{m})_m$
at order
$\delta $
if
$\rho (G_m)>\delta $
for all m. A state
$\rho $
is said to pass the q-ML test
$(G_{m})_m$
at order
$\delta $
if it does not fail the test at order
$\delta $
. A state
$\rho $
is said to be quantum Martin-Löf random if it passes all q-ML tests at all
$\delta>0$
.
3 Measuring a state
We formalize the notion of qubit-wise measurement of a state which involves measuring each qubit of the state in order. We first define the measurement operators used in the measurement of a state.
3.1 The measurement system
Definition 3.1. A computable measurement system
$B= ((b^{n}_{0},b^{n}_{1}))_{n=1}^{\infty }$
(‘system’ for short) is a computable sequence of orthonormal bases of
$\mathbb {C}^{2}$
such that
$b^{n}_{i}$
has complex algebraic entries for all n and i.
Recall the notion of a measurement from Definition 2.2. Given a system
$B= ((b^{n}_{0},b^{n}_{1}))_{n}$
, let
$\Pi ^{B}_i(n) = |b^{n}_{i}\rangle \langle b^{n}_{i}|$
for
$i \in \{0,1\}$
and set
$$ \begin{align} \Pi^B(n) := \{\Pi^{B}_0(n), \Pi^{B}_1(n)\}. \end{align} $$
Note that
$ (\Pi ^B(n))_n$
is a sequence of measurements on
$\mathbb {C}^2$
. Measuring
$\rho $
qubit-wise in the system B will correspond to measuring the first qubit of
$\rho $
using
$\Pi ^B(1)$
, followed by measuring the second qubit using
$\Pi ^B(2)$
, followed by measuring the third qubit using
$\Pi ^B(3),$
and so on for all n. As each measurement will have an outcome in
$\{0,1\}$
, this sequence of qubit-wise measurements will yield an element of
$2^\omega $
in the limit. Fix a
$\tau \in 2^n$
and consider the projection
$ \bigotimes _{i=1}^{n} \big (|b^{i}_{\tau (i)}\big> \big < b^{i}_{\tau (i)}|\big ) = \bigotimes _{i=1}^{n} \Pi ^B_{\tau (i)}(i).$
For notational simplicity, define
$$ \begin{align} \Pi^B (\tau):=\bigotimes_{i=1}^{n} \Pi^{B}_{\tau(i)}(i). \end{align} $$
(This notation should not cause any confusion with the object
$\Pi (n)$
as n is a natural number while
$\tau $
is a bitstring.) For each i,
$|b^i_{\tau (i)}\big>$
is an element of
$\mathbb {C}^2$
(a column vector) and
$\big <b^i_{\tau (i)}|$
is its dual (a row vector). Thus,
$\Pi ^B_{\tau (i)}(i)=|b^{i}_{\tau (i)}\big> \big < b^{i}_{\tau (i)}|$
is the projection operator onto the one dimensional subspace span
$ (|b^i_{\tau (i)}\big>)$
.
$\Pi ^B(\tau )$
is thus the tensor product of n many projection operators such that the
$i^{th}$
projection in the product is
$\Pi ^B_{\tau (i)}(i)$
. Note that
$\{\Pi ^B(\tau ):\tau \in 2^n\}$
is a measurement on
$\mathbb {C}^{2^n}$
(Definition 2.2) since
$\{|\bigotimes _{i=1}^n b^i_{\tau (i)}\rangle : \tau \in 2^n\}$
is an orthonormal basis of
$\mathbb {C}^{2^n}$
.
3.2 Premeasures
A (probability) premeasure p is a function from
$ 2^{<\omega }$
to
$[0,1]$
satisfying
$p(\emptyset )=1$
and
$\forall \tau , p(\tau )=p(\tau 0)+p(\tau 1)$
(Section 6.12.1 in [Reference Downey and Hirschfeldt7]). A premeasure p uniquely extends to a probability measure
$\mathcal {D}$
on
$2^\omega $
such that 
for all
$\tau \in 2^{<\omega }$
. A random process whose outcome is in
$2^\omega $
induces a premeasure p where
$p(\tau )$
is the probability that the outcome of the random process extends
$\tau $
. For example, an infinite sequence of independent tosses of a fair coin induces a premeasure which maps
$\tau \mapsto 2^{-|\tau |}$
. An infinite sequence of independent tosses of a coin which has a
$0.75$
probability of turning up ‘
$1$
’ induces a premeasure which maps
$$\begin{align*}\tau \mapsto (0.75^{|\{i:\tau(i)=1\}|})(0.25^{|\{i:\tau(i)=0\}|}).\end{align*}$$
We will be interested in the premeasure induced by the random process of measuring a state.
3.3 Informal description
We first give an intuitive, informal description of the process of qubit-wise measuring a state in the system B. The reader may choose to skip this section and move directly to Section 3.4 without loss of continuity.
Consider the random process of tossing a fair coin independently, infinitely often. This is represented by a sequence
$(Z_n)_n$
of
$\{0,1\}$
-valued, Bernoulli-
$1/2$
, i.i.d. random variables, where
$Z_n$
is the outcome of the
$n^{th}$
toss. This process induces a premeasure
$p_u$
, where
$p_u(\sigma )=2^{-|\sigma |}$
is the probability that
$Z_i =\sigma (i)$
for all
$i \leq |\sigma |$
.
The random process of measuring a state is similarly represented by a sequence of random variables and the induced premeasure, which we now describe.
Fix a state
$\rho =(\rho _{n})_n$
and a system
$B=(\Pi ^B(n))_n$
(see Definition 3.1). Measuring
$\rho $
qubit-wise using B can be formalized by a sequence of
$\{0,1\}$
-valued random variables
$(X_n)_n$
, where
$X_n$
is the outcome of the measurement of the
$n^{th}$
qubit of
$\rho $
. The details are as follows. We drop the superscript B from
$\Pi ^B$
as B is fixed henceforth.
Measure the first qubit
$\rho _1$
of
$\rho $
using
$\Pi (1)$
and let
$X_1 \in \{0,1\}$
be the outcome of this measurement. This measurement causes
$\rho _2$
to change to the post-measurement state represented by the density matrix
$\hat {\rho }_2$
(see Definition 2.2). Now measure the second qubit of
$\hat {\rho }_2$
using
$\Pi (2)$
and let
$X_2 \in \{0,1\}$
be the outcome of this second measurement. The first and second measurements cause
$\rho _3$
to change to the post-measurement state represented by the density matrix
$\hat {\rho }_3$
and so on. This process, referred to as the ‘qubit-wise measurement of
$\rho $
using B’, generates an infinite bitstring whose
$n^{th}$
bit is
$X_n$
.
We now describe the premeasure
$p^B_\rho $
(denoted simply by p) induced by this process. For any
$\tau \in 2^{<\omega }$
,
$p(\tau )$
will be the probability that qubit-wise measuring the first
$|\tau |$
many qubits of
$\rho $
using B gives
$\tau $
.
