Distinguishability

	Distinguishability
	Computes the maximum probability of distinguishing quantum states
Other toolboxes required	CVX
Related functions	ChannelDistinguishability; LocalDistinguishability
Function category	Distinguishing objects

Distinguishability is a function that computes the maximum probability of distinguishing two or more quantum states. That is, this function computes the maximum probability of winning the following game: You are given a complete description of a set of $k$ quantum states $\rho_1, \ldots, \rho_k$, and then are given one of those $k$ states, and asked to determine (via quantum measurement) which state was given to you.

Syntax

DIST = Distinguishability(X)
DIST = Distinguishability(X,P)
[DIST,MEAS] = Distinguishability(X,P)

Argument descriptions

Input arguments

X: The quantum states to be distinguished. X can either be a cell containing 2 or more density matrices, or X can be a matrix whose columns are pure vector states.
P (optional, default [1/k, 1/k, ..., 1/k], where k is the number of quantum states): A vector whose j-th entry is the probability that the state $\rho_j$ is given to you in the game described above. All entries must be non-negative, and the entries of this vector must sum to 1.

Output arguments

DIST: The maximum probability of distinguishing the states specified by X.
MEAS (optional): A cell containing optimal measurement operators that distinguish the states specified by X with probability DIST.

Examples

Orthogonal states can be perfectly distinguished

Any number of quantum states can be perfectly distinguished (i.e., distinguished with probability 1) if they are mutually orthogonal. The following code generates a random $6\times 6$ unitary matrix (i.e., a matrix with orthogonal pure states as columns) and verifies that those pure states are perfectly distinguishable:

>> Distinguishability(RandomUnitary(6))
 
ans =
 
     1

Two states

The maximum probability of distinguishing two quantum states $\rho$ and $\sigma$ is exactly $\frac{1}{2} + \frac{1}{4}\|\rho - \sigma\|_1$^[1], where $\|\cdot\|_1$ is the trace norm. We can verify this in a special case as follows:

>> rho = RandomDensityMatrix(4);
>> sigma = RandomDensityMatrix(4);
>> Distinguishability({rho, sigma})
 
ans =
 
    0.7762
 
>> 1/2 + TraceNorm(rho - sigma)/4
 
ans =
 
    0.7762

Three or more states

We can also compute the maximum probability of distinguishing three or more states, but no simple formula is known in this case.

>> for j = 1:6
       rho{j} = RandomDensityMatrix(4);
   end
>> Distinguishability(rho)
 
ans =
 
    0.4156

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

%%  DISTINGUISHABILITY    Computes the maximum probability of distinguishing quantum states

%   This function has one required input argument:

%     X: a cell containing density matrices or a single matrix containing

%        pure states as its column vectors

```
%
```

%   DIST = Distinguishability(X) is the maximum probability of

%   distinguishing the quantum states specified by X. X can either be a

%   cell containing 2 or more density matrices to be distinguished, or X

%   can be a matrix whose columns are pure vector states to be

```
%   distinguished.
```
```
%
```

%   This function has one optional input argument:

%     P (default [1/k, ..., 1/k], where k is the number of states)

```
%
```

%   [DIST,MEAS] = Distinguishability(X,P) is the maximum probability of

%   distinguishing the quantum states specified by X, where the vector P

%   contains the probability that each state is chosen (by default, the

%   states are chosen uniformly at random). MEAS is a cell containing the

```
%   optimal measurement operators.
```
```
%
```

%   URL: http://www.qetlab.com/Distinguishability

```
 
```

%   requires: cvx (http://cvxr.com/cvx/), kpNorm.m, opt_args.m, TraceNorm.m

```
%
```

%   author: Nathaniel Johnston (nathaniel@njohnston.ca)

```
%   package: QETLAB
```
```
%   last updated: October 6, 2014
```
```
 
```

function [dist,meas] = Distinguishability(X,varargin)

```
 
```
```
if(iscell(X))
```
```
    num_ops = length(X);
```
```
    dim = length(X{1});
```
```
else
```
```
    [dim,num_ops] = size(X);
```
```
end
```
```
 
```

% set optional argument defaults: p = [1/k,1/k,...,1/k], where k = # of states

[p] = opt_args({ ones(1,num_ops)/num_ops },varargin{:});

```
 
```

if(abs(sum(p) - 1) > num_ops^2*eps || length(p) ~= num_ops)

    error('Distinguishability:InvalidP','The vector P must be a probability distribution of the same length as the number of states: its elements must be non-negative and they must sum to 1.');

```
end
```
```
 
```

if(num_ops == 1 || max(p) >= 1) % of course we can distinguish 1 object

```
    dist = 1;
```
```
    if(nargout > 1) 
```

        meas = eye(dim); % optimal measurements is trivial in this case

```
    end
```
```
    return
```
```
end
```
```
 
```
```
% X is a cell of density matrices
```
```
if(iscell(X))
```

    for j = 1:num_ops % make sure that the density operators are scaled

```
        X{j} = X{j}/trace(X{j});
```
```
    end
```
```
 