Measure
$\rho _1$
in the system
$\Pi (1)= \{\Pi _0(1), \Pi _1(1)\}$
(recall the notation (3.1)) and define
$X_1$
to be the
$i \in \{0,1\}$
such that
$\Pi _i(1)$
is the outcome of the measurement. Let
$\hat {\rho _{2}}$
be the density matrix corresponding to the post-measurement state of
$\rho _2$
given that
$\rho _2$
yields
$\Pi _{X_1}(1) \otimes I$
when measured in the system
$$\begin{align*}\{\Pi_{0}(1) \otimes I, \Pi_{1}(1) \otimes I \},\end{align*}$$
which is the system for measuring the first qubit. The measurement axioms (recall Section 2.2) imply that
$$\begin{align*}\hat{\rho_{2}} = \dfrac{(\Pi_{X_1}(1) \otimes I) \rho_2 (\Pi_{X_1}(1) \otimes I)}{\text{Tr} (\Pi_{X_1}(1) \otimes I \rho_2 \big)}. \end{align*}$$
To define
$X_2$
, measure
$\hat {\rho _{2}}$
in the system
$$\begin{align*}\{ I \otimes \Pi_{0}(2) , I \otimes \Pi_{1}(2) \},\end{align*}$$
which is the system for measuring the second qubit. Set
$X_2:= i$
, where
$i \in \{0,1\}$
is such that
$I \otimes \Pi _{i}(2)$
is obtained after the measurement. (The measurement when defining
$X_2$
is on
$\hat {\rho _{2}}$
and not on
$\rho _2$
as the state after the first measurement is
$\hat {\rho _{2}}$
and not
$\rho _2$
.)
The random variable
$X_n$
is defined similarly. The function p is: For any n and for any
$\tau \in 2^{n}$
,
$p(\tau )= \mathbb {P}(X_1 = \tau (1),\ldots ,X_{n}=\tau (n))$
. For any
$\tau \in 2^n$
standard calculations show that
$$ \begin{align} p(\tau) =\mathbb{P}(X_1=\tau(1), \ldots,X_n=\tau(n)) = \text{Tr} \big[ \rho_n \Pi(\tau) \big]. \end{align} $$
We see that for any
$\tau \in 2^{<\omega }$
,
$p(\tau )$
is the probability that
$X_i = \tau (i)$
for all
$i \leq |\tau |$
. We show in Lemma 3.4 that
$p=p^B_\rho $
is a premeasure and thus extends to a unique distribution
$\mu ^B_\rho $
on
$2^{\omega }$
.
Note that while the random variables
$Z_1, Z_2, \ldots $
are mutually independent, the random variables
$X_1, X_2, \ldots $
need not be independent as the density matrices
$\rho _1, \rho _2, \ldots $
could be entangled tensors in general.
Remark 3.2. We have shown in Equation (3.3) that
$p(\tau ) = \text {Tr} \big [ \rho _n \Pi ^B(\tau ) \big ]$
is the probability of obtaining
$\tau \in 2^n$
when the first n many qubits of
$\rho $
are measured sequentially in increasing order. (That is, the first qubit is measured, followed by a measurement of the second qubit of the resulting post-measurement state, followed by a measurement of the third qubit on the post-measurement state after the first and second measurements and so on.) Obviously this probability should equal the probability of obtaining
$\tau $
when the first n qubits of
$\rho $
are measured simultaneously. We verify this. Recall from Section 3.1 that
$\Lambda _n := \{\Pi ^B(\tau ):\tau \in 2^n\}$
is the measurement corresponding to simultaneously measuring the first n qubits using B. By the axioms in Section 2.2, the probability of obtaining
$\tau $
when measuring
$\rho _n$
using
$\Lambda _n$
is equal to
$\text {Tr} \big [\rho _n \Pi ^B(\tau )\big ]$
. This is equal to the value of
$p(\tau )$
in Equation (3.3) which is the probability of obtaining
$\tau $
when the qubits of
$\rho _n$
are measured sequentially in increasing order. This shows that for any n, the sequential process of qubit-wise measuring
$\rho _n$
in B is equivalent to measuring
$\rho _n$
using
$\Lambda _n$
.
3.4 Formal details
We formalize the above intuitive description of measurement of a state. Take a state
$\rho $
and a system B. We will first define a premeasure (see Definition 3.3)
$p^B_\rho $
depending on
$\rho $
and B. Letting
$\mu ^B_\rho $
be the unique extension of
$p^B_\rho $
to
$2^\omega $
we will show that for any measurable
$S\subseteq 2^\omega $
,
$\mu ^B_\rho (S)$
is the probability that the qubit-wise measurement of
$\rho $
in the system B yields an element from S.
Definition 3.3. Let
$\rho =(\rho _n)_{n}$
be a state and
$B= ((b^{n}_{0},b^{n}_{1}))_{n}$
be a system. The function
$p^B_\rho $
on
$2^{<\omega }$
is defined as:
$p^B_\rho (\emptyset )=1$
. For all
$n>0$
and all
$\tau \in 2^{n}$
,
$$ \begin{align} p^B_\rho(\tau)=\text{Tr}[\rho_n \Pi^B(\tau)]. \end{align} $$
As explained in Remark 3.2,
$p^B_\rho (\tau )$
is the probability of obtaining
$\tau $
when the first
$|\tau |$
many qubits of
$\rho $
are measured in the system B, where the measurement can be thought of as a simultaneous measurement of all n qubits or as a sequential, qubit-wise measurement. We now show the following lemma.
Lemma 3.4. The function
$p^B_\rho $
is a premeasure.
Proof. As B will be fixed henceforth, we drop the B superscript to keep the notation readable. We show that
$\forall \tau \in 2^{<\omega }, p_\rho (\tau )=p_\rho (\tau 0)+p_\rho (\tau 1)$
. If
$\tau = \emptyset $
, then it can be easily checked that
$p_\rho (0)+p_\rho (1)=1$
. Let
$n>0$
be arbitrary and fix a
$\tau \in 2^n$
. Recall the notation in (3.2). Note that
$\Pi (\tau j)=\Pi (\tau )\otimes \Pi _j (n+1)$
for
$j\in \{0,1\}$
. So
$$\begin{align*}p_\rho(\tau 0) + p_\rho(\tau 1) = \text{Tr} \big[ \rho_{n+1} (\Pi(\tau) \otimes \Pi_0 (n+1))] + \text{Tr} \big[ \rho_{n+1} (\Pi(\tau) \otimes \Pi_1 (n+1))].\end{align*}$$
By the linearity of
$\text {Tr}$
, this equals
$$\begin{align*}\text{Tr}\big[\rho_{n+1}\big(\Pi(\tau)\otimes \{\Pi_0 (n+1) + \Pi_1 (n+1)\}\big) ]= \text{Tr}\big[ \rho_{n+1}(\Pi(\tau)\otimes I) ].\end{align*}$$
By the definition of the partial trace, this equals
$$\begin{align*}\text{Tr}\big[ \text{Tr}_{\mathbb{C}^2}(\rho_{n+1}) \Pi(\tau) ] = \text{Tr}[\rho_n \Pi(\tau) ] = p_\rho(\tau).\\[-34pt] \end{align*}$$
As
$p^B_\rho $
is a premeasure, we can make the following definition.
Definition 3.5. Let
$\mu ^B_\rho $
be the probability measure on
$2^\omega $
which is the unique extension of
$p^B_\rho $
.
As
$p^B_\rho (\tau )$
is the probability of obtaining
$\tau \in 2^{<\omega }$
when the first
$|\tau |$
many qubits of
$\rho $
are measured in B,
$\mu ^B_\rho $
is the probability distribution induced on
$2^\omega $
by the qubit-wise measurement of
$\rho $
in B. That is, for any measurable
$S \subseteq 2^\omega $
,
$\mu ^B_\rho (S)$
is the probability that qubit-wise measurement of
$\rho $
in B yields an infinite bitstring in S.
4 Quantum measurement randomness
Recall that
$\textbf {ML}\subseteq 2^\omega $
is the set of Martin-Löf random bitstrings. If
$\rho $
is a state and B is a system,
$\mu ^{B}_{\rho } (\textbf {ML})$
is the probability of getting a Martin-Löf random bitstring by a qubit-wise measurement of
$\rho $
in B as described in Section 3.
Definition 4.1. A state
$\rho $
is quantum measurement random if
$\mu ^{B}_{\rho } (\textbf {ML})=1$
for any B.
Nies and Stephan define a measure
$\mu $
on
$2^\omega $
to be Martin-Löf absolutely continuous if
$\mu (\textbf {ML})=1$
(see [Reference Nies and Stephan11, Definition 2.3]). So a state
$\rho $
is quantum measurement random if
$\mu ^{B}_{\rho }$
is Martin-Löf absolutely continuous for any B. Any quantum measurement random state hence yields a countably infinite collection of Martin-Löf absolutely continuous measures.
4.1 Relationship with other randomness notions
We give an overview of the relationships of quantum measurement randomness with other notions of randomness which will be established in this article. We first study how quantum measurement randomness relates to Martin-Löf randomness. Recall the state
$\rho _X$
from Section 2.3. We now show that quantum measurement randomness agrees with Martin-Löf randomness when restricted to states of the form
$\rho _X$
for
$X \in 2^\omega $
.
Lemma 4.2.
$X \in 2^\omega $
is Martin-Löf random iff
$\rho _X$
is quantum measurement random.
Proof. (
$\Longrightarrow $
): Let X be Martin-Löf random. Then
$\rho _X$
is quantum Martin-Löf random by [Reference Nies and Scholz10, Theorem 3.9]. By Theorem 4.3,
$\rho _X$
is quantum measurement random.
(
$\Longleftarrow $
) Let X not be Martin-Löf random and let
$\rho :=\rho _X$
. Let
$B=((b^n_0, b^n_1))_n$
be the system such that
$b^n_i = |i\rangle $
for all n and i.
For any n and any
$\tau \in 2^n$
,
$\mu ^B_\rho (\tau ) = \text {Tr} (|X\upharpoonright n \big>\big <X\upharpoonright n |\Pi ^B (\tau ))= |\langle \tau |X\upharpoonright n \rangle |^2$
which is 1 if
$\tau = X\upharpoonright n$
and is 0 otherwise. So,
$\mu ^B_\rho (X)=1$
. As X is not Martin-Löf random this shows that
$\rho $
is not quantum measurement random.
The result [Reference Nies and Scholz10, Theorem 3.9] (which says that
$X \in 2^\omega $
is Martin-Löf random iff
$\rho _X$
is quantum Martin-Löf random) together with the above result imply that quantum measurement randomness is equivalent to quantum Martin-Löf randomness for states of the form
$\rho _X$
for
$X \in 2^\omega $
. Also since both quantum measurement randomness and quantum Martin-Löf randomness are equivalent to Martin-Löf randomness when restricted to states of the form
$\rho _X$
for
$X \in 2^\omega $
, quantum measurement randomness and quantum Martin-Löf randomness may be thought of as generalizations of Martin-Löf randomness from
$2^\omega $
to the quantum realm. We note that quantum Schnorr randomness does not imply and is not implied by quantum measurement randomness: Let
$X \in 2^\omega $
be Schnorr random and not Martin-Löf random. By [Reference Bhojraj3, Lemma 3.9]
$\rho _X$
is quantum Schnorr random and by the above it is not quantum measurement random. By Lemma 6.7 there is a quantum measurement random state which is not quantum Schnorr random. This shows that quantum Schnorr randomness and quantum measurement randomness are incomparable notions. Both quantum Schnorr randomness and quantum measurement randomness imply a quantum version of the law of large numbers by [Reference Bhojraj3, Theorem 4.2] and Section 4.3. We will show later that quantum Martin-Löf randomness strictly implies quantum measurement randomness (Lemma 4.3 and Theorem 6). Let q-MLR, q-MR, and q-SR denote quantum Martin-Löf random, quantum measurement random, and quantum Schnorr random, respectively. Let q-LLN denote the property of satisfying the quantum law of large numbers. Then the following diagram summarizes all implications existing between these four notions. All implications are strict. There are no implications between q-MR and q-SR.
4.2 Quantum Martin-Löf randomness implies quantum measurement randomness
Lemma 4.3. If a state is quantum Martin-Löf random, then it is quantum measurement random.
Proof. Let
$\rho =(\rho _n)_n$
be quantum Martin-Löf random. Suppose towards a contradiction that there is a
$\delta>0$
and a system
$B= ((b^{n}_{0},b^{n}_{1}))_n$
such that
$\mu ^{B}_{\rho } (2^{\omega }\backslash \textbf {ML})>\delta $
.
The idea is to derive a contradiction by using the universal ML-test to construct a q-ML-test that
$\rho $
fails at order
$\delta $
. Let
$(S^{m})_m$
be the universal ML-test [Reference Nies8] and let for all m,
where the
$A^{m}_{i}$
s satisfy the conditions of Definition 2.7. Since
$\lambda (S^m)\leq 2^{-m}$
, we have
$$\begin{align*}A^{m}_{i}= \{\tau^{m,i}_{1},\dots,\tau^{m,i}_{k^{m,i}}\} \subseteq 2^i\end{align*}$$
for some
$0 \leq k^{m,i} \leq 2^{i-m}$
for all m and
$i\geq m$
.
For all m and all
$i\geq m$
, define the special projection:
$$ \begin{align} p^{m}_{i}= \sum_{a\leq k^{m,i}} \Pi^B(\tau^{m,i}_a). \end{align} $$
Let
$P^{m}:=(p^{m}_{i})_{i=m}^\infty $
. It is easy to see that
$(P^{m})_{m}$
is a q-ML-test.Footnote
6
We now show that
$\rho $
fails
$(P^m)_m$
.
As
$(S^m)_m$
is the universal ML-test, we have for all m that
$\mu ^B_\rho (S^m)\geq \mu ^{B}_{\rho }(2^{\omega }\backslash \textbf {ML})>\delta $
. Since (4.1) is an increasing union, for all m there exists an
$i(m)>m$
such that
Fix an m and
$i=i(m)$
. Let
$p^m_i$
be as in (4.2). By (3.4) we have that
As m was arbitrary, we have shown that
$\rho $
fails the q-ML test
$(P^{m})_m $
at order
$\delta $
thus contradicting that
$\rho $
is quantum Martin-Löf random.
Theorem 6.4 shows that the implication in Lemma 4.3 is strict.
4.3 The quantum law of large numbers
A quantum version of the law of large numbers (q-LLN) has been formulated ([Reference Bhojraj3, Definition 4.1] and [Reference Nies9, Section 6.6]). Satisfying the law of large numbers is a weak form of randomness and thus ought to be implied by any reasonable notion of classical randomness [Reference Downey and Hirschfeldt7].Footnote 7 Similarly, quantum measurement randomness can qualify as a plausible notion of quantum randomness only if it implies satisfaction of the q-LLN.Footnote 8 Theorem 4.4 shows this.
Theorem 4.4. All quantum measurement random states satisfy the q-LLN.
The proof of Theorem 4.4 is almost identical to the proof of [Reference Bhojraj3, Theorem 4.2] and is omitted.
5 General computable measures and Kronecker products
We prove two general mathematical results that may be inherently interesting beyond their specific applications in Section 6. The first (Theorem 5.2) concerns Martin-Löf randomness for general computable measures and is invoked in the proof of Lemma 6.5. The second (Lemma 5.3) is a multi-linear algebraic result characterizing the entries of vectors which are Kronecker products of vectors in
$\mathbb {C}^2$
. It is crucially used in the proof of Lemma 6.3.
Definition 5.1. Let
$\mu $
be any computable measure on
$2^\omega $
. A
$\mu $
-ML-test is a computable sequence
$(S_{m})_{m=1}^\infty $
of
$\Sigma _{1}^{0}$
classes such that
$\mu (S_m) \leq 2^{-m}$
for all m. A
$X\in 2^\omega $
fails
$(S_m)_m$
if
$X \in S_m$
for all m and X is
$\mu $
-Martin-Löf random if it does not fail any
$\mu $
-ML-test.Footnote
9
Let
$\textbf {ML}(\mu )$
denote the set of infinite bitstrings Martin-Löf random with respect to
$\mu $
.