```

    % There is a closed-form expression for the distinguishability of two

```
    % density matrices.
```
```
    if(num_ops == 2)
```

        dist = 1/2 + TraceNorm(p(1)*X{1} - p(2)*X{2})/2;

        if(nargout > 1) % construct optimal measurements, if requested

            [v,d] = eig(p(1)*X{1} - p(2)*X{2});

```
            d = diag(d);
```
```
            pind = find(d >= 0);
```

            meas{1} = v(:,pind)*v(:,pind)';

            meas{2} = eye(dim) - meas{1};

```
        end
```
```
        return
```
```
    end
```
```
 
```

    % Check to see if the states are mutually orthogonal, and return 1 if

```
    % they are.
```
```
    if(num_ops <= dim)
```
```
        mut_orth = 1;
```
```
        for j = 1:num_ops
```
```
            for k = j+1:num_ops
```

                if(max(max(abs(X{j}*X{k}))) > eps*dim^2)

```
                    mut_orth = 0;
```
```
                    break;
```
```
                end
```
```
            end
```
```
        end
```

        if(mut_orth == 1) % states are mutually orthogonal

```
            dist = 1;
```

            if(nargout > 1) % construct optimal measurements, if requested

```
                meas_sum = zeros(dim);
```

                for j = num_ops:-1:2 % pre-allocate for speed

```
                    oX = orth(X{j});
```
```
                    meas{j} = oX*oX';
```

                    meas_sum = meas_sum + meas{j};

```
                end
```

                meas{1} = eye(dim) - meas_sum;

```
            end
```
```
            return;
```
```
        end
```
```
    end
```
```
 
```

% X is a matrix whose columns are pure states

```
else
```

    X = normalize_cols(X); % make sure that the columns have unit length

```
 
```

    % There is a closed-form expression for the distinguishability of two

```
    % pure states.
```
```
    if(num_ops == 2)
```

        dist = 1/2 + sqrt(2*(p(1)^2 + p(2)^2) - 4*p(1)*p(2)*abs(X(:,1)'*X(:,2))^2)/2;

        if(nargout > 1) % construct optimal measurements, if requested

            [v,d] = eig(p(1)*X(:,1)*X(:,1)' - p(2)*X(:,2)*X(:,2)');

```
            d = diag(d);
```
```
            pind = find(d >= 0);
```

            meas{1} = v(:,pind)*v(:,pind)';

            meas{2} = eye(dim) - meas{1};

```
        end
```
```
        return
```
```
    end
```
```
 
```

    % Check to see if the states are mutually orthogonal, and return 1 if

```
    % they are.
```
```
    if(num_ops <= dim)
```
```
        X2 = X'*X;
```

        if(max(max(abs(X2 - diag(diag(X2))))) < eps*dim^2)

            dist = 1; % the states are orthogonal, so perfectly distinguishable

            if(nargout > 1) % construct optimal measurements, if requested

```
                meas_sum = zeros(dim);
```

                for j = num_ops:-1:2 % pre-allocate for speed

                    meas{j} = X(:,j)*X(:,j)';

                    meas_sum = meas_sum + meas{j};

```
                end
```

                meas{1} = eye(dim) - meas_sum;

```
            end
```
```
            return
```
```
        end
```
```
    end
```
```
 
```

    % Turn pure states into density operator form.

```
    Y = X; X = {};
```

    for j = num_ops:-1:1 % pre-allocate for speed

```
        X{j} = Y(:,j)*Y(:,j)';
```
```
    end
```
```
end
```
```
 
```

% For 3 or more density matrices, we have to use semidefinite programming.

```
cvx_begin sdp quiet
```
```
    cvx_precision default;
```

    variable P(dim,dim,num_ops) hermitian

```
 
```
```
    P_tr = 0;
```
```
    P_sum = zeros(dim);
```
```
    for j = 1:num_ops
```

        P_tr = P_tr + p(j)*trace(P(:,:,j)*X{j});

```
        P_sum = P_sum + P(:,:,j);
```
```
    end
```
```
    P_tr = P_tr + P_tr';
```
```
 
```
```
    maximize P_tr
```
```
    subject to
```
```
        P_sum == eye(dim);
```
```
        P >= 0;
```
```
cvx_end
```
```
dist = real(cvx_optval)/2;
```
```
 
```

% Also return the optimal measurements, if requested.

```
if(nargout > 1)
```

    meas = mat2cell(reshape(P,dim,dim*num_ops),dim,dim*ones(1,num_ops));

```
end
```

References

↑ John Watrous. Theory of Quantum Information lecture notes, Fall 2011.

[1] John Watrous. Theory of Quantum Information lecture notes, Fall 2011.

[1]

Distinguishability

Contents

Syntax

Argument descriptions

Input arguments

Output arguments

Examples

Orthogonal states can be perfectly distinguished

Two states

Three or more states

Source code

References

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