Note that
$\textbf {ML}(\lambda )=\textbf {ML}$
using Definition 2.10. Theorem 5.2 which concerns the inclusion between
$\textbf {ML}(\nu )$
and
$\textbf {ML}(\pi )$
when
$\nu $
and
$\pi $
are related in a specific way extends a result of V. Vovk [Reference Vovk14]. We use the notation: Let f be a strictly positive computable function on
$\omega $
and let
$\beta \in \omega - \{0\}$
. Any
$X\in 2^\omega $
may be written as a concatenation of bitstrings of lengths
$f(\beta ), f(\beta + 1), f(\beta + 2), \ldots $
in order from left to right. Formally, X can be written as
$X= \sigma _{\beta } \sigma _{\beta + 1} \sigma _{\beta +2} \dots $
, where
$\sigma _i \in 2^{f(i)}$
for
$i\geq \beta $
.
Theorem 5.2. Let f be a strictly positive computable function on
$\omega $
and let
$\beta $
be a positive natural number. Let
$(\nu _i)_{i=\beta }^\infty $
and
$(\pi _i)_{i=\beta }^\infty $
be uniformly computable (in i) sequences such that
$\nu _i$
and
$\pi _i$
are measures on
$2^{f(i)}$
such that
$2 \nu _i(\tau )- \pi _i(\tau )>0$
for all
$\tau \in 2^{f(i)}$
. Let
$\nu = \prod _{i=\beta }^\infty \nu _i$
and
$\pi = \prod _{i=\beta }^\infty \pi _i$
be the corresponding product measures on
$2^\omega $
. Suppose that there is a constant C such that for all
$X \in \textbf {ML}(\nu )$
written as a concatenation
$X= \sigma _{\beta } \sigma _{\beta +1} \sigma _{\beta +2} \dots \sigma _{i} \dots $
as above
$$\begin{align*}\prod_{i=\beta}^\infty \big(1- \dfrac{(\nu_i(\sigma_i)-\pi_i(\sigma_i))^{2}}{\nu_i (\sigma_i)^2 }\big) = C.\end{align*}$$
Then
$\textbf {ML} (\nu ) \subseteq \textbf {ML} (\pi )$
.
The proof of Theorem 5.2 borrows ideas used in the proof of [Reference Shen, Uspensky and Vereshchagin13, Theorem 196(a)] which is as follows.
Theorem. Let
$\varepsilon>0$
and
$(p_i)_i$
and
$(q_i)_i$
be computable sequences of reals in
$(\varepsilon, 1-\varepsilon )$
. Let
$\nu _i$
and
$\pi _i$
be the measures on
$\{0,1\}$
associated with the Bernoulli(
$p_i)$
distribution and Bernoulli(
$q_i$
) distribution, respectively. Let
$\nu =\prod _i \mu _i$
and let
$\pi = \prod _i \pi _i$
be the corresponding product measures on
$2^\omega $
. Then
$\textbf {ML}(\nu )= \textbf {ML}(\pi )$
if
$\sum _i (p_i - q_i)^2 < \infty $
.Footnote
10
We use the notions of prefix-free complexity (K) and monotone complexity (KM) for strings. The definitions of K and KM, which can be found in [Reference Shen, Uspensky and Vereshchagin13, Chapter 4 and Section 5.5], respectively, are not relevant to my arguments. The symbols
$\leq ^+$
and
$\leq ^\times $
will denote an inequality which holds up to an additive constant and up to a multiplicative constant, respectively.
Proof. Let
$X \in \textbf {ML}(\nu )$
. We aim to show that
$X \in \textbf {ML}(\pi )$
. Recall the notation from above which writes X as a concatenation of bitstrings
$\sigma _i \in 2^{f(i)}$
. For
$n\geq \beta $
, let
$S_n:= \sigma _{\beta } \sigma _{\beta +1} \dots \sigma _n$
. For all i, let
$r_{i}:= \nu _{i}(\sigma _{i})$
and
$\delta _{i}:= \pi _i(\sigma _{i})-r_{i}$
. We outline the main idea of the proof before giving the details.
Proof sketch.
Note that
$\nu $
and
$\pi $
are computable. As
$X \in \textbf {ML}(\nu )$
and
$\nu $
is computable, the Levin–Schnorr theorem (see, e.g.,[Reference Shen, Uspensky and Vereshchagin13, Theorem 90, Section 5.6]) implies that
$$ \begin{align} \forall n \geq \beta, -\log(\nu(S_{n})) \leq^+ KM(S_{n}). \end{align} $$
Showing that
$\nu (S_n) \leq ^\times \pi (S_n)$
will imply that
$-\log (\pi (S_n)) \leq ^+ -\log (\nu (S_n)) \leq ^+ KM(S_n)$
. This shows that
$-\log (\pi (S_n))-KM(S_n)\leq O(1)$
. Theorem 93 from [Reference Shen, Uspensky and Vereshchagin13] then implies that
$X \in MLR(\pi )$
, as required.
We hence desire to show that
$ \nu (S_n)\leq ^\times \pi (S_n)$
or equivalently that
$ \prod _{i=\beta }^n r_i \leq ^\times \prod _{i=\beta }^n (r_i + \delta _i)$
. We observe that
$$ \begin{align*} \prod_{i=\beta}^{n} (r_i - \delta_i)\prod_{i=\beta}^{n}(r_i + \delta_i) = \prod_{i=\beta}^{n} r^{2}_{i} \prod_{i=\beta}^{n} \big(1-\dfrac{\delta^{2}_{i}}{r^{2}_{i}}\big) = C \prod_{i=\beta}^{n} r^{2}_{i} \end{align*} $$
since by assumption we have
$\prod _{i=\beta }^{n} \big (1-(\delta ^{2}_{i}/r^{2}_{i})\big )=C$
.
The desired result (namely that
$\prod _{i=\beta }^n r_i \leq ^\times \prod _{i=\beta }^n (r_i + \delta _i)$
) can hence be proven by showing that
$\forall n\geq \beta $
, we have
$$ \begin{align*} \prod_{i=\beta}^{n} (r_{i}- \delta_{i}) \leq^\times \prod_{i=\beta}^{n} r_{i}. \end{align*} $$
We show this (see (5.3)) by defining a computable measure,
$\nu '$
such that
$\nu '(S_n)=\prod _{i=\beta }^n r_i - \delta _i$
and then relating it to
$\nu $
. Proof details: Define a family,
$(\nu ^{\prime }_i)_{i=\beta }^\infty $
, where
$\nu ^{\prime }_i$
is a function on
$2^{f(i)}$
. For all
$i\geq \beta $
, define
$\nu ^{\prime }_i$
on
$2^i$
by
$\nu ^{\prime }_i (\tau ):=2 \nu _i(\tau ) - \pi _i(\tau )$
. Informally,
$\nu ^{\prime }_i$
is a ‘reflection of
$\pi _i$
around
$\nu _i$
’. Each
$\nu ^{\prime }_i$
is a measure on
$2^{f(i)}$
since
$\sum _{\tau \in 2^i} \nu ^{\prime }_i (\tau ) = 1$
and as
$2 \nu _i(\tau )- \pi _i(\tau )\geq 0$
for all
$\tau \in 2^{f(i)}$
by assumption. Let
$\nu '$
be the product measure,
$\Pi _{i=\beta }^\infty \nu ^{\prime }_i$
. In particular, for all
$n\geq \beta $
we have that
$$ \begin{align} \nu'(S_{n})= \prod_{i=\beta}^{n} \nu^{\prime}_{i}(\sigma_{i}). \end{align} $$
Note that
$\nu ^{\prime }_i(\sigma _i)= \nu _i(\sigma _i) - (\pi _i(\sigma _i) - \nu _i(\sigma _i))= r_i - \delta _i$
for all i. So, for all
$n\geq \beta $
we have that
$$\begin{align*}\nu'(S_{n}) = \prod_{i=\beta}^{n} r_{i}-\delta_{i}.\end{align*}$$
As
$\nu '$
is computable (as
$\nu $
and
$\pi $
are), Theorem 89 in Section 5.6 in [Reference Shen, Uspensky and Vereshchagin13] gives that for all
$n\geq \beta $
,
$$ \begin{align*} KM(S_{n}) \leq^+ -\log(\nu '(S_{n})). \end{align*} $$
Combining this with (5.1) and taking exponents, we see that for all
$ n\geq \beta ,$
$$\begin{align*}\nu'(S_{n}) \leq^\times \nu (S_{n}),\end{align*}$$
and hence that, for all
$n\geq \beta $
,
$$ \begin{align} \prod_{i=\beta}^{n} r_{i}- \delta_{i} \leq^\times \prod_{i=\beta}^{n} r_{i}. \end{align} $$
Now, for all
$n\geq \beta $
we have that
$$ \begin{align} \prod_{i=\beta}^{n} (r_i - \delta_i)\prod_{i=\beta}^{n}(r_i + \delta_i) = \prod_{i=\beta}^{n} r^{2}_{i}- \delta^{2}_{i}= \prod_{i=\beta}^{n} r^{2}_{i} \prod_{i=\beta}^{n} \big(1-\dfrac{\delta^{2}_{i}}{r^{2}_{i}}\big) = C \prod_{i=\beta}^{n} r^{2}_{i}. \end{align} $$
The last equality followed by noting that
$\delta ^{2}_i = (\nu _i(\sigma _i)-\pi _i(\sigma _i))^2$
and then using the assumption. This gives that for all
$n\geq \beta $
,
$$ \begin{align} \prod_{i=\beta}^{n} r^{2}_{i} \leq^\times \prod_{i=\beta}^{n} (r_i - \delta_i)\prod_{i=\beta}^{n}(r_i + \delta_i). \end{align} $$
Divide the right side by
$\prod _{i=\beta }^{n} (r_i - \delta _i)$
Footnote
11
and the left side by
$ \prod _{i=\beta }^{n} r_i $
and use (5.3) to see that for all
$n\geq \beta $
,
$$\begin{align*}\prod_{i=\beta}^{n} r_{i}\leq^\times \prod_{i=\beta}^{n} r_{i}+ \delta_{i}. \end{align*}$$
Recalling the definitions of
$r_i$
and
$\delta _i$
we have that for all
$n\geq \beta $
,
$$\begin{align*}\nu(S_n)\leq^\times \pi(S_n). \end{align*}$$
Together with (5.1), this gives that for all
$n\geq \beta $
,
$$\begin{align*}-\log(\pi(S_n)) \leq^+ -\log(\nu(S_n)) \leq^+ KM(S_n), \end{align*}$$
and thus that for all
$n\geq \beta $
,
$$\begin{align*}-\log(\pi(S_n)) - KM(S_n) \leq O(1).\end{align*}$$
As
$|S_n|$
is increasing in n, this shows that
$-\log (\pi (S_n)) - KM(S_n)$
is bounded from above over all n. That is, there is a uniform bound on
$-\log (\pi (x)) - KM(x)$
for infinitely many prefixes x of X. [Reference Shen, Uspensky and Vereshchagin13, Theorem 93, Section 5.6] then implies that
$X \in \textbf {ML}(\pi )$
. As X was an arbitrary element of
$\textbf {ML}(\nu )$
, Theorem 5.2 is proved.
The following combinatorial lemma concerns the entries of vectors which are Kronecker products of vectors in
$\mathbb {C}^2$
.
Lemma 5.3. Let
$\{[a_{i}, b_{i}]^{T}\}_{i=1}^{n} $
be a set of column vectors in
$\mathbb {C}^2$
. Let
$V=\bigotimes _{i=1}^{n}[a_{i}, b_{i}]^{T}$
be their Kronecker product. If
$V=[v_{1},v_{2},\dots ,v_{2^n}]^{T}$
, then for all
$1\leq k \leq 2^{n-1}$
, we have that
$$\begin{align*}|v_{k}||v_{2^{n}-k+1}| = \prod_{i=1}^{n} |a_{i}||b_{i}|.\end{align*}$$
This lemma can be proved easily using induction and we only give a sketch the proof. We use the following convention for the Kronecker product [Reference Regalia and Mitra12]:
$$\begin{align*}\begin{bmatrix} a_1\\ b_1\\ \end{bmatrix} \otimes \begin{bmatrix} a_2\\ b_2\\ \end{bmatrix} = \begin{bmatrix} a_1 a_2\\ b_1 a_2\\ a_1 b_2\\ b_1 b_2 \end{bmatrix} .\end{align*}$$
For natural numbers u and q, let
$[u]_{q}$
denote the remainder obtained by dividing u by q. For any
$k\leq 2^{n-1}$
,
$v_k$
has the form
$v_k= \prod _{i=1}^{n} c^{k}_{i}$
, for some
$c^{k}_{i} \in \{a_{i}, b_{i}\}$
and
$v_{2^{n}-k+1}$
has the form
$v_{2^{n}-k+1} = \prod _{i=1}^{n} e^{k}_{i}$
, for some
$e^{k}_{i} \in \{a_{i}, b_{i}\}$
. Note that
$c^{k}_{1}=a_{1}$
iff k is odd iff
$e^{k}_{1}=b_{1}$
. Similarly, we have that
$c^{k}_{2}=a_{2}$
iff
$[k]_{2^{2}} \in \{1,2\}$
iff
$e^{k}_{2}=b_{2}$
. In general, for
$i\leq n$
, for all
$k\leq 2^{n-1}$
,
$$\begin{align*}c^{k}_{i}=a_{i} \iff [k]_{2^{i}} \in \{1,\dots,2^{i-1}\} \iff e^{k}_{i}=b_{i}.\end{align*}$$
6 A quantum measurement random state that is not quantum Martin-Löf random
The state will be constructed as an infinite tensor product of the matrices
$d_n$
defined below.
Definition 6.1. For any
$n>0$
, let
$r_{n}:=\lfloor 2^n/n \rfloor $
and let
$d_n$
be a
$2^n$
by
$2^n$
matrix with
$2^{-n}$
along the diagonal and
$r_{n}$
many
$2^{-n}$
’s on the extreme ends of the anti-diagonal. That is,
$d_n$
is the symmetric matrix such that its non-zero entries are given as:
-
(1) For first
$r_n$
rows: For
$i \leq r_n $
,
$d_{n}(i,j)= 2^{-n}$
if
$j=i$
or
$j=2^{n}-i+1$
. -
(2) For the rows from
$r_n + 1$
to
$2^{n}-r_{n}-1$
: For
$ r_n < i < 2^{n}-r_{n}$
,
$d_{n}(i,j)= 2^{-n}$
if
$j=i$
. -
(3) For the last
$r_n$
many rows: For
$i \geq 2^n - r_n $
,
$d_{n}(i,j)= 2^{-n}$
if
$j=i$
or
$j=2^{n}-i+1$
.
We illustrate the case
$n=3$
by an example. We have
$r_{3}=2$
and
$$\begin{align*}d_3 = \begin{bmatrix} 2^{-3} & 0 & 0 & 0 & 0 & 0 & 0 & 2^{-3}\\ 0 & 2^{-3} & 0 & 0 & 0 & 0 & 2^{-3} & 0\\ 0 & 0 & 2^{-3} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2^{-3} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2^{-3} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2^{-3} & 0 & 0\\ 0 & 2^{-3} & 0 & 0 & 0 & 0 & 2^{-3} & 0\\ 2^{-3} & 0 & 0 & 0 & 0 & 0 & 0 & 2^{-3}\\ \end{bmatrix}. \end{align*}$$
Define
$\theta _4:=0$
and for natural numbers
$i\geq 5$
, define
$$ \begin{align} \theta_{i} := \sum_{j=5}^{i}j. \end{align} $$
Definition 6.2. For all
$N\geq 5$
, let
$S_{N}:= \bigotimes _{n=5}^{N} d_{n}$
. Note that
$S_N$
is a density matrix on
$\theta _N$
qubits. Let
$\rho := (\rho _n)_n$
be the state such that
$\rho _{\theta _N}= S_N$
for all N.
Informally,
$\rho $
is the state: ‘
$\bigotimes _{n=5}^\infty d_{n}$
’. Formally,
$\rho $
is the state such that for all N, its finite initial segment of length
$\theta _N$
is given by the density matrix
$S_N$
. The reader may be curious about the reason for the choice of
$5$
when constructing
$\rho ^e$
. This reason will become clear in the proof of Lemma 6.5 (see Remark 6.6). We first establish a property of
$d_n$
which is vital to showing that
$\rho $
is quantum measurement random.
Lemma 6.3. Let
$n>0$
. Let
$\{[a_{i}, b_{i}]^{T}\}_{i=1}^{n} $
be such that for all i,
$[a_{i}, b_{i}]^{T}$
is a unit vector in
$\mathbb {C}^2$
and let
$W=\bigotimes _{i=1}^{n}[a_{i}, b_{i}]^{T}$
. Then
$|\big <W|d_{n}|W\big>| \in [2^{-n}(1-2n^{-1}),2^{-n}(1+2n^{-1})]$
.
The lemma roughly says that for all n, if W is a tensor product of n many unit norm vectors in
$\mathbb {C}$
, then
$|\big <W|d_{n}|W\big>|$
is ‘close to’
$2^{-n}$
. This will be used to later show that
$\mu ^B_{\rho }$
is ‘close’ to
$\lambda $
for any B. The proof crucially relies on Lemma 5.3.
Proof. Fix n and W as in the statement and write
$d_n$
as a block matrix with each block of size
$2^{n-1}$
by
$2^{n-1}$
:
$$\begin{align*}d_n = \begin{bmatrix} A & B\\ B^{T} & A\\ \end{bmatrix}. \end{align*}$$
Let
$V=\bigotimes _{i=1}^{n-1}[a_{i}, b_{i}]^{T}$
. Then in block matrix form,
$$\begin{align*}W=[a_{n}V^{T}, b_{n}V^{T}]^{T}.\end{align*}$$
Let
$V=[v_{1},v_{2},\dots ,v_{2^{n-1}}]^{T}$
. It is easily checked that
$$\begin{align*}\big<W|d_{n}|W\big>= 2^{-n} + a_{n}^{*} b_{n} V^{\dagger}BV+ a_{n}b_{n}^{*} V^{\dagger}B^{T}V.\end{align*}$$
By the definition of
$d_n$
, the block B has
$2^{-n}$
at the
$r_n$
many top-right entries along the antidiagonal and has zeroes everywhere else. This form of B implies that
$$\begin{align*}V^{\dagger}BV &= 2^{-n} [v^{*}_{1},v^{*}_{2},\dots,v^{*}_{2^{n-1}}][v_{2^{n-1}},v_{2^{n-1}-1},\dots,v_{2^{n-1}-r_{n}+1},0,\dots ,0]^{T}.\\&= 2^{-n}\sum_{k=1}^{r_{n}}v^{*}_{k}v_{2^{n-1}-k+1}. \end{align*}$$
By Lemma 5.3,
$$\begin{align*}|V^{\dagger}BV| \leq 2^{-n}\sum_{k=1}^{r_{n}}|v_{k}||v_{2^{n-1}-k+1}| = 2^{-n} r_{n} \prod_{i=1}^{n-1} |a_{i}||b_{i}| = 2^{-n} r_{n} \prod_{i=1}^{n-1} |a_{i}|\sqrt{1-|a_{i}|^{2}}. \end{align*}$$
Since the maximum value of
$x\sqrt {1-x^{2}}$
on
$x \in [0,1]$
is
$1/2$
and recalling the definition of
$r_n$
we see that
$$\begin{align*}|V^{\dagger}BV| \leq 2^{-n}\dfrac{1}{2^{n-1}}\dfrac{2^{n}}{n} = \dfrac{2^{1-n}}{n}. \end{align*}$$
Similarly,
$|V^{\dagger }B^{T}V| \leq \dfrac {2^{1-n}}{n}.$
Noting that
$|a_{n}^{*} b_{n}|, |a_{n} b_{n}^{*}| \leq 1/2$
,
$$\begin{align*}|\big<W|d_{n}|W\big>| \leq 2^{-n} + |a_{n}^{*} b_{n} V^{\dagger}BV| + |a_{n}b_{n}^{*} V^{\dagger}B^{T}V|\leq 2^{-n} + \dfrac{2^{1-n}}{n},\end{align*}$$
and
$$\begin{align*}|\big<W|d_{n}|W\big>| \geq 2^{-n} - |a_{n}^{*} b_{n} V^{\dagger}BV| - |a_{n}b_{n}^{*} V^{\dagger}B^{T}V|\geq 2^{-n} - \dfrac{2^{1-n}}{n}.\\[-34pt] \end{align*}$$
The following is our central result. Recall the state
$\rho $
from Definition 6.2.
Theorem 6.4. The state
$\rho $
is quantum measurement random and not quantum Martin-Löf random.
Proof. It is not hard to show that
$\rho $
is not quantum Martin-Löf random (Lemma 6.7). Showing that
$\rho $
is quantum measurement random (Lemma 6.5) is more involved and crucially relies on two general mathematical results from Section 5. The first (Theorem 5.2) extends a result of V.Vovk concerning Martin-Löf randomness for computable measures. The second (Lemma 5.3) is a combinatorial result about Kronecker products and gets used here through Lemma 6.3, a property of the
$d_n$
’s proved earlier.
Lemma 6.5. The state
$\rho $
is quantum measurement random.
Proof. Let
$B = ((b^{n}_{0},b^{n}_{1}))_{n=1}^{\infty }$
be an arbitrary computable measurement system. We need to show that
$\mu ^B_\rho (\textbf {ML})=1$
.
Proof sketch.
We prove that
$ \textbf {ML}(\mu ^{B}_{\rho }) \subseteq \textbf {ML}$
. Since
$\mu ^{B}_{\rho }[\textbf {ML}(\mu ^{B}_{\rho })]=1$
, this implies that
$\mu ^{B}_{\rho }(\textbf {ML})=1$
as desired.
To simplify notation, denote
$\mu ^{B}_{\rho }$
by
$\mu $
. To show
$ \textbf {ML}(\mu ) \subseteq \textbf {ML}$
, we show that
$\mu $
and
$\lambda $
satisfy the conditions of Theorem 5.2 under the replacements:
$\nu \mapsto \mu $
,
$\pi \mapsto \lambda $
,
$\beta \mapsto 5$
, and
$f \mapsto $
the identity function. Theorem 5.2 then implies that
$\textbf {ML}(\mu )\subseteq \textbf {ML}(\lambda )= \textbf {ML}$
as required.
Proof details: Recall the definition of
$\theta _i$
(6.1). Let
$\mu _{i}$
and
$\lambda _i$
be functions on
$2^i$
such that for all
$\tau \in 2^i$
,
$$\begin{align*}\mu_{i}(\tau):= \text{Tr}\big[d_{i}(|\bigotimes_{q=1}^{i} b^{q+\theta_{i-1}}_{\tau(q)}\big> \big<\bigotimes_{q=1}^{i} b^{q+\theta_{i-1}}_{\tau(q)}|)\big] = \big<\bigotimes_{q=1}^{i} b^{q+\theta_{i-1}}_{\tau(q)}| d_{i}|\bigotimes_{q=1}^{i} b^{q+\theta_{i-1}}_{\tau(q)}\big> ,\end{align*}$$
and
$$\begin{align*}\lambda_i (\tau):=2^{-i}.\end{align*}$$
Note that
$\sum _{\tau \in 2^i} \mu _i(\tau ) = 1$
and hence that
$\mu _i$
is a measure on
$2^i$
for each i.
$\mu _i$
is the measure on
$2^{i}$
induced by measuring the
$i^{th}$
block of
$\rho $
(namely,
$d_{i}$
) when
$\rho $
is measured in
$B=((b^n_0, b^n_1))_{n=1}^\infty $
. This measurement hence uses the
$i^{th}$
block of the system B, when B is divided into a sequence (indexed by
$i=5,6,\ldots $
) of blocks with the
$i^{th}$
block in the sequence having length i. The indexing of the b’s in the definition of
$\mu _i$
starts from
$1+\theta _{i-1}$
as the measurement of the
$i^{th}$
block of
$\rho $
uses the bases
$B=((b^n_0, b^n_1))_{n=1+\theta _{i-1}}^{\theta _i}$
. By the cyclicity of the trace and by Lemma 6.3,
$$\begin{align*}\mu_{i}(\tau)= \big<\bigotimes_{q=1}^{i} b^{q+\theta_{i-1}}_{\tau(q)}|d_{i}|\bigotimes_{q=1}^{i} b^{q+\theta_{i-1}}_{\tau(q)}\big> \in [2^{-i}(1-2i^{-1}),2^{-i}(1+2i^{-1})].\end{align*}$$
By (3.3) and by the form of
$\rho $
we see that
$\mu $
is the product measure
$\Pi _{i=5}^\infty \mu _i$
.
Note that
$(\mu _i)_{i=5}^\infty $
and
$(\lambda _i)_{i=5}^\infty $
are uniformly computable (in i) sequences where
$\mu _i$
and
$\lambda _i$
are measures on
$2^{f(i)}$
where f is the strictly positive function on
$\omega - \{0\}$
given by
$f(i)=i$
.
Fix a
$\tau \in 2^i$
. By Lemma 6.3,
$2 \mu _i(\tau )- \lambda _i(\tau )\geq 2^{1-i}- 2^{2-i}i^{-1} - 2^{-i}$
. As
$i\geq 5$
, this is strictly greater than
$ 2^{1-i}- 2^{2-i}4^{-1} - 2^{-i}=2^{1-i}>0$
. This shows that
${2\mu _i(\tau )- \lambda _i(\tau )>0}$
for all
$\tau \in 2^i$
.Footnote
12
Let X be an arbitrary element of
$ \textbf {ML}(\mu )$
. Write X as a concatenation of finite bitstrings:
$$\begin{align*}X=\sigma_{5}\sigma_{6}\dots \sigma_{i}\dots,\end{align*}$$
where
$\sigma _{i} \in 2^i$
for all
$i \in \omega $
. Let
$S_n := \sigma _{5}\sigma _{6}\dots \sigma _{n}$
be the concatenation up to n. In particular, for all n,
$$ \begin{align} \mu(S_{n})= \prod_{i=5}^{n} \mu_{i}(\sigma_{i}). \end{align} $$
As
$\lambda $
is the uniform measure, for all n,
$$ \begin{align} \lambda(S_{n})= \prod_{i=5}^{n} \lambda_{i}(\sigma_{i}). \end{align} $$
For all i, let
$r_{i}:= \mu _{i}(\sigma _{i})$
and
$\delta _{i}:= \lambda _i(\sigma _{i})-r_{i} = 2^{-i}- r_i$
. By Lemma 6.3 and by recalling the definition of
$\mu _i$
, we have that
$ r_{i}\in [2^{-i}(1-2i^{-1}),2^{-i}(1+2i^{-1})]$
for all i.Footnote
13
So
$|\delta _{i}|=|r_{i}-2^{-i}| \in [0,2^{-i+1}i^{-1}]$
. Hence,
$$\begin{align*}\dfrac{|\delta_{i}|}{r_{i}} \leq \dfrac{2^{-i+1}i^{-1}}{2^{-i}(1-2i^{-1})} = 2(i-2)^{-1}.\end{align*}$$
For all
$n\geq 5$
, we have that
$$ \begin{align*} 1 \geq \prod_{i=5}^{n} \big(1-\dfrac{\delta^{2}_{i}}{r^{2}_{i}}\big) \geq \prod_{i=5}^{\infty} \big(1-\dfrac{\delta^{2}_{i}}{r^{2}_{i}}\big) \geq \prod_{i=5}^{\infty} \big(1-4(i-2)^{-2}\big)=O(1)>0. \end{align*} $$
This shows that there is a constant
$C>0$
independent of X such that
$$\begin{align*}\prod_{i=5}^\infty \big(1- \dfrac{(\mu_i(\sigma_i)-\lambda_i(\sigma_i))^{2}}{\mu_i (\sigma_i)^2 }\big) = C.\end{align*}$$
We can thus apply Theorem 5.2 with the following replacements:
$\nu \mapsto \mu $
,
$\pi \mapsto \lambda $
,
${\beta \mapsto 5}$
, and
$f \mapsto $
the identity function to conclude that
$\textbf {ML}(\mu ) \subseteq \textbf {ML}$
. As
$\mu [\textbf {ML}(\mu )]=1$
, this implies that
$\mu [\textbf {ML}]=1$
. This shows that
$\rho $
is measurement random since the B used to define
$\mu $
was an arbitrary measurement system.
Remark 6.6. A reason for using
$i\geq 5$
in constructing
$\rho $
is to allow for the existence of
$C>0$
as above. As
$(1-4(4-2)^{-2})=0$
,
$$\begin{align*}\prod_{i=1}^\infty \big(1- \dfrac{(\mu_i(\sigma_i)-\lambda_i(\sigma_i))^{2}}{\mu_i (\sigma_i)^2 }\big) = 0\end{align*}$$
and the constant
$C>0$
as above thus does not exist if
$i\geq 1$
. Another reason, as mentioned above, was to ensure
$2 \mu _i(\tau )- \lambda _i(\tau )\geq 2^{1-i}- 2^{2-i}i^{-1} - 2^{-i}> 2^{1-i}- 2^{2-i}4^{-1} - 2^{-i}=2^{1-i}>0$
.
It remains to show that the state
$\rho $
is not quantum Martin-Löf random. We prove this by showing the stronger result that
$\rho $
is quantum Schnorr random (recall that quantum Martin-Löf randomness implies quantum Schnorr randomness [Reference Bhojraj3]). A quantum Schnorr test is a computable sequence of special projections
$(S^{m})_{m=1}^\infty $
such that
$\sum _{m}\tau (S^{m}) $
is a computable real number [Reference Bhojraj3]. A state
$\rho $
fails the quantum Schnorr test
$(S^{m})_{m}$
at order
$\delta>0$
if
$\rho (S^{m})>\delta $
for infinitely many m. A state
$\rho $
passes the test
$(S^{m})_{m}$
at order
$\delta>0$
if it does not fail it at
$\delta $
. A state is quantum Schnorr random if it passes all quantum Schnorr tests at all
$\delta>0$
.
Lemma 6.7. The state
$\rho $
is not quantum Schnorr random.
Proof. We denote
$S^1_N$
(Definition 6.2) by
$S_N$
and
$\theta _i(1)$
(6.1) by
$\theta _1$
. It is easy to see that
$d_n$
has the eigenvalues zero with multiplicity
$r_n$
,
$2^{-n}$
with multiplicity
$2^n - 2r_n$
and
$2^{-n+1}$
with multiplicity
$r_n$
. So
$d_n$
has
$q_n:= 2^{n}-r_{n}$
many non-zero eigenvalues.
So the eigenpairs of
$d_n$
can be listed as
$\{\alpha ^{n}_{i}, v^{n}_i\}_{i=1}^{2^{n}}$
, where
$\alpha ^{n}_{i}=0 $
for exactly those i’s in the range
$ q_{n}+1 \leq i \leq 2^{n} $
and
$(v^{n}_i)_{i=1}^{2^{n}}$
is an orthonormal basis of
$\mathbb {C}^{2^{n}}$
. That is, the list puts the
$r_n$
many eigenvectors with zero eigenvalue at the last
$r_n$
places in the list.
Fix an
$N>5$
. Recall that
$\theta _N = \sum _{n=5}^{N} n$
for any
$N>5$
. By properties of the Kronecker product,
$S_{N}$
is a density matrix on
$\mathbb {C}^{2^{\theta _N}}$
and has an orthonormal basis of eigenvectors:
$$\begin{align*}\{ \bigotimes_{n=5}^{N} v^{n}_{l(n)}: (l(n))_{n=5}^{N} \text{ is a sequence such that } \forall n, l(n) \in \{1, \ldots 2^n\} \} \subseteq \mathbb{C}^{2^{\theta_N}}.\end{align*}$$
The eigenvector
$\bigotimes _{n=5}^{N} v^{n}_{l(n)}$
has eigenvalue
$ \prod _{n=5}^{N} \alpha ^{n}_{l(n)} $
. Let
$M_N$
be the elements of the above eigenbasis having non-zero eigenvalues. Then the order of listing the eigenpairs implies that
$$ \begin{align} M_{N}=\{ \bigotimes_{n=5}^{N} v^{n}_{l(n)}: (l(n))_{n=5}^{N} \text{ is a sequence such that for all } n, l(n) \leq q_{n} \} \subseteq \mathbb{C}^{2^{\theta_N}}. \end{align} $$
By the definition of
$q_n$
,
$$\begin{align*}|M_N|&= \prod_{n=5}^{N} 2^{n}-\lfloor 2^n/n \rfloor \leq \prod_{n=5}^{N} 2^{n}- (2^n/n) + 1 = \prod_{n=5}^{N} 2^{n}(1 - n^{-1} + 2^{-n})\\ &= \prod_{n=5}^{N} 2^{n} \prod_{n=5}^{N} (1 - n^{-1} + 2^{-n}).\end{align*}$$
Note that
$\prod _{n=5}^{\infty } (1 - n^{-1} + 2^{-n}) =0 $
. Define a quantum Schnorr test,
$(T_{m})_{m}$
as follows. Given m, we describe the construction of
$T_{m}$
. Find
$N=N(m)$
such that
$\prod _{n=5}^{N} (1 - n^{-1} + 2^{-n})< 2^{-m}$
. The idea is to use the elements of
$M_N$
to define a special projection. As
$M_N$
contains all the eigenvectors of
$\rho _{\theta _N}$
with non-zero eigenvalues,
$\rho _{\theta _N}$
will have a ‘large projection’ onto span
$(M_N)$
(see (6.5)).
Let
$$\begin{align*}T_{m}= \sum_{v \in M_{N(m)}} |v \big>\big<v |.\end{align*}$$
This completes the construction of
$T_m$
for a fixed m. As
$\rho $
is computable,
$T_m$
can be computed uniformly in m. Note that
$T_m$
is a projection on
$\mathbb {C}^{2^{{\theta _{N(m)}}}}$
having rank equal to
$|M_{N(m)}|$
. The choice of
$N(m)$
implies that
$|M_{N(m)}| < 2^{\theta _{N(m)}}2^{-m}$
and hence that
$\tau (T_{m}) < 2^{-m}$
. So,
$\sum _m \tau (T_m)$
is a computable real number. This shows that
$(T_m)_m$
is a quantum Schnorr test. This test demonstrates that
$\rho $
is not quantum Schnorr random as follows. Fix an m. Recalling that
$M_{N(m)}$
is the set consisting of all eigenvectors of
$S_{N(m)}$
with non-zero eigenvalue, we have that
$$ \begin{align} \rho(T_{m}) = \text{Tr}(\rho_{\theta_{N(m)}}T_m) = \text{Tr}(S_{N(m)}T_m)= \text{Tr}(S_{N(m)})=1. \end{align} $$
As m was arbitrary,
$\rho $
fails the quantum Schnorr test
$(T_m)_m$
at order
$1$
.
Given
$S\in 2^{\omega }$
, let
$\textbf {ML}^{S} \subset 2^{\omega }$
be the set of Martin-Löf randoms with respect to the oracle S.Footnote
14
Observe that the state
$\rho $
(from Theorem 6.4) is such that
$\mu ^{B}_{\rho }(\textbf {ML}^{S})=1$
for any
$S \in 2^{\omega }$
and any computable measurement system B. We can see this as follows. Fix an
$ S \in 2^{\omega } $
and a computable measurement system B. Relativizing the proof of Lemma 6.5 to S shows that
$\textbf {ML}^{S}(\mu ^{B}_{\rho }) \subseteq \textbf {ML}^{S}$
(Take an
$X\in \textbf {ML}^{S}(\mu ^{B}_{\rho })$
. Relativizing [Reference Shen, Uspensky and Vereshchagin13, Theorems 85 and 90] and [Reference Nies8, Proposition 3.2.14] to S and noting that
$KM^{S}(.) \leq KM(.)$
and following the proof of Lemma 6.5 shows that
$X \in \textbf {ML}^{S}$
. So
$\textbf {ML}^{S}(\mu ^{B}_{\rho }) \subseteq \textbf {ML}^{S}$
). This shows that
$\mu ^{B}_{\rho }(\textbf {ML}^{S})\geq \mu ^{B}_{\rho }(\textbf {ML}^{S}(\mu ^{B}_{\rho }))=1$
.
Remark 6.8. The state
$\rho $
(from Theorem 6.4) is such that
$\mu ^{B}_{\rho }$
is absolutely continuous with respect to
$\lambda $
for any computable measurement system B. To see this, let B be an arbitrary computable measurement system and let
$N \subset 2^\omega $
be such that
$\lambda (N)=0$
. It is easy to see that there is some oracle
$ S \in 2^{\omega } $
such that
$\textbf {ML}^S \subseteq (2^{\omega } \backslash N ) $
. Together with the previous observation, this implies that
$\mu ^{B}_{\rho }(2^{\omega } \backslash N )\geq \mu ^{B}_{\rho }(\textbf {ML}^{S}) =1$
. This shows that
$\mu ^{B}_{\rho }(N)=0$
.Footnote
15
Acknowledgments
The article was written during my PhD studies under Joe Miller at the Department of Mathematics, University of Wisconsin-Madison and during my post-doctoral fellowship at the School of Computer Science, University of Auckland under André Nies.Footnote 16 J.M. and Peter Cholak (independently) asked if a state can be ‘measured’. I thank J.M. and A.N. for reading the article in detail and suggesting key improvements.
Funding
The post-doctoral fellowship at the School of Computer Science, University of Auckland was funded by the Marsden grant UOA1931 awarded by the Royal Society of New Zealand to A.N.
, where 
.
